lilt*' MASARYK UNIVERSITY w i IMI1 Faculty of Science Wavelets on the Interval and Their Applications Habilitation Thesis Dana Cernä Brno 2019 Abstract This habilitation thesis is concerned with constructions of new spline-wavelet bases on the interval and product domains, their adaptations to boundary conditions, and their applications. This thesis is a collection of eight previously published papers [10, 11, 13, 14, 15, 17, 19, 23]. First, we introduce the concept of a wavelet basis on a bounded interval and on tensor product domains. Then, we review the wavelet-Galerkin method and adaptive wavelet methods for the numerical solution of operator equations. Finally, we discuss constructions of quadratic and cubic spline wavelets and we comment on the collected papers. Papers [10, 11, 15] are focused on the construction of well-conditioned biorthogonal spline-wavelet bases on the interval where both primal and dual wavelets have compact support. In [13, 14, 19, 23], a local support of dual wavelets is not required which enables the construction of wavelets that have smaller supports and significantly smaller condition numbers than wavelets of the same type but with local duals. Another advantage is the simplicity of the construction. In [17], we constructed wavelets where the corresponding matrices, arising from discretization of second-order differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids, are sparse and not only quasi-sparse as for most wavelet bases. We used the constructed bases for solving various types of operator equations, e.g. Poisson's equation, the Helmholtz equation, a fourth-order boundary value problem, and the Black-Scholes equation with two state variables. We also applied the constructed bases for option pricing under Kou's double exponential jump-diffusion option pricing model which is represented by a partial integro-differential equation. i Contents List of Symbols v Introduction 1 1 Wavelet Basis 5 1.1 Wavelet Bases on Bounded Interval.................... 5 1.2 Construction of Wavelet Bases....................... 7 1.3 Proofs of Riesz Basis Property....................... 8 1.4 Fast Wavelet Transform .......................... 10 1.5 Wavelet Bases on Product Domains.................... 12 1.6 Wavelet-Galerkin Method ......................... 14 1.7 Adaptive Wavelet Methods......................... 17 2 Constructions of Spline-Wavelet Bases 23 2.1 Quadratic Spline-Wavelet Bases...................... 23 2.2 Cubic Spline-Wavelet Bases........................ 25 2.3 Applications of Constructed Bases..................... 29 Conclusions and Further Research 33 References 35 Selected papers 41 iii List of Symbols N the set of all positive integers Z the set of all integers IR the set of all real numbers I bounded interval I C IR Q bounded domain □ hyperrectangle |-| absolute value of a number or a level for an index A [•J floor function Sij Kronecker delta, S^i = 1, Sij = 0 for i ^ j span linear span of a set supp support of a function diam diameter of a bounded set cond spectral condition number of a matrix or condition number of a basis © direct sum of two spaces VL orthogonal complement of V V closure of the subspace V L2 (Q) the space of all square integrable functions defined on Q Hs (Q) Sobolev space of order s G IR on Q Hq1 (Q) Sobolev space of Hm functions satisfying homogeneous Dirichlet boundary conditions of order m Cm (IR) the space of m-times continuously differentiable functions I2 (J) the space of (infinite) vectors v with finite /2-norm Um (Q) the space of all algebraic polynomials on Q of degree at most m ||-|| the /2-norm of an (infinite) vector, the L2-norm of a function, or the spectral norm of a matrix or an operator ||-||# norm in the space H \-\Hs(ty seminorm in Hs (Q) (•, •) the L2-inner product or a dual product (•, ■)H inner product in H jo the coarsest level in a multiresolution analysis in a given context A wavelet index, usually A = (j, k) v 0, j,k scaling functions ^,^j,k wavelets or more generally functions in a wavelet basis (scaling functions on the coarsest level and wavelets) ^max eigenvalue that is maximal in the absolute value eigenvalue that is minimal in the absolute value # cardinality of a set A Laplace operator C generic constant V for all MT transpose of a matrix M I identity matrix or identity operator 0 zero matrix vi Introduction Wavelet bases and the fast wavelet transform are a powerful and useful tool for signal and image analysis, detection of singularities, data compression, and also for the numerical solution of partial differential equations, integral equations, and integro-differential equations. One of the most important properties of wavelets is that they have vanishing moments. Vanishing wavelet moments ensure the so-called compression property of wavelets. This means that integrals of a product of a function and a wavelet decay exponentially, dependent on the level of the wavelet if the function is smooth enough in the support of the wavelet. This enables the obtainment of sparse representations of functions as well as sparse representations of some operators, see e.g. [2, 28, 72]. There are two main classes of wavelet based methods for the numerical solution of operator equations. The first method is the wavelet-Galerkin method. Due to vanishing moments, the wavelet-Galerkin method leads to sparse matrices not only for differential equations but also for integral and integro-differential equations while the Galerkin method with the standard B-spline basis leads to full matrices if the equation contains an integral term. Another important property of wavelet bases is that they form Riesz bases in certain spaces, such as Lebesgue, Sobolev or Besov spaces. Due to this property, the diagonally preconditioned matrices arising from discretization using the Galerkin method with wavelet bases have uniformly bounded condition numbers for many types of operator equations. The second class of methods are adaptive wavelet methods. We focus on adaptive wavelet methods that were originally designed in [29, 30] and later modified in many papers [43, 52, 67]. For a large class of operator equations, both linear and nonlinear, it was shown that these methods converge and are asymptotically optimal in the sense that the storage and the number of floating point operations, needed to resolve the problem with desired accuracy, depend linearly on the number of parameters representing the solution. Moreover, the method enables higher-order approximation if higher-order spline-wavelet bases are used. The solution and the right-hand side of the equation have sparse representations in a wavelet basis, i.e. they can be represented by a small number of numerically significant parameters. Similarly as in the case of the wavelet-Galerkin method, the differential and integral operators can be represented by sparse or quasi-sparse matrices. For a large class of problems, the matrices arising from a discretization using wavelet bases can be simply preconditioned by a diagonal preconditioner, and the condition numbers of these preconditioned matrices are uniformly bounded. For more details about adaptive wavelet methods, see [7, 29, 30, 43, 52, 67, 72]. 1 2 Introduction The first wavelet methods used orthogonal wavelets, e.g. Daubechies wavelets or coiflets. Their disadvantages are that the most orthogonal wavelets are usually not known in an explicit form and their smoothness is typically dependent on the length of the support. In contrast, spline wavelets are known in a closed form, are smoother, and have shorter support than orthogonal wavelets with the same polynomial exactness and the same number of vanishing moments. Therefore, they are preferable in numerical methods for operator equations. This habilitation thesis is concerned with constructions of new spline-wavelet bases on the interval and product domains, their adaptations to boundary conditions, and their applications. The thesis is conceived as a collection of the following eight previously published articles supplemented by commentary. [10] Černá, D.; Finěk, V.: Construction of optimally conditioned cubic spline wavelets on the interval, Adv. Comput. Math. 34(2), (2011), pp. 219-252. My contribution to this paper was 60%. [11] Černá, D.; Finěk, V.: Cubic spline wavelets with complementary boundary conditions, AppL Math. Comput. 219(4), (2012), pp. 1853-1865. My contribution to this paper was 60%. [13] Černá, D.; Finěk, V.: Quadratic spline wavelets with short support for fourth-order problems, Result. Math. 66(6), (2014), pp. 525-540. My contribution to this paper was 60%. [14] Černá, D.; Finěk, V.: Cubic spline wavelets with short support for fourth-order problems, Appl. Math. Comput. 243, (2014), pp. 44-56. My contribution to this paper was 60%. [15] Černá, D.; Finěk, V.: Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions, Int. J. Wavelets Multiresolut. Inf. Process. 13(3), (2015), article No. 1550014. My contribution to this paper was 60%. [17] Černá, D.; Finěk, V.: Sparse wavelet representation of differential operators with piecewise polynomial coefficients, Axioms 6, (2017), article No. 4. My contribution to this paper was 60%. [19] Černá, D.; Finěk, V.: Quadratic spline wavelets with short support satisfying homogeneous boundary conditions, Electron. Trans. Numer. Anal. 48, (2018), pp. 15-39. My contribution to this paper was 90%. [23] Černá, D.: Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model, Int. J. Wavelets Multiresolut. Inf. Process. 17(1), (2019), article No. 1850061. Papers [10, 11, 15] are focused on constructions of well-conditioned biorthogonal spline wavelet bases on the interval where both primal and dual wavelets have compact support. In [13, 14, 19, 23] we do not require local support of dual wavelets, Introduction 3 which enables us to construct wavelet bases that have smaller support and have significantly smaller condition number than wavelet bases with local duals. Moreover, their construction is significantly simpler than constructions of wavelets with local duals, which are typically quite long and technical. In [18], we constructed wavelets that are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Due to this property, matrices arising from discretization of second-order differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. We use the constructed bases for solving various types of operator equations, e.g. Poisson's equation, the Helmholtz equation, fourth-order differential equations, and the Black-Scholes equation with two state variables. We also applied the constructed bases for option pricing under Kou's double exponential jump-diffusion option pricing model. Other applications are presented in Chapter 2. This thesis is organized as follows. In Chapter 1 we briefly review a concept of a wavelet basis on a bounded interval, the fast wavelet transform on a bounded domain, and two constructions of wavelet bases on product domains that are based on tensorizing univariate wavelet bases. We also describe basic principles of the wavelet-Galerkin method and adaptive wavelet methods. Since all the papers collected in this thesis are concerned with constructions of quadratic or cubic spline wavelets on the interval, in Chapter 2 we present existing constructions of such types of wavelets and their applications. Most of the papers presented in this thesis comes from a collaboration with my colleague Vaclav Finek. I would like to thank him for this friendly and very helpful collaboration. Most of the presented work was done at Technical University of Liberec. I want to express my gratitude to all of my colleagues there for all the inspiration and help. In particular, I am grateful to Jirka Hozman and Prof. Jan Picek, who allowed me to collaborate with them on interesting projects. Introduction Chapter 1 Wavelet Basis In this chapter, we introduce the concept of a wavelet basis on the interval and product domains, the fast wavelet transform, the wavelet-Galerkin method, and adaptive wavelet methods. Wavelet bases were originally constructed as orthonormal bases for the space L2 (M.) and later as Riesz bases for this space. One of the possibilities of how to use these bases and the fast wavelet transform on a bounded domain is the extension of a function or a signal near the boundary using for example zero padding, periodiza-tion, or symmetrization, see e.g. [7, 8]. However, this approach can lead to boundary effects, and it is not suitable for the numerical solution of operator equations in which a basis has to be adapted to boundary conditions. We present another approach, one where the wavelet bases on the real line are adapted to the interval with special boundary functions that have to be constructed, such that the resulting basis is a Riesz basis for a chosen space and the locality of the support, smoothness of basis functions, the number of vanishing wavelet moments, and polynomial exactness are preserved. 1.1 Wavelet Bases on Bounded Interval In this section, we briefly recall the concept of a wavelet basis on a bounded interval / C 1; for more details refer to [7, 28, 31, 37, 41, 72]. Let J be at most countable index set such that each index A G J takes the form A = (j, k), where |A| = j G Z denotes a level. We define for v = {vx}XeJ,vxeR, (1.1) xej and I2 (J) = {v : v = {vx}XeJ ,vxER, || v || < oo} . (1.2) We use the standard notation L2 (J) for the space of all square-integrable functions defined on J, ||-|| for the L2-norm, and (•, •) for the L2-inner product. Let H C L2 (J) be a real separable Hilbert space equipped with the inner product (•, -)H and the norm \\-\\H- For example, H can be the Sobolev space Hq (J) of functions whose first weak 5 6 CHAPTER 1. WAVELET BASIS derivatives are in L2 (J) and that vanish at boundary points. First, a wavelet basis ^ = {0a, A G J} has to be a Riesz basis for H. Definition 1. A family ^ = A G J} is called a Riesz basis of iJ, if the span of ^ is dense in H and there exist constants c, C G (0, oo) such that cllbll < We refer to the constants j0, are called wavelets on the level j. Wavelets in the inner part of the interval called inner wavelets are typically translations and dilations of one function ip or several functions ip1,..., ipp also called wavelets (or mother wavelet, wavelet generator), i.e. i/jjjk (x) = 2i/y (2jx - m) , (1.9) 1.2. CONSTRUCTION OF WAVELET BASES 7 for some / G {1,... ,p} and some k,m G Z, k dependent on m and /. Similarly the wavelets near the boundary are derived from functions called boundary wavelets. Another desired property is a polynomial exactness of order M > 1. This means that the multiscale basis j *J = *io u U *i> (1-10) jo < J, is such that span^>J contains all polynomials of degree at most M — 1. Polynomial exactness determines the convergence rate of methods for the numerical solution of operator equations. Finally, we require that there exists L > 1, such that all functions ip\ G tyj, jo < j, have L vanishing moments, i.e. Jxktpx (x)dx = 0, k = 0, ...,L- 1. (1.11) Vanishing wavelet moments are important for sparse representation of functions and operators. The concept of a wavelet basis is not unified in the mathematical literature and some of the above conditions can be omitted or generalized. 1.2 Construction of Wavelet Bases The wavelet basis \l/ is typically constructed using a multiresolution analysis. Definition 2. A sequence {Vj}^=-Q of closed linear subspaces Vj C H is called a multiresolution analysis, if these subspaces are nested and their union is dense in H, i-e-, _ oo V]0 c V]0+1 c ... c vs c vs+1 c ... c u, (J 1} //. i=io Let the set $j = { join [10, 11], we constructed dual bases that are local. However, in some applications such as solving linear PDEs, the dual basis is not directly used. Therefore, in [13, 14, 15, 17, 19, 23] we were concerned with constructions of wavelet bases without requiring locality of duals, but with a shorter support or some special properties. 1.3 Proofs of Riesz Basis Property While one can employ the Fourier transform to prove the Riesz basis property (1.3) for the space L2 (IR), the proof of the Riesz basis property for the space H C L2 (J) is usually more complicated. We present here several possible approaches that we used in papers collected in this thesis. In [10, 11, 15, 17], the proof of the Riesz basis property (1.3) for \l/ is based on the following theorem [28, 36, 45]. Theorem 3. Let j0 G N and for j > j0 let Vj and Vj be subspaces of the space H C L2 (J) such that Vj C Vj+i, Vj C Vj+i, and dim Vj = dim Vj < oo. Let be bases ofVj, be bases ofVj, and tyj be bases of Vj1 fl Vj+i, where Vj- denotes the L2-orthogonal complement ofVj in H. Moreover, let the Riesz bounds with respect to the L2-norm of'Qj, and ^j, be uniformly bounded. Let ^ be composed of$>j0 and ^j, j > jo, as in (1-8)- Furthermore, we assume that r, = ($,,$,) (i.i3) is invertible and that the spectral norm of Tj1 is bounded independently on j. In addition, for some positive constants C, 7 and d, such that 7 < d, let inf \\v - Vj\\L2(I) < C2-jt \\v\\Ht(I) , v E Hl (J) fl H, 0 < t < d, (1.14) and IMIff'(i) < C23S IMIl2(/) > v3 e Vj, 0 jo, form a multiresolution analysis for the space L2 (Q). Let Hq for fixed q > 0 be a linear subspace of L2 (Q) that is itself a normed linear space and assume that there exist positive constants A\ and A2 such that a) If f G Hq has decomposition f = Y2j>j0 fj> fj ^ Vj then \\f\\2Hq < ^i£2«- \\fjf; (1.17) J>J0 b) For each f G Hq there exists a decomposition f = Y2j>j0 fj> fj ^ Vj, such that J2^\\fi\\2J0 Furthermore, suppose that Pj is a linear projection from Vj+i onto Vj, Wj is the kernel space of Pj, $j = {0j,fc, k G Ij} are Riesz bases ofVj with respect to the L2-norm with uniformly bounded condition numbers, and tyj = {ipj^jk G Ij} are Riesz bases of Wj with uniformly bounded condition numbers. If there exist constants C and p such that 0 < p < q and \\PmPm+1...Pn-1\\jo,k, k G Ij0} U {2 !"cjM..i > jo, kel,} (1.20) is a Riesz basis of Hq. To employ Theorem 4, one has to find appropriate projectors Pj and prove the inequality (1.19). The advantage of this approach is that it enables proving the Riesz 10 CHAPTER 1. WAVELET BASIS basis property in the Sobolev spaces Hs for values of s in some range (si, S2), where s± can be positive. Therefore, it is possible to use this theorem to prove the Riesz basis property in Hs even if the Riesz basis property does not hold in L2. For example in [19], we used this theorem to prove that the constructed set is a Riesz basis in the spaces Hq (J) and Hq (J2) for I = (0,1), but numerical experiments show that the L2-condition numbers increase with the level and they seem to be unbounded. This suggests that the basis on I is not a Riesz basis in the space L2 (J). In [23], we used a completely different approach and derived a condition under which a union of Riesz sequences is also a Riesz sequence. Theorem 5. Let I and J be at most countable index sets, {fk\kex ^e a R^esz sequence with a Riesz lower bound Cf, and {gi}lej be a Riesz sequence with a Riesz lower bound cg. Furthermore, let the matrix G with entries Gk,i = (fk,9i), k e 1, I E J, satisfy ||G||/(c/C3) 4 /1 — il^li . min (cf, cg). (1.22) C.fC9 In [23], we proved the Riesz basis property separately for inner wavelets and for boundary wavelets, and then we verified the condition (1.21) to show that their union is also a Riesz basis. 1.4 Fast Wavelet Transform As we have already mentioned, we view the sets of functions such as $^ and ^j also as columns vectors. The nestedness of the spaces Vj implies the existence of a refinement matrix M^0 such that Mjo^+i. Since Wj C Vj+i, there exists a matrix M^i such that (1.23) (1.24) Applying (1.23) and (1.24) several times we find out that the multiscale basis \E'J defined by (1.10) and the scaling basis $j are interrelated by the transform Tj such that JO io+1 (1.25) and the transform Tj can be expressed by Tj = Tj5j_i ... TJJo, where Tjj ™S ?)> M, = (M,,0,M,,1J, (1.26) 1.4. FAST WAVELET TRANSFORM 11 where 0 and I are zero and identity matrices, respectively, of appropriate sizes. This transform is often called the fast wavelet transform (FWT). Since Vj = span $ j = sp&n^J, any function / G Vj has a single-scale representation f = cJ$i = cJ,kJ,k, cj = {cj,k}keXj , (1-27) keij and also a multiscale representation j-i keXJ0 j=jo kejj where Cj0 = {cj0,k}kex an<^ di = i^i^kej'•• Then, the vectors cj and dJ= (cJo,dJo,...dJ_1)T (1-29) are also interrelated by the fast wavelet transform Tj, i.e. cj = TjdJ, (1.30) because from (1.23) and (1.24) we obtain cj$j + dJVj = (MjfiCj + M^djf = cj+1$j+1. (1.31) Schematically Tj applied on dJ can be visualized as a pyramid scheme, Mj_ii0 Cj0 > Cj0+1 > cj0+2 > ■ ■ ■ CJ-1 > CJ • Mj-1,1 d, Due to the local support of basis functions, the matrices Mj are sparse and they can be applied in O (Nj) operations, where Nj = dim Vj. Thus, the fast wavelet transform can be applied in O (Nj) operations when using a pyramid scheme. Since the matrix Mj represents a basis transformation, its inverse exists. Let us define G, = (0,0,0,!) =Mj\ (1.32) where the matrix G^o is °f the size #XJ+1 x #Tj and the matrix G^i is of the size #Xj+i x Then ' ci = Gj)0Cj+i, dj = Gjtlcj+1. (1.33) Thus, the inverse fast wavelet transform (IFWT) Tj1 has the form TV T/..;., • • • T/../ , where T,.; °) . (1.34) The corresponding pyramid scheme is then M3Q + 2,0 cio+l Cio+2 M30,i M30+i d30 dJ0- n djo+2 12 CHAPTER 1. WAVELET BASIS gj-i,o Cj-l gj-2,0 Cj-2 gj-3^,0 \Gj_3.1 dj-i dj-2 dj-3 Cj ->• Cj_i -> Cj_2 ->" • • • cj0+l -> Cjo ■ \Gio.i dio Clearly, different wavelet bases lead to different fast wavelet transforms. As mentioned above, FWT can be used to transform a scaling basis to a multiscale wavelet basis and vectors of multi-scale coefficients to vectors of single-scale coefficients. Furthermore, if A is a differential or an integral operator, then FWT can be used to transform the discretization matrix in a scaling basis (AQj, $j) to a discretization matrix with respect to a multiscale basis (A$j, ^j) by (A*j,*j) = Ttj (A$j,$j)Tj. In signal processing FWTs and IFWTs are widely used for signal analysis, signal compression and decompression. FWTs corresponding to a wavelet basis adapted to a bounded interval have an advantage that the boundary wavelets also have vanishing moments and thus the boundary effects that can occur when using the standard approach based on symmetrization of the signal are reduced. We studied this issue in [8, 9], where we used the fast wavelet transforms corresponding to wavelet bases that we constructed in [10] for image compression. 1.5 Wavelet Bases on Product Domains There are several approaches for constructing a multi-dimensional wavelet basis on a tensor product domain, for example an isotropic approach [57, 72], an anisotropic approach [35, 45] or a sparse tensor product [44]. In this section, we recall an isotropic and an anisotropic approach. Both constructions are based on tensorizing univariate wavelet bases and they preserve their important properties. We consider a product domain □ = (ai, &i) x (a2, 62)x • • •x (ad, bd), where a^, 6« G IR, di < bi, i = 1,..., d, d G N. The construction usually starts with a Riesz basis = {(f>30,k, k G Ij0} U k G Jj,j > jo} (1.35) for the space H C Hs (0,1) for s in some interval (sl,s2). First, we use a simple linear transformation to obtain a wavelet basis for the space Hs (ai} bi). Let us define 4>),k (x) = j,k (5, _ q.) ' ^},k (x) = ^J\k (5, _ ' x e (ai'hi) ' (L36) then V® = (4,fc, k G 1J0} U {$,fc, k G Jj,j > jo} (1.37) forms a Riesz basis in Hs (ai, bi). Isotropic wavelet bases. We define the multivariate scaling functions by (f)j:k (x) = Uf=1(f)ljM (xi), x = (xu ...,xd)eD, (1.38) 1.5. WAVELET BASES ON PRODUCT DOMAINS 13 with k = (ki,..., kd) now being a multi-index, k G = Ij x ... x Xj. We introduce the abbreviation i.e. the parameter e allows distinguishing between scaling functions and wavelets. Furthermore, we denote JE = {e=(e1,...,ed),e,G{0,l},e^(0,0,...,0)}, (1.40) and J^e = Jj,ei X ... X J},ed, = Jj,e- (1-41) eeE For any e = (ei,...,e j0, and A; = (k1,...,k(i) G J^n, we define the multivariate wavelet ipx(x)=Ut1ipljteitkl(xi), x = (Xl,...,xd) G □, A = (j,e,fc), (1.42) where ^elh = { fy' 6Z = i' (1-43) 3,1,1 I ez = 0. v ' The wavelet basis on the hyperrectangle □ is then given by = e G E, k G J,n, j > Jo} U {j0tk, k G if) . (1.44) We denote the multiscale basis containing wavelets up to level J as ^J = {ipj,e,k, eeE, ke J^,j0 < J < J} U {(/>jo>k, fc G X°} . (1.45) If we start with a Riesz basis in the space L2 (0,1), then the resulting basis is a Riesz basis in the space L2 (Q). The Riesz basis property in the space Hs (Q) can be verified using e.g. Theorem 4. Furthermore, this approach preserves the regularity of basis functions, the full degree of polynomial exactness, vanishing wavelet moments, as well as locality of bases functions. For more details see e.g. [57, 72]. In this thesis, we constructed isotropic wavelet bases and used them for the numerical solution of differential equations in [10, 11, 13, 14, 15, 19]. Anisotropic wavelet basis. Let be a wavelet basis on the interval (aj,&j) defined by (1.36) and (1.37). For notational simplicity, we denote J}0-i = Tj0 and tf0-i,k = 4,*> k e Zo-u J = {(J, k), J > Jo ~ 1, k G Jj} . (1.46) Then can also be expressed as *W = M*, J > Jo ~ 1, k G Jj} = {^, A G J} . (1.47) Recall that for the index A = (j, k) we denote |A| = j. We use u f to denote the tensor product of functions -u and v, i.e. (if ® v) (x1} x2) = u (xi) v (x2). For d > 1 we generalize the definition of the index set J: J={\ = (A1; ...,Xd):Xi = (ji, ki) ,ji > jo -l,he JH} . (1.48) 14 CHAPTER 1. WAVELET BASIS We define multivariate basis functions as i>x = ®tA, X = (Xi,--.,Xd)eJ. (1.49) Then |A| = maxj=lr. ^ \ \\ represents a level. We also denote [A] = mhij=lv. ^ \ \\. Due to locality of the one-dimensional basis functions, i.e. diam supp ^ < Cj2~'Ai', we have diam supp ip\ < \ VC|2-2W< CVd2-[x], C=maxd. (1.50) i=i,...,d i=l In this case, basis functions are not local in the sense that diam supp < C2 'A', but only in the sense that (1.50) holds. We define the set \l/ = A G J}, and the set ^J = {^A:A = (A1,...,Ad), \W < J} . (1.51) If we start with a univariate Riesz basis in the space L2 (0,1), then the set \l/ is a Riesz basis of the space L2 (Q), see e.g. [48]. This approach also preserves the properties of the univariate basis, such as polynomial exactness, smoothness of basis functions, and vanishing moments, but as already mentioned the resulting functions are local only in the sense of (1.50). For more details see [35, 45, 48]. In this thesis, we used anisotropic wavelet bases in [15, 17, 19]. 1.6 Wavelet-Galerkin Method In this section, we recall the wavelet-Galerkin method for solving operator equations. Let Q be a bounded domain, and let H C L2 (Q) be a separable Hilbert space with the norm ||'||^. We denote a dual space to H as H' and by (•, •) we denote a duality product. For an operator A : H —>• H' and given / G H' we consider an operator equation Au = f. (1.52) We define a corresponding bilinear form a : H x H —> IR by a (u, v) = (Au, v) \/u,vEH. (1.53) The variational problem becomes: Given / G H', find u G H such that a(u,v) = (f,v) VveH. (1.54) Let $ be a family of functions such that ^ normalized in the iJ-norm is a wavelet basis of H. Let C ^ be a multiscale basis of the form (1.10) that contains scaling functions at a coarsest level jo and wavelets up to level k. Let us assume that the spaces Xk = span ^k form a multiresolution analysis in H. The Galerkin formulation of (1.54) reads as: Find uk G Xk such that a{uk,v) = (f,v) Wv G Xk. (1.55) 1.6. WAVELET-GALERKIN METHOD 15 We focus on the case where the bilinear form a is continuous and coercive. Recall that a bilinear form a : H x H —> IR is called continuous if there exists a constant C such that \a{u,v)\ < C \\u\\H \\v\\H Wu,v G H, (1.56) and a is called coercive if there exists a constant a > 0 such that a (if, if) > a ||w||^- \/u £ H. (1-57) Under these assumptions, the existence and uniqueness of the solutions of equations (1.54) and (1.55) are a consequence of the Lax-Milgram theorem, see [27, 62]. Theorem 6. Lax Milgram Let H be a Hilbert space, let the bilinear form a : H x H —>• IR be continuous and coercive with constants C and a as in (1.56) and (1.57), respectively, and let f G H'. Then the solution u of the equation a(u,v) = (f,v) VveH (1.58) exists and is unique, and the stability estimate Nl*<^ll/ll*' (1-59) holds. The Lax-Milgramn theorem guaranties existence and uniqueness for solution u of the variational problem (1.54) as well as the existence and uniqueness of the approximate solution Uk by the Galerkin method. Now, we study the convergence rate of the Galerkin method. Theorem 7. Cea's lemma If the bilinear form a : H x H —^ IR is continuous and coercive with constants C and a as in (1.56) and (1.57), then C \\u — Uk\\H< — inf \\u — v\\H . (1.60) a vexk Hence, Cea's lemma shows that the convergence rate of the Galerkin method depends on the approximation power of the spaces Xk- The term Ek{u) = inf \\u — v\\H (1.61) vexk is known as the error of the best approximation in H. The study of this error is a subject of approximation theory. Nowadays, approximation order is known for several kinds of spaces Xk- For instance, one starts with a univariate wavelet basis corresponding to a mul-tiresolution analysis formed by spaces Vk = {ve CT (0,1) : v\^m G Hr (1, i±l\ , / = 0,..., 2k - l] , (1.62) 16 CHAPTER 1. WAVELET BASIS where 0 < m < r, Ur (a, b) is the space of all polynomials on (a, b) of degree less than r, and Cm (0,1) is the space of m-times continuously differentiable functions on (0,1). If then multiscale bases ^k on Q are constructed using an isotropic or anisotropic tensor product of bases of the spaces then the spaces X^ = span\l/fc satisfy inf \\u - v\\Hs < C2-(r-s)k \u\Hr, (1.63) vexk for any u G Hr (Q) provided that 0 < s < r and Xk is contained in Hs (Q). Here, we view Hs (Q) for s = 0 as the space L2 (Q). Similar results hold for spaces of piecewise polynomial functions incorporating boundary conditions. Hence, r = 3 for the Galerkin method with quadratic spline wavelets from [13, 19], and r = 4 for the Galerkin method with cubic spline wavelets from [11, 14, 15, 17, 23]. From Theorem 6 and Theorem 7, the convergence rate depends on the chosen discretization spaces and not directly on the chosen bases of these spaces. Since a scaling basis generates the same spaces as a multiscale basis tyk, it can be expected that the error will be similar. However, the Galerkin method with a wavelet basis, called the wavelet-Galerkin method, has several advantages. This method seems to be superior to classical methods especially for operator equations with an integral term, because the discretization matrices can be approximated by sparse matrices while most other methods lead to full matrices, see [2, 23, 25]. The second advantage is that a simple diagonal preconditioner is optimal in the sense that diagonally rescaled discretization matrices have uniformly bounded condition numbers. This affects the number of iterations needed to resolve the problem with a desired accuracy. Finally, the solution has a sparse representation in a wavelet basis, which can be used for adaptive versions of the wavelet-Galerkin method that are based on analysis of the size of wavelet coefficients, see e.g. [72], or on a priori knowledge of singularity regions as we did in [20]. We write the function u\. as Let the matrix Ak and the vector ffc have entries A*A = a(>A,^), fk = (f,^), ^A,^G^, (1.65) and ck be the column vector of coefficients c\. Substituting (1.64) into (1.55), we obtain the system Afccfc = ffc. (1.66) Preconditioning. We apply the standard Jacobi diagonal preconditioning to the system (1.66). Let Dfc be a diagonal matrix with diagonal elements D*A = ^/A^ = y/atyxM. (1.67) Then, we obtain the preconditioned system 1.7. ADAPTIVE WAVELET METHODS 17 with Ak = (Dfc)_1 Ak (Dfc)_1, ffc=(Dfc)_1ffc, ck = r>kck. (1.69) The resulting system can be large, and therefore it is usually solved by an appropriate iterative method such as the method of generalized residuals or, in the case where the system matrix is symmetric and positive definite, one can use the conjugate gradient method. Due to coercivity of the bilinear form a, the matrices Ak have uniformly bounded condition numbers (cond), i.e. there exists a constant C such that condAfc < C for all k > j0, see [28, 35, 52]. Sparsity of discretization matrices. If A is a differential operator, then it may be convenient to compute the discretization matrix (AQj, in a scaling basis first, because this matrix is banded. However, the condition numbers of these matrices are not uniformly bounded. Therefore, we use the fast wavelet transform on its rows and columns to transform it to the matrix Ak for a wavelet basis. Using this approach, wavelets are not used directly in the computation, and the fast wavelet transform can be viewed together with diagonal rescaling (1.69) as optimal preconditioning of the system. For more details see e.g. [15, 72]. If A is an integral or integro-differential operator, then a discretization matrix in a scaling basis is typically full, and in this case, it is more convenient to compute entries of the matrix Ak directly rather than to use FWT. For a large class of integral operators this matrix can be approximated by a sparse or quasi-sparse matrix. Several estimates for decay estimates of these matrices are known [2, 23] that make it possible to compute only significant entries of these matrices. We used the wavelet-Galerkin method in [15, 19, 23]. In [15, 19] we used a modification of the wavelet-Galerkin method called multilevel Galerkin method which first computates the solutions of (1.68) on some coarse scale and then use this solution to define an initial vector of the iterative method when solving the discrete problem on some finer scale. In [23] we used the Crank-Nicolson scheme for time discretization and the wavelet-Galerkin method for spatial discretization of the parabolic partial integro-differential equation representing Kou's model for option pricing. In [20], we proposed an adaptive version of the wavelet-Galerkin method for the numerical solution of differential equation with the Dirac measure on the right-hand side. 1.7 Adaptive Wavelet Methods We briefly review a class of adaptive methods that were originally designed by A. Cohen, W. Dahmen, and R. DeVore in [29, 30] and later modified in many papers [7, 12, 33, 34, 41, 43, 53, 67, 72]. The results presented in this section are known and fuller details can be found in these papers. While the classical adaptive methods use refining a mesh according to a posteriori local error estimates, the wavelet approach is different and comprises the following steps: 18 CHAPTER 1. WAVELET BASIS • We start with a variational formulation but instead of its finite dimensional approximation as in the case of the wavelet-Galerkin method, we expand the solution in a wavelet basis and transform the continuous problem into an infinite-dimensional /2-problem. • We propose an iteration scheme for the infinite-dimensional problem. • We replace all infinite-dimensional quantities by finitely supported ones, and we design the routine for an approximate multiplication of an infinite matrix and a finite vector. As in the previous section, we consider the problem (1.52) and the corresponding variational formulation (1.54). We focus on the case that the bilinear form a : HxH —> IR is symmetric, continuous, and coercive. Then, by the Lax-Milgram theorem, the problem (1.54) has a unique solution. Let when normalized with respect to the iJ-norm, be a wavelet basis in H. Let D be a bi-infinite diagonal matrix with diagonal elements Da,a = y/a tyxM, G (1-70) Then the original equation (1.52) can be reformulated as an equivalent bi-infinite matrix equation Ac = f, (1.71) where A = D 1 (A^f, D 1 is a diagonally preconditioned discretization matrix, u = cTD-1^, and f = D 1 (f,ty). Then u solves (1.52) if and only if c solves the matrix equation (1.71). Moreover, the condition number of the matrix A is finite. The simplest convergent iteration for the /2-problem (1.71) is a Richardson iteration which has the following form c0 = 0, cn+i = cn + u) (f - Acn), n = 0,1,---- (1-72) The method is convergent if 0 < u < 2/Xmax (A), where Xmax (A) is the largest eigenvalue of A. It is known that the optimal relaxation parameter u and the corresponding estimate of the error reduction are given by 2 = cond (A) - 1 (A) + Xmax (A)' cond (A) + 1' where \min (A) is the smallest eigenvalue of A. Then, ||c„+i -c\\ < p \\cn - c|| . (1.74) Hence, the small condition number of A, which depends on the chosen wavelet basis, guaranties the small value of a reduction parameter p. Structure of the discretization matrix. Since the matrices Ak defined by (1.69) arising from discretization using the wavelet-Galerkin method are submatrices of biinfinite matrix A, the matrices Ak and A have similar structure. For differential 1.7. ADAPTIVE WAVELET METHODS 19 equations the discretization matrices typically have a so-called finger pattern, see e.g. [17]. Therefore these matrices are quasi-sparse, i.e. they have 0(N\ogN) nonzero entries, where iV x iV is the size of the matrix. For equations containing the integral term the discretization matrices can be approximated by sparse or quasi-sparse matrices. In some papers, e.g. [17, 45], a construction of a wavelet basis was proposed which leads to discretization matrices that are truly sparse, i.e. they have O (N) nonzero entries. Coarsening of vectors. To control the number of degrees of freedom in the algorithm, one needs a routine for approximation of a vector v by its iV-term approximation, i.e. v^v is obtained by retaining the iV largest components of v. It can be done simply by sorting and thresholding as in the following algorithm. COARSEfv, N] -+ vN 1. Sort \v\\, A G 3, in descending order and denote the resulting vector as v. 2. Denote the iV-th element of v as P. 3- If \v\\ > P then set (vn)a = v\, else set (vn)\ = 0. The sorting of all nonzero elements of v requires iVv log iVv arithmetic operations, where iVv = #supp v. However, it is possible to avoid sorting to obtain the algorithm with linear complexity. Such algorithm uses so-called binning and can be found in Stevenson [67]. Approximation of the right-hand side. We assume that it is possible to compute the vector f = D 1 (/, of wavelet coefficients of the right-hand side / G H' with a desired accuracy. More precisely, we require that for any e > 0, there exists a finitely supported vector fe G I2 (J), such that ||f-f£|| ie. This can be realized by computing a highly accurate approximation to f as a preprocessing step and then applying the routine COARSE to this finitely supported array of coefficients. Matrix vector multiplication. Solution of the equation (1.71) by some iterative method requires a multiplication of the infinite-dimensional matrix A with a finitely supported vector v = {v\}Xej. There are several routines available. Here, we present the routine APPLY that we proposed in [12]. The idea is the following: We truncate A in scale by zeroing its entries AA ^ whenever ||A| — > k, k G NU{0}, and denote the resulting matrix by A&. Let us denote Sa* = max{|A,\,M|, ||A| — = k}. Then we multiply the matrix Ao with vector entries that are greater than given tolerance e, the matrix Ai — Ao with vector entries that are greater than c/Sai, ■ ■ ■, and the matrix Ak — Ax-i with vector entries that are greater than e/Sak- In the case that Sau = 0 for some k, we can formally define e/SAk = oo and no multiplications with 20 CHAPTER 1. WAVELET BASIS matrix — Ak-i are necessary. More precisely, let The parameter K is the smallest number such that \ek is an empty set for all k > K. In [12] we proposed two algorithms based on these ideas. We present one of them below. APPLY [A, v, e] -> we For j G N U {0}, let Cj be such that ||A — Aj\\ < Cj. 1. Set SAfe := max{|AAjM|, ||A| — = k}. 2. Set G = 1.1 and 5 = [logc-ej, where |_-J denotes the floor function. Compute wi := w and W2 := w according to (1.76). 3. While ||wi — W2II > e 5:= 5- 1 Compute wi := wG and w2 := w2G using (1.76). end while. 4. we := wi. Algorithm SOLVE. Since we consider here a class of adaptive methods, there are many algorithms representing these methods. We present one example of such an algorithm that we used in [19]. The method insists in solving the infinite preconditioned system (1.71) with Richardson iterations. We compute the relaxation parameter u and the error reduction factor p by (1.73). Then we set 9 = 0.3 and K G N such that 2pK/9 < 0.6. The resulting algorithm is of the form: SOLVE [A, f, e] ce 1. Set j := 0, Uo := 0, and eo > ||c||2. 2. While ej > e do zo := ci? For / = 1,...,K do zz := zz_x + uj (RHS[f, - APPLY [A, zz_1; |^]), end for, j ■= J + 1 Cj := COARSE[zx, (1 - 9) e,], end while, ce := Cj. It is known that the coefficients of a function in the wavelet basis are small in regions where the function is smooth and large in regions where the function has some singularity or a large gradient. Since we work with a sparse representation of the right-hand side and a sparse representation of the vector representing the solution, the method is adaptive. 1.7. ADAPTIVE WAVELET METHODS 21 For analysis of the method we refer to e.g. [30, 41, 72]. Roughly speaking, the methods converge with the same rate as the wavelet-Galerkin method, but for a wider class of functions, because the error estimates are derived in Besov spaces and not only in Sobolev spaces. Other advantages are a small number of parameters representing the solution with desired accuracy, asymptotical optimality in the sense that the number of floating point operations depend linearly on the number of degrees of freedom, optimality of diagonal preconditioner, a sparse structure of matrices also for equations containing an integral term, and a higher-order convergence if higher-order basis functions are used. In this thesis, we used adaptive wavelet methods in [10, 11, 13, 14, 17, 19]. CHAPTER 1. WAVELET BASIS Chapter 2 Constructions of Quadratic and Cubic Spline-Wavelet Bases Since this thesis is a collection of eight articles that are all concerned with constructions of quadratic or cubic spline-wavelet bases on the interval, we review here existing constructions of such types of bases, discuss their advantages and disadvantages, and comment on the papers presented in this thesis. We also review applications of these bases. We focus on concrete quadratic and cubic spline wavelet bases for which the Riesz basis property was proven. There also exist general methods for construction of wavelet bases on the interval, e.g. [51], spline wavelets without adaptation to boundary conditions and without the proof of the Riesz basis property, e.g. [59, 66], and quadratic and cubic finite element wavelets [39]. 2.1 Quadratic Spline-Wavelet Bases In [38, 40], W. Dahmen, A. Kunoth and K. Urban proposed a construction of a spline-wavelet biorthogonal wavelet basis on the interval. The inner wavelets were the same as wavelets from [32], where wavelet bases were constructed on the whole real line. The order of spline is any N > 1, and the number of vanishing moments is L > N such that N + L is even. Both the primal and dual wavelets are local. A disadvantage of these bases is their relatively large condition number. Therefore many modifications of this construction were proposed, see e.g. [1, 3, 5, 70]. The construction by M. Primbs [63] outperforms previous constructions for the linear and quadratic spline-wavelet bases with respect to their conditioning. In [8, 10, 43] the construction was significantly improved in the case of cubic spline wavelet basis, but the condition numbers of quadratic spline-wavelet bases was comparable to those constructed by M. Primbs. In the case of quadratic spline wavelet basis adapted to homogeneous Dirichlet boundary conditions, these bases are even the same up to a normalization, see also the comparison in [19]. In [46], a method for a construction of the L2-orthogonal wavelet basis on the real line was proposed starting from a non-orthogonal wavelet basis. In [64], the L2-orthogonal spline-wavelet bases on the unit 23 24 CHAPTER 2. CONSTRUCTIONS OF SPLINE-WAVELET BASES interval were constructed using this method. Quadratic spline wavelet bases with nonlocal duals have also been constructed and adapted to some types of boundary conditions [26, 56]. The main advantages of these types of bases in comparison to bases with local duals are usually the shorter support of wavelets, the lower condition numbers of the bases and the corresponding stiffness matrices, and also the simplicity of the construction. In [13, 19], we also constructed quadratic spline-wavelet bases on the interval. Here, we comment on these constructions. [13] Černá, D.; Finěk, V.: Quadratic spline wavelets with short support for fourth-order problems, Result. Math. 66(6), (2014), pp. 525 540. In [13], we proposed two constructions of quadratic spline-wavelet bases for the space Hq (0,1). The inner wavelets have one vanishing moment and boundary wavelets are of two types, wavelets with one vanishing moment and wavelet with shorter support but without vanishing moments. Since we did not require local support of dual wavelets, we were able to construct wavelets with a short support of length 2, which is the shortest possible support for wavelets with one vanishing moment corresponding to the quadratic B-spline multiresolution analysis. We used the isotropic tensor product to obtain a wavelet basis for the space Hq ((0, l)2). We studied the quantitative behaviour of the adaptive wavelet method for the numerical solution of the fourth-order differential equation A2u = / on the unit square, A being the Laplace operator. Due to the short support, the discretization matrices are sparser than for other quadratic spline wavelets of the same type. The condition numbers of discretization matrices are uniformly bounded and small, e.g. for the stiffness matrix of the size 64516 x 64516 the condition number is 11.1. [19] Černá, D.; Finěk, V.: Quadratic spline wavelets with short support satisfying homogeneous boundary conditions, Electron. Trans. Numer. Anal. 48, (2018), pp. 15 39. In [19], we constructed wavelets of the similar type as in the previous paper, but adapted to the first-order homogeneous Dirichlet boundary conditions, i.e. quadratic spline wavelets on the interval and on a unit square with one vanishing moment and the shortest possible support. The matrices arising from discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers and the condition numbers are small, e.g. the condition number was 2.84 for a matrix of the size 1024 x 1024 corresponding to one-dimensional Poisson's equation, and it was 18.3 for the matrix of the size 1048576 x 1048576 corresponding to a two-dimensional Poisson's equation. We also provided numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis require a smaller number of iterations than methods with other quadratic spline wavelet bases of the same type, i.e. bases from [3, 26, 43, 63]. Moreover, due to the short support of our wavelets, one iteration requires a smaller number of floating-point operations than for these bases. In [20], we propose post-processing for the Galerkin method with this basis, such that the resulting method has a convergence rate the same as the rate of convergence for the Galerkin method with cubic spline wavelets under the assumption that the 2.2. CUBIC SPLINE-WAVELET BASES 25 solution is smooth enough. We show theoretically as well as numerically that the presented method outperforms the Galerkin method with many other quadratic or cubic spline wavelets with respect to the number of floating point operations needed to compute a sufficiently accurate solution. Furthermore, we proposed local postprocessing for example with an equation with Dirac measure on the right-hand side. In Table 2.1, we list parameters and properties for several constructions of quadratic spline-wavelet bases such as the order of homogeneous Dirichlet boundary conditions (bound, cond.), the number of vanishing moments (vanish, moments), the maximal length of the support of generators of inner scaling functions (supp seal.), the maximal length of the support of generators of inner wavelets (supp wav.), locality of duals (loc. duals), the number of generators of inner scaling functions (seal, gen.), the number of generators of inner wavelets (wav. gen.), and special properties. The property short, sup. means that a wavelet basis is such that wavelets have the shortest possible support among all wavelets with the same number of vanishing moments corresponding to the same scaling basis. The other very important parameters that characterize wavelet bases are the condition number of the basis and the condition numbers of discretization matrices. These numbers are problem dependent and can be found in the attached papers. Table 2.1: Parameters characterizing quadratic spline-wavelet bases. wavelet bound. vanish. supp supp loc. seal. wav. special basis cond. moments seal. wav. duals gen. gen. property DKU [38] 0 L > 3 odd 3 L + 2 loc. 1 1 D [43] > o L > 3 odd 3 L + 2 loc. 1 1 B[3] 0 L > 3 odd 3 L + 2 loc. 1 1 P [63] 0-1 L > 3 odd 3 L + 2 loc. 1 1 CF [10] 0-1 L > 3 odd 3 L + 2 loc. 1 1 R [64] 1 3 2 2 loc. 6 6 L2-orth. CQ [26] 0 3 3 5 glob. 1 1 semiorth. J [56] 2 1 3 3 glob. 1 1 J [56] 2 3 3 5 glob. 1 1 CF [13] 2 1 3 2 glob. 1 1 short, sup. CF [19] 1 1 3 2 glob. 1 1 short, sup. 2.2 Cubic Spline-Wavelet Bases As already mentioned in the previous section, biorthogonal cubic spline-wavelet bases with local support of primal and dual wavelets were constructed in [38, 40], and this construction was modified in several papers. In [10, 43] the construction was significantly improved with regard to conditioning of the bases. Biorthogonal cubic Hermite spline multiwavelet bases on the interval with local duals were designed in 26 CHAPTER 2. CONSTRUCTIONS OF SPLINE-WAVELET BASES [40, 65]. The L2-orthogonal piecewise cubic basis was constructed in [64] using the method from [46]. Several cubic spline wavelet and multiwavelet bases with nonlocal duals have been constructed and adapted to various types of boundary conditions in [26, 45, 55, 56, 57]. Their properties are summarized in Table 2.2. Similarly to the case of quadratic spline wavelets the main advantages of these types of bases in comparison with bases with local duals are usually shorter supports of wavelets, lower condition numbers of the bases and corresponding discretization matrices, and also simplicity of the construction. Below we comment on the constructions of cubic spline wavelets that are presented in the papers [10, 11, 14, 15, 17, 23] collected in this thesis. [10] Černá, D.; Finěk, V.: Construction of optimally conditioned cubic spline wavelets on the interval, Adv. Comput. Math. 34(2), (2011), pp. 219 252. In [10], we constructed biorthogonal spline-wavelet bases such that both primal and dual wavelets are local and they have the desired number of vanishing wavelet moments. Inner wavelets are translated and dilated versions of the well-known wavelets designed by A. Cohen, I. Daubechies, and J.-C. Feauveau in [32]. Our objective was to construct interval spline-wavelet bases with condition numbers close to the condition numbers of spline wavelet bases on the real line, especially in the case of cubic spline wavelets. We showed that the constructed set of functions is indeed a Riesz basis for the space L2 (0,1) and for the Sobolev space Hs (0,1) for a certain range of s. Then we adapted the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to complementary boundary conditions. We compared the efficiency of an adaptive wavelet scheme for our wavelets and cubic spline wavelets constructed in [63] by M. Primbs and we showed the superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson's equations where the solution has steep gradients. [11] Černá, D.; Finěk, V.: Cubic spline wavelets with complementary boundary conditions, Appl. Math. Comput. 219(4), (2012), pp. 1853 1865. In [11], we focused on a construction of a cubic spline-wavelet basis on the interval with local duals satisfying complementary boundary conditions of the second order. This means that a primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions of the second order, while the dual wavelet basis preserves the full degree of polynomial exactness. We showed superiority of our construction in comparison to spline wavelet bases of the same type, i.e. those from [63, 65], with respect to conditioning of wavelet bases and the number of iterations in an adaptive wavelet method for the numerical solution of the partial differential equation A2u = f in two dimensions. For example, the discretization matrix for this problem in one dimension when a basis with seven levels of wavelets is used has the condition number 66.7 for our basis, 693.0 for the basis from [65], and 1117.0 for the basis from [63]. Hence, this basis can be recommended for problems where local duals are needed and second-order Dirichlet boundary condition are prescribed. 2.2. CUBIC SPLINE-WAVELET BASES 27 [14] Černá, D.; Finěk, V.: Cubic spline wavelets with short support for fourth-order problems, Appl. Math. Comput. 243, (2014), pp. 44 56. In [14], we proposed a construction of new cubic spline-wavelet bases on the unit cube satisfying homogeneous Dirichlet boundary conditions of the second order. Wavelets have short supports of the length 3 and two vanishing moments. In this paper, we were inspired by the construction of cubic spline-wavelet basis satisfying similar type of boundary conditions proposed by R.Q. Jia and W. Zhao in [57], where the wavelets have no vanishing moments. They used their basis for solving fourth-order problems, and they showed that the Galerkin method with this basis has superb convergence and it outperforms the Galerkin method with cubic splines preconditioned using BPX preconditioner or multigrid method. The discretization matrices for the equation A2u = f on a unit square have very small and uniformly bounded condition numbers. In our paper [14], we designed wavelet bases with the same scaling functions, but with different wavelets. We showed that our basis has an even smaller condition number than the basis in [57] and additionally the wavelets have vanishing moments. For example, the condition number of the discretization matrix of the size 65025 x 65025 was 18.6. Vanishing moments enable the use of this wavelet basis in adaptive wavelet methods and wavelet-based methods for equations with an integral term. [15] Černá, D.; Finěk, V.: Wavelet basis of cubic splines on the hy-percube satisfying homogeneous boundary conditions, Int. J. Wavelets Multiresolut. Inf. Process. 13(3), (2015), article No. 1550014. In [15], we proposed a construction of new cubic spline wavelets on the hyper-cube that have two vanishing moments and satisfy first-order homogeneous Dirichlet boundary conditions. In comparison with [14] where the duals are not discussed, here we defined dual spaces as linear spline spaces. We defined bases of dual scaling spaces that have compact support and used them for the proof of the Riesz basis property. The biorthogonal wavelet basis contains functions with global support. The matrices arising from discretization of second-order elliptic problems using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are relatively small. We constructed wavelet bases on the hypercube using both isotropic and anisotropic tensor product, studied condition numbers of discretization matrices corresponding to the Helmholtz equation with various parameters, and we provided a numerical example to show the efficiency of the multilevel-Galerkin method using the constructed basis. This basis was studied by L. Calderón, M.T. Martin, and V. Vampa in [6]. They used our basis in numerical experiments and showed that the additional advantage is that the stiffness matrix corresponding to the one-dimensional Poisson's equation is banded. This also affects the structure of discretization matrices for the Helmholtz equation in higher dimensions, because they are computed using tensor products of the stiffness matrix and the mass matrix. 28 CHAPTER 2. CONSTRUCTIONS OF SPLINE-WAVELET BASES [17] Černá, D.; Finěk, V.: Sparse wavelet representation of differential operators with piecewise polynomial coefficients, Axioms 6, (2017), article No. 4. In [17], we proposed a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. We focused on the structure of discretization matrices rather than on the length of the support and the condition number of the basis as in the previous papers. Here, the wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore the wavelets have eight vanishing moments and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse and not only quasi-sparse as for most wavelet bases. This greatly simplifies the routine APPLY needed for computation of the multiplication of a biinfinite matrix with a finitely supported vector in adaptive wavelet methods. Numerical examples showed the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving a second-order elliptic equation and the Black-Scholes equation with two state variables and quadratic volatility. [23] Černá, D.: Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model, Int. J. Wavelets Multiresolut. Inf. Process. 17(1), (2019), article No. 1850061. As in our paper [14], our aim was to construct cubic spline wavelets with the shortest possible support corresponding to B-spline multiresolution analysis, but with a larger number of vanishing moments, namely four vanishing moments. We constructed bases that satisfies no boundary conditions and bases that satisfy first-order homogeneous Dirichlet boundary conditions. Inner wavelets are the same as inner wavelets for a wavelet basis on the real line constructed in [24, 50]. To illustrate the applicability of the constructed bases we used the wavelet-Galerkin method with our bases to option pricing under the double exponential jump-diffusion model represented by a partial integro-differential equation. We used the Crank-Nicolson scheme for time discretization and the wavelet-Gal erkin method for spatial discretization. We compared the results with B-spline bases and cubic spline wavelet bases from [10], because they are adapted to the same type of boundary conditions. Since the equation contains an integral term, most classical methods lead to full matrices. Hence the advantage of the proposed method is the quasi-sparse structure of the discretization matrices. In comparison with methods from [49, 54, 58, 61, 71], the presented method required significantly smaller number of degrees of freedom needed to compute the solution with desired accuracy. In Table 2.2, we list parameters and properties for cubic spline-wavelet bases. The property sparse Lapl. means that matrices arising from discretization of Laplacian are truly sparse and the property sparse diff. means that the discretization matrices are sparse for some class of differential operators with piecewise polynomial coefficients. 2.3. APPLICATIONS OF CONSTRUCTED BASES 29 Table 2.2: Parameters characterizing cubic spline-wavelet bases. wavelet bound. vanish. supp supp loc. seal. wav. special basis cond. moments seal. wav. duals gen. gen. property DKU [38] 0 L > 4 even 4 L + 3 loc. 1 1 D [43] m > 0 L > 4 even 4 L + 3 loc. 1 1 B[3] 0 L > 4 even 4 L + 3 loc. 1 1 P [63] 0-1 L > 4 even 4 L + 3 loc. 1 1 CF [10] 0-1 L > 4 even 4 L + 3 loc. 1 1 CF [11] 2 6 4 9 loc. 1 1 S [65] 2 2 2 3 loc. 2 2 R [64] 1 4 2 2 loc. 6 6 L2-orth. CQ [26] 0 4 4 7 glob. 1 1 semiorth. J [56] 3 2 4 5 glob. 1 1 J [56] 3 4 4 7 glob. 1 1 JZ [57] 2 0 4 3 glob. 1 1 JL [55] 1 2 2 2 glob. 2 2 CF [14] 2 2 4 3 glob. 1 1 short, sup. CF [15] 1 2 4 5 glob. 1 1 CF [16] 1 4 2 4 glob. 2 4 sparse Lapl. CF [17] 1 8 2 8 glob. 2 8 sparse diff. CF [23] 0-1 4 4 4 glob. 1 1 short, sup. D [45] 1 4 2 4 glob. 2 4 sparse Lapl. 2.3 Applications of Constructed Bases Wavelet bases on the interval and product domains are useful in a wide range of applications including signal and image analysis, data compression, and numerical solution of various types of operator equations. In this section, we mention several concrete examples where wavelets constructed in the enclosed papers were used. First, we mention applications from these papers. Second-order linear elliptic equations These equations represent a wide range of applications, typically governing equilibrium problems in physics such as displacement of a membrane, electric potential, gravity fields, or pressure fields. We focused on the Poisson and Helmholtz equations, which we solved by the wavelet-Galerkin method in [15, 17] and by the adaptive wavelet method in [10]. Fourth-order linear elliptic problems These equations arise for example in linear elasticity theory, mechanics of elastic plates, or slow flows of viscous fluids. We solved these equations in [11, 13, 14]. As already mentioned, we were motivated by the results in [57], where a cubic spline-wavelet basis adapted to second-order boundary conditions was constructed. The 30 CHAPTER 2. CONSTRUCTIONS OF SPLINE-WAVELET BASES wavelet-Galerkin method has superb convergence and outperformed BPX precondi-tioner and multigrid methods, but wavelets did not have vanishing moments. We improved their results in [14], where we constructed wavelets of similar type, but even better conditioned and with vanishing moments. This enabled us to reduce the number of iterations needed to find a sufficiently accurate solution and to apply these wavelets in adaptive wavelet methods. Option pricing under the Black-Scholes model In [17], we solved the Black-Scholes equation with two state variables and quadratic volatility by an adaptive wavelet method. The advantages of an adaptive wavelet approach were not only the small number of degrees of freedom needed to find a solution with desired accuracy, but also that the routine APPLY was greatly simplified due to the fact that the discretization matrices have uniformly bounded number of nonzero entries in each row. This is not the case for most other wavelet bases, which have a so-called finger pattern. Option pricing under a double exponential jump diffusion model In [23], we studied option pricing under a double exponential jump diffusion model proposed by Kou in [60]. Since this model is represented by partial integro-differential equation, most classical methods suffer from the fact that discretization matrices are full. We used the wavelet-Gal erkin method combined with the Crank-Nicolson scheme and showed that the discretization matrices can be approximated by quasi-sparse matrices. Furthermore, we showed that our method enables a solution to the problem with desired accuracy and smaller number of degrees of freedom than methods from [49, 54, 58, 61, 71]. Hence, smaller matrices are involved in computation. We also used the wavelets from this thesis in applications in our other works. Several of them are mentioned below. Option pricing under stochastic volatility models Wavelet methods are a very promising tool for option pricing, for a survey see [52, 64]. We used an adaptive wavelet method with bases constructed in this thesis for option pricing under stochastic volatility models that are improvement of the famous Black and Scholes model, where volatility is a constant or deterministic function. For instance in [21], we used wavelets from [19] for option pricing under the Heston model. Valuation of Asian options In [18], we used the adaptive wavelet method with a linear spline-wavelet basis from [10] for valuation of two-asset Asian options with a floating strike. We compared this method with the wavelet-Galerkin method with the same basis, and we found that the adaptive method required a significantly smaller number of degrees of freedom to compute the solution with a desired accuracy. Moreover, the optimal convergence rate with respect to the L2-norm was achieved for the adaptive wavelet method, while this was not the case of the wavelet-Galerkin method. 2.3. APPLICATIONS OF CONSTRUCTED BASES 31 Sensitivity analysis of options In the book [53], we were concerned with option pricing and the numerical computation of the Greeks, i.e. derivatives of the option price with respect to underlying parameters such as the underlying asset price, time to expiration, volatility, and interest rates. The Greeks measure the sensitivity of the option price to these parameters and their computation is important for hedging. Since Greeks are defined as derivatives of option price with respect to these parameters, the convergence rate for the methods for their computation is typically smaller than the convergence rate for the methods for computation of the price of the option. Therefore, it is beneficial to use a higher-order method such as the adaptive method with quadratic and cubic spline-wavelet basis. In [53], comparison with other methods such as the finite difference method, a discontinuous Galerkin method and a fuzzy method was provided. Our method was superior in the sense that it enabled achieving a significantly smaller error for the same number of degrees of freedom as these methods. Differential equations with Dirac right-hand side In [20], we proposed an adaptive method that uses wavelets constructed in [19] for the numerical solution of the partial differential equation with the right-hand side that contains the Dirac delta function. Sparse representation of images and image compression As already mentioned in Section 1.4, the wavelet bases on the interval lead to the fast wavelet transforms that use special boundary filters. In [8, 9], we used the FWT corresponding to spline wavelets from [10] to sparse representation of images and image compression. We compared our method with methods based on the signal extension such as zero padding, symmetrization, periodization, etc. and we showed that the error near the boundary is significantly smaller for our method and that the method enables to reduce boundary artefacts. Singularly perturbed boundary value problems We also used the adaptive wavelet method for the solution of singularly perturbed boundary value problems in [4]. In summary, wavelets on the interval can be used directly in methods for the numerical solution of operator equations and in signal and image processing for decomposition, analysis, and compression. In addition, constructions of wavelets on the interval can be used as the first step of constructions of wavelets on more general domains and constructions of wavelets satisfying some special conditions. For example as mentioned in [69], a construction of divergence-free wavelets starts with the pair of two biorthogonal wavelet bases such as those from [11, 45, 63]. Divergence free wavelets then can be used for the numerical solution of the Navier-Stokes equations representing the flow of viscous fluid. Moreover, wavelets on the interval can be used in many engineering applications, e.g. the method from [13] was used as the part of the algorithm for building venting system on complex surfaces of injection molds in [73]. For other applications of wavelets on the interval we refer to [25, 28, 37, 41, 72]. CHAPTER 2. CONSTRUCTIONS OF SPLINE-WAVELET BASES Conclusions and Further Research The previous text summarizes the results presented in the attached papers. In these papers, we constructed several quadratic and cubic spline wavelets on the interval and product domains and compared them to existing constructions. All constructed bases were well-conditioned. Bases from [13, 14, 23] have the shortest possible support, therefore discretization matrices are sparse and one iteration of the method for the solution of the resulting discrete system requires less floating point operations than for other wavelets of the same type applied to the same equations. Bases from [17] lead to truly sparse matrices for a class of differential operators with polynomial coefficients, while other wavelet bases lead to only quasi-sparse matrices. For applications, where global support of dual functions is needed we can recommend bases from [10, 11]. We used the constructed basis for the numerical solution of many types of equations, and we presented other possible applications of the bases. In terms of further research, we would like to extend our previous results to higher-dimensional problems (dimension d > 4), especially for solving partial integro-differential equations representing pricing multi-asset options under jump diffusion models. Furthermore, we recently constructed wavelet dictionaries for ECG signal modelling in [22]. Here, the dictionaries were constructed from wavelets on the real line simply by restriction. 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Comput. 30, 2008, pp. 1949-1970. [72] Urban, K.: Wavelet Methods for Elliptic Partial Differential Equations, Oxford University Press, Oxford, 2009. [73] Zhang, Y.; Hou, B.; Wang, Q.; Huang, Z.; Zhou, H.: Automatic generation of venting system on complex surfaces of injection mold, Int. J. Adv. Manuf. Technol. 98, 2016, pp. 1379-1389. BIBLIOGRAPHY Selected papers This chapter contains articles [10, 11, 13, 14, 15, 17, 19, 23]. Due to copyright restrictions, we present postprints of the papers [10, 11, 13, 14, 17, 19] and preprints of the papers [15, 23]. 41 Reprinted by permission from Springer Nature: Springer Nature, Advances in Computational Mathematics 34(2), 2011, pp. 219-252, https://doi.org/10.1007/sl0444-010-9152-5 Copyright 2010, https://link.springer.com/journal/10444 Construction of Optimally Conditioned Cubic Spline Wavelets on the Interval Dana Černá • Václav Finěk Abstract The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L2 ([0,1]) and for the Sobolev space Hs ([0,1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep gradients. Keywords Biorthogonal wavelets • Interval • Spline • Condition number Mathematics Subject Classification (2000) 65T60 65N99 1 Introduction Wavelets are by now a widely accepted tool in signal and image processing as well as in numerical simulation. In the field of numerical analysis, methods based on wavelets are successfully used especially for preconditioning of large systems arising from discretization of elliptic partial differential equations, sparse representations of some types of operators and adaptive solving of operator equations. The quantitative performance of such methods strongly depends on a choice of a wavelet basis, in particular on its condition number. Wavelet bases on a bounded domain are usually constructed in the following way: Wavelets on the real line are adapted to the interval and then by a tensor product technique to the ^-dimensional cube. Finally by splitting the domain into subdomains which are images of (0,1)" under appropriate parametric mappings one can obtain wavelet bases on a fairly general domain. Thus, the properties of the employed wavelet basis on the interval are crucial for the properties of the resulting bases on general a domain. Biorthogonal spline-wavelet bases on the unit interval were constructed in [16]. The disadvantage of them is their bad condition which causes problems in practical applications. Some D. Černá Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic Tel.: +420-48-535-2407 Fax: +420-48-535-2332 E-mail: dana.cerna@tul.cz V. Finěk Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic 2 modifications which lead to better conditioned bases were proposed in [2], [17], [24], and [33]. The recent construction by M. Primbs, see [12], [24], or [25], seems to outperform the previous constructions with respect to the Riesz bounds as well as spectral properties of the corresponding stiffness matrices in the case of linear and quadratic spline-wavelets. In this paper, we focus on cubic spline wavelets and we construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line. It is known that the condition number of the wavelet basis on the real line is less than or equal to the condition number of the interval wavelet basis, where the inner functions are restrictions of scaling functions and wavelets on the real line. First of all, we summarize the desired properties: - Riesz basis property. The functions form a Riesz basis of the space L2 ([0,1]). - Locality. The basis functions are local. Then the corresponding decomposition and reconstruction algorithms are simple and fast. - Biorthogonality. The primal and dual wavelet bases form a biorthogonal pair. - Polymial exactness. The primal MRA has polynomial exactness of order ./V and the dual MRA has polynomial exactness of order N. As in [9], N + N has to be even and N >N. - Smoothness. The smoothness of primal and dual wavelet bases is another desired property. It ensures the validity of norm equivalences, for details see below. - Closed form. The primal scaling functions and wavelets are known in the closed form. It is a desirable property for the fast computation of integrals involving primal scaling functions and wavelets. - Well-conditioned bases. Our objective is to construct wavelet bases with an improved condition number, especially for larger values of ./V and N. From the viewpoint of numerical stability, ideal wavelet bases are orthogonal wavelet bases. However, they are usually avoided in the numerical treatment of partial differential and integral equations, because they are not accessible analytically, the complementary boundary conditions can not be satisfied and it is not possible to increase the number of vanishing wavelet moments independent from the order of accuracy. Moreover, sufficiently smooth orthogonal wavelets typically have a large support. Biorthogonal wavelet bases on the unit interval derived from B-splines were constructed also in [8] and [19] and they were adapted to homogeneous Dirichlet boundary conditions in [20]. These bases are well-conditioned, but have globally supported dual basis functions. Another construction of spline-wavelets was proposed in [4], but the corresponding dual bases are unknown so far. We should also mention the construction of spline multiwavelets [15], [22], and [28], though the dual wavelets have a low Sobolev regularity. The paper is organized as follows. Section 2 provides a short introduction to the concept of wavelet bases. Section 3 is concerned with the construction of primal multiresolution analysis on the interval. The primal scaling functions are B-splines defined on the Schoenberg sequence of knots, which have been used also in [4], [8], and [24]. In Section 4 we construct dual multiresolution analysis. There are two types of boundary scaling functions. The functions of the first type are defined in order to preserve the full degree of polynomial exactness as in [1] and [10]. The construction of the scaling functions of the second type is a delicate task, because the low condition number and nestedness of the multiresolution spaces have to be preserved. Section 5 is concerned with the computation of refinement matrices. In Section 6 wavelets are constructed by the method of stable completion proposed in [18]. The construction of initial stable completion is along the lines of [16]. In Section 7 we show that the constructed set of functions is indeed a Riesz basis for the space L2 ([0,1]) and for the Sobolev space Hs ([0,1]) for a certain range of s. In Section 8 we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented in Section 9. Finally, in Section 10, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for one-dimensional and two-dimensional Poisson equations where the solution has steep gradients. 3 2 Wavelet bases This section provides a short introduction to the concept of wavelet bases. Let us introduce some notation. We use N, Z, q, and R to denote the set of positive integers, integers, rational numbers, and real numbers, respectively. Let Nj0 denote the set of integers which are greater than or equal to jQ. We consider a domain Q C M.d and the space L2 (Q) with the inner product (•,•) and the induced norm ||-||. Let J? be some index set and let each index 2 6/ takes the form A = (j,k), where A | = j G Z is a scale or a level. Let I2 (j?) be a space of all sequences b = \bx\x^J such that \\nkU)-= i l nr i <~- (i) Definition 1. A family f := {y^ G C L2 (Q) is called a vravefe basis of L2 (£2), if /) W is a basis for L2 (£2), it means that the linear span of W is dense in L2 (Q) and there exist constants c, C G (0, °°) such that cWhWh(J)< X^f j,k,keSj}, %-={Wi,k,ke/}}, (7) where J^-, J?j are finite or at most countable index sets. We refer to (j)j^ as scaling functions and \l/j:k as wavelets. The multiscale basis is given by ^ = ^U U *J (8) 4 and the overall wavelet basis of L2 (Q) is obtained by ^ = *i.u[J^ (9) The single-scale and the multiscale bases are interrelated by the wavelet transform TjiS: I (Ij+S) —>■ I2 (Ij+s), 1/'m T,.s*, s- (10) The dual wavelet system W generates a dual multiresolution analysis S with a dual scaling basis 0. Polynomial exactness of order JVeN for the primal scaling basis and of order N £ N for the dual scaling basis is another desired property of wavelet bases. It means that nN-i(n)cSj, n^ifycSj, j>j0, (li) where TIm (Q) is the space of all algebraic polynomials on Q of a degree at most m. 3 Primal Scaling Basis The primal scaling bases will be the same as bases designed by Chui and Quak in [8], because they are known to be well-conditioned. A big advantage of this approach is that it readily adapts to the bounded interval by introducing multiple knots at the endpoints. Let ./V be the desired order . / .\V+N-l of the polynomial exactness of the primal scaling basis and let tJ = [ tk J be a Schoenberg sequence of knots defined by k=-N+l lk ■ The corresponding B-splines of order N are defined by tJk:=0, k = -N+l,...,0, (12) J — k lr- J tj . — r. k — 1,.... 2 1. k V tl:=l, k = 2j,...,2j +N-1. B{,n (x) :— yk+n lk t1 t1 'klk+N (t-x)N+-\ xe(0,l), (13) where (x)+ := max{0,x}. The symbol [tk,.. -tk+N\ f is the ,/V-th divided difference of / which is recursively defined as {[h+1 v■ ■ A+m]f-[tk,—A+m-l]/ -f f i t fW{tk) ^ h + t^ (14) if tk = tk+N, with / = /(**)■ The set &j = {j k,k = —N + 1,..., 2J — 1} of primal scaling functions is then simply defined by 0^ = 2^^, k=-N+l,...,V-I, j>0. (15) Thus there are 2J— N + 1 inner scaling functions and N — I functions at each boundary. Figure 1 shows the primal scaling functions for ./V = 4 and j = 3. The inner scaling functions are translations and dilations of one function (j) which corresponds to the primal scaling function constructed by Cohen, Daubechies, and Feauveau in [9]. In the following, we consider (j) from [9] which is shifted so that its support is [0,N]. We define the primal multiresolution spaces by Sj := span <£>j. (16) 5 3 2.5 Fig. 1 Primal scaling functions for N = 4 and j = 3 without boundary conditions. Lemma 3. Under the above assumptions, the following holds: i) For any jo £ N the sequence 5^ = {Sj} .q forms a multiresolution analysis ofL2 ([0,1]). ii) The spaces Sj are exact of order N, i.e. njv-i([0,l])cSj, j>\. (17) The proof can be found in [8], [24], [29]. 4 Dual Scaling Basis The desired property of the dual scaling basis & is the biorthogonality to 0 and the polynomial exactness of order N. Let 0 be the dual scaling function which was designed by Cohen, Daubechies, and Feauveau in [9] and which is shifted so that = 0, i.e. its support is [—N + 1,N + N — l]. In this case N > N and N + N has to be an even number. It is known that there exist sequences {hk}k€% and {hk}k^z such that the functions (j) and satisfy the refinement equations 0(jc) = J^Ak0(2jc-Jk), 0 (jc) = Ak0 (2x - ik), xel. (18) The parameters hk and hk are called scaling coefficients. By biorthogonality of 0 and 0, we have 2 52 h2m+kh = So,m, mj0:= flog2 (JV + 2JV-3)] (20) so that the supports of the boundary functions are contained in [0,1]. We define inner scaling functions as translations and dilations of 0: 0j: k ■= 2j/2$ (2j- -k), k = N-l,...,2j-N-N+l. (21) There will be two types of basis functions at each boundary. In the following, it will be convenient to abbreviate the boundary and inner index sets by ,yfA = {-N + l,...,-N + N}, (22) J^1 = {-N + N+\,...,N-2}, (23) = {N-l,...,2j-N-N + l}, (24) jf2 = {2j-N-N + 2,...2j-N-l}, (25) Jfl = {2i-N,...,V-l}, (26) 6 and Sf'lutf'2 = {-N+l,...,N-2}, (27) jf2\JjfX = {2j-N-N + 2,...,2j -1}, (28) yL,i u yL,2 u yo J ^R,2 J = {—Ar + l,...,2-/ — 1}. (29) Basis functions of the first type are defined to preserve polynomial exactness by the same way as in [1], [10]: n-2 6j,k = 2j/2 E (pk+JV_i,0(--Z))0(2^-Ol[o,i]. keJ?f'\ (30) /=-7V-iV+2 where {/?o5 • • • ,Pn-i} is a basis of n^_l ([0,1]). In Lemma 6 we show that the resulting dual scaling functions do not depend on the choice of the polynomial basis. In our case, are the Bernstein polynomials defined by wW^-»+f-'y^r- *=o,....*-., <31) The Bernstein polynomials were used also in [16]. On the contrary to [16], in our case the choice of polynomials does not affect the resulting dual scaling basis but it has only the effect of stabilization of the computation, for details see Lemma 6 and the discussion below. The definition of basis functions of the second type is a delicate task, because the low condition number and the nestedness of the multiresolution spaces have to be preserved. This means that the ~ ~ L 2 L 2 relation 0^ C V/ C Vj+\, k G J^-' , should hold. Therefore we define 0^, k G J^- ' , as linear combinations of functions which are already in V)+i. To obtain well-conditioned bases, we define functions 0j:k, k e J^L'2, which are close to . Then 0^ is close to 0]Pfe|[o,i], because by (18) we have 0^|[Oil] = 2* 1 hrf ■ -2k-I)\m, k G J]>2. (33) k=-n+l Figure 2 shows the functions 0^ and ^k and 0A-N-l > k, IE Jf. (43) ii) The following holds: (Qj)Kl = 5kj, keSj,leSf. (44) Hi) The following holds: (Q;)k,/ = 0, ^e^/e^u^. (45) 8 Proof Due to (35) and by substitution we have for k, I E <0;,*A*> = (2^0;o,fc(2;'-;°.) ,2^0^ (2^-)) = (<^Ao,/>- (46) Therefore, Q^£, = Q;o,l = Ql> i-e. the matrix Qj:l is independent of j. Due to (36) Qj:r is independent of j too. The property (43) is a direct consequence of the reflection invariance (34). The property ii) follows from the biorthogonality of {(j) (• — k)}k€% and {0 (• — 0}/G2' ^ a^so implies (45) for ik e J^°, Z € J^'1 U jf^. It remains to prove (45) for k E , I E U jf1. By the definition of the dual scaling functions of the second type (32), the refinement relation (18) for the dual scaling function , and (19), we have for k E J??, I E J^, ' , / N+N-l \ U(--k),V2 £ ht$(2- -21 -m) |[0il] ) (47) \ m=N-l-2k / 1 N N+N- 1 \ 2/ £m>(2--2£-«), £ hm^{2- -2l-m)\[0:1] ) (48) \"=0 m=N-l -2k 1 N N+N-l N+N-l 2 £ £ hnhm52k+n,2l+m = z 2 £ h2l-2k+mhm (49) n=0m=N-l-2k m=N-l-2k 2 £ h2l-2k+mnm = 0. (50) By (34), the relation (45) holds also for keSfje Jy . Thus, we can write (ql yT (QIt \ *j:=Qf0j=\ hs? ] ®j=\ \&h (51) V Or/ V Q-/J Since the matrix Q; is symmetric in the sense of (43), the properties (34), (35), and (36) hold for tyjjc as well. Fig. 3 Boundary dual scaling functions for N — 4 and N — 6 without boundary conditions. Remark 5. It is known that the scaling function 0 has typically a low Sobolev regularity for smaller values of N. Thus the functions Gj^ have a low Sobolev regularity for smaller values of N, too. Therefore, we do not obtain the sufficiently accurate entries of the matrix Qj directly by 9 classical quadratures. Fortunately, we are able to compute the matrix Qj precisely up to the round off errors. For k 6 j^f'1 U j^f'2, / € j^f'1 we have m j j j <0;,*,e;,*>= I' <(•)", 0(--«))<0(--*),0(--«)>L2( p\-\} are two different bases of the space 71^ ([0,1]) and for i = 1,2 we define the sets ©j = |0j kj by [Ojjc, k£^'2UJ??ij^2. Furthermore, we define Q) = (\ (58) Therefore, we have ^2=(q2rr®2=(q!) >•=(59) Although a choice of a polynomial basis does not influence the resulting dual scaling basis, it has an influence on the stability of the computation and the preciseness of the results, because some choices of the polynomial bases lead to the critical values of the condition number of the biorthogonalization matrix. We present the condition numbers of the matrix for the monomial basis 11 ,x,x2,.. .x^_11 and Bernstein polynomials (31) with the parameters b = 4 and b = 10 in Table 4. In our numerical experiments in Section 9 we choose b = 10. 10 Remark 7. In the case of linear primal basis, i.e. N = 2, there are no boundary dual functions of the second type. In [24] the primal scaling functions and the inner dual scaling functions are the same as ours. The boundary dual functions before biorthogonalization are defined by (30) with the same choice of polynomials po,... ,/?jv-i as ^n [10]- Due to tne I^emma 6, for N = 2 the wavelet basis in [24] is identical to the wavelet basis constructed in this section. The main difference of the construction by M. Primbs [24] in comparison with our construction is the definition of dual basis functions of the second type Q^mbst k = —N + 1,..., —2. Note that they correspond to different indexes than ours. These functions are defined as linear combination of functions 0?"™^, n > k, in order to be already biorthogonal to the primal scaling functions. The refinement coefficients for them are obtained by solving certain system of linear algebraic equations. In case N > 3, the functions Q?^imbs) k = —N + 1,..., — 2, take much larger values than primal scaling functions and than the inner dual scaling functions. Then some of the boundary wavelets take much larger values than inner wavelets which probably causes bad conditioning of wavelet bases. Furthermore, the dual boundary functions of the first type which are defined to preserve the polynomial exactness correspond to the first N scaling functions in our case and they correspond to the primal scaling functions indexed by —l,...,N — 2 in case of the construction from [24]. It leads to better matching of the supports and values of the primal and dual functions in our construction. This better localization and 'almost biorthogonality' of the dual functions of the second type to the primal scaling functions lead to optimally conditioned wavelet bases for N < 4 and to an improvement of the condition number also for N = 5, see Section 9. The constructions of primal and dual boundary scaling functions in [16] and [17] is based on the relation (30) with various choices of polynomials. There are no boundary generators of the second type. This construction also leads to some boundary functions which take larger values than the inner functions and the condition number of wavelet bases is bad for N > 3, see figures in [16], [17], and [35]. For the proof of Theorem 9 below and also for deriving of refinement matrices we will need the following lemma. Lemma 8. For the left boundary functions of the first type there exist refinement coefficients mn^, k e -/) |[0,1], ^jf1. (61) l=-N-N+2 Then (f^L,l,mon\ q0+1 J ' <62) Table 1 Condition numbers of the matrices Qi N N mon. b = 4 b=10 N N mon. b = 4 b = 10 2 2 6.68e+00 9.94e+00 3.16e+01 4 4 2.46e+04 6.75e+02 1.33e+04 2 4 4.66e+02 1.94e+01 9.48e+02 4 6 1.30e+07 2.94e+04 7.34e+04 2 6 1.40e+05 1.00e+02 4.47e+03 4 8 1.24e+10 6.24e+06 9.42e+04 2 8 1.03e+08 8.52e+03 5.81e+03 4 10 1.92e+13 2.26e+09 5.24e+04 2 10 1.48e+ll 1.67e+06 1.58e+03 5 5 5.34e+06 3.29e+04 1.26e+05 3 3 2.18e+02 1.07e+02 1.00e+03 5 7 5.62e+09 6.91e+06 3.73e+05 3 5 3.73e+04 1.88e+02 1.05e+04 5 9 9.39e+12 2.57e+09 3.47e+05 3 7 1.64e+07 1.20e+04 2.26e+04 6 6 1.20e+09 3.68e+06 6.81e+05 3 9 1.54e+10 2.90e+06 1.33e+04 6 8 2.97e+12 1.92e+09 1.81e+06 11 where the refinement matrix Mmo" = < m™°,n \ ,, ,, is given by 1 J ' J (j=22-\ k = n,ne^1, j=2 ^((■)k+N-\$(--q))hn-2q, ke^\ne^f, (63) , n-N-N+l H~ I 2 0, otherwise. For deriving of Mmo" see [16]. It is known that the coefficients of Bernstein polynomials in a monomial basis are given by f(. = /(-')'-(V)C)*-. <64) I 0, otherwise. Hence, the matrix C = {cin}l7„=NN+1 is an upper triangular matrix with nonzero entries on the diagonal which implies that C is invertible. We denote O^'1 = j 0j:k, k 6 j and we obtain @fl = c@^mon = C (Mmon)T ^{m°n j = C (Mmon)T ^CQ 1 fj j . (65) Therefore, the refinement matrix M = {mn^\nGJ^L,iUJ^3 k€yU\ is given by M = ^CQr ^ Mmo"Cr. (66) We define the dual multiresolution spaces by Sj := span &j. (67) Theorem 9. Under the above assumptions, the following holds i) The sequence 5^ = {Sj} forms a multiresolution analysis ofL2 ([0,1]). ii) The spaces Sj are exact of order N, i.e. %_1([0,1])C5J-, j>j0. (68) Proof To prove /) we have to show the nestedness of the spaces Sj, i.e. Sj C Sj+i- Note that Sj = span j0 where M denotes the closure of the set M in L2 ([0,1]). It is known [26] that for the spaces generated by inner functions Sj:={^^/?} (71) we have _ U^° = L2([0,1]). (72) j>j0 12 Hence, (70) holds independently of the choice of boundary functions. To prove ii) we recall that the scaling function 0 is exact of order N, i.e. 2^+i/2)Jcr= ^aKr2j/2$(2jx-k), i£Ra.«, r = 0, 1, where (73) (74) Ofc,r=((O*,0(-r))-It implies that for r = 0,... ,N — 1, x 6 (0,1), the following holds 2''(r+1/V|(o,i) = if «^/20 - *) |(0,i) + 2 jf+* «^'/20 -k) |(0,i) k=-n-n+2 fc=iV-l v+n-2 + £ ak,r2j/2$(2jx-k) |(o,i). fe=2J-iV-JV+2 By (30), (34), and (69), we immediately have %-i([0,l]) Cspan{0;ik,*e jf'WjWf1} C^. (75) 5 Refinement Matrices Due to the length of the support of the primal scaling functions, the refinement matrix Mj:o corresponding to

j',-i / it could be computed by solving the system Ml / j+l,-n+l\ \ o,-n+i (2N - 3)\ 4>0,-n+2 (0) 4>0,-n+2 (1) • • • 4>0,-n+2 (2N ~ 3) V 0o,-i (0) 00,-1 (1) ... 00,-1 (2tf-3) / (76) (V-N + 2) (77) (78) (79) (80) 13 and /01,-iv+i (0) 01,-^+1(1) ... ■ fa (Ij+i) is called a stable completion of M^o, if 0(1), ;^oo, (86) where Mj := (Mj:0,Mj:i). The idea is to determine first an initial stable completion and then to project it to the desired complement space Wj determined by {Vj}j>j0 - This is summarized in the following theorem [6]. Theorem 11. Let &j and E Hs (R)} , f := sup {s : 0 E Hs (R)} . (112) 18 It is known that y = N — ^. The Sobolev exponent of smoothness y can be computed by the method from [21]. The functions in jo, have the Sobolev regularity at least y, because the primal scaling functions are B-splines and the primal wavelets are finite linear combinations of the primal scaling functions. Similarly, the functions in jo, have the Sobolev regularity at least y. Theorem 14. i) The sets {&j} '■= {^j} j>j0 anc^ {^j} := i^Aj>j0 are uniformly stable, i.e. ^P\\h(^)< E bkfo> < C \\b\\h^ far all b = {bk}k^. e I2 (Jj), j > jo- (113) ii) For all j > jo, the Jackson inequalities hold, i.e. inf ||v-y/||<2 sj ||v||tf^r0 n\ far all v € Hs ([0,1]) ands jo, the Bernstein inequalities hold, i.e. v||^([o,i]) ~2°J Wvj\\ forallvJ £Sjands< y, and ";'ll#s([o,i]) ~2°J \\vj\\ farallvi ESjands < f. (114) (115) (116) (117) Proof i) Due to Lemma 2.1 in [16], the collections {<£/} := {0j}j>jo and {jo are uniformly stable, if j0, (118) and 0j and &j are locally finite, i.e. #{*/eJ^:fl^nflM^0}j0, (119) and # {k1 e J-3 ■ Qhk, n nLk + 0} < 1, for all k e Jb j > jo, (120) where Qj:k := supp (j)j:k and Qj:k := supp j^. By (40) the sets 0. + £ 22sJ J=Jo (121) where v € Hs ([0,1]) ands € (~f,y). The norm equivalence for 5 = 0, Theorem 11, and Lemma 13, imply that CO CO •F := tf* U |J ¥5 ^d V:=0jou\JVj J=J0 J=J0 are biorthogonal Riesz bases of the space L2 ([0,1]). Let us define D = (D (122) (123) The relation (121) implies that D slF is a Riesz basis of the Sobolev space Hs ([0,1]) for s £ (-7,7)- 19 8 Adaptation to Complementary Boundary Conditions In this section, we introduce a construction of well-conditioned spline-wavelet bases on the interval satisfying complementary boundary conditions of the first order. This means that the primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions of the first order, whereas the dual wavelet basis preserves the full degree of polynomial exactness. This construction is based on the spline-wavelet bases constructed above. As already mentioned in Remark 7, in the linear case, i.e. N = 2, our bases are identical to the bases constructed in [24]. The adaptation of these bases to complementary boundary conditions can be found in [24]. Thus, we consider only the case N >3. Let 0j = {(j>j,k,k = —N + 1,..., 2J — 1} be defined as above. Note that the functions 0/,-at+i, tyj,2i-i are me only two functions which do not vanish at zero. Therefore, defining *7mp = { hk, k=-N + 2,...,2j~2} (124) we obtain the primal scaling bases satisfying complementary boundary conditions of the first order. 2 Fig. 6 Primal scaling functions for N = 4 and j = 3 satisfying complementary boundary conditions of the first order. On the dual side, we also need to omit one scaling function at each boundary, because the number of the primal scaling functions must be the same as the number of the dual scaling functions. Let 0j = {Oj^,k 6 J^} be the dual scaling basis on the level j before biorthogonalization from Section 4. There are the boundary functions of two types. Recall that the functions Oj-n+i, ■ ■ 6j _N+ft are the left boundary functions of the first type which are defined to preserve polynomial exactness of the order N. The functions 0- _N+ft+i, ..., Qj n-2 are me left boundary functions of the second type. The right boundary scaling functions are then derived by the reflection of the left boundary functions. Since we want to preserve the full degree of polynomial exactness, we omit one function of the second type at each boundary. Thus, we define fe,-,fc-i, k=-N + 2,...,-N + N + l, 0jTP = \ 0J,k, k=-N + N + 2,...,2j-N-2, (125) k = 2J-N-l,...,2j-2. Since the set ®c.omp •= j 6j°kmp : k = —N + 2,..., 2J— 2 j is not biorthogonal to .k QcompX\ , then J J W / / k,l=-n+2 viewing j,k,j,i))ktlGj?, and Xmin(Gj), Xmax{Gj) denote the smallest and the largest eigenvalue of G;-, respectively. The Riesz bounds of &j, Wj and Wj are computed in a similar way. The condition numbers of the constructed bases are presented in Table 2. To improve it further we provide a diagonal rescaling in the following way: (t>ftk=-j=JM=, 0Ji = 0M*^<0M,0M>, keSj, j>j0, (129) Vf,k= r-^^=, vf,k = Vj,k*\/(Vj,k,Vj,k), ke^j, j>j0. (130) Then the new primal scaling and wavelet bases are normalized with respect to the L2-norm. As already mentioned in Remark 7, the resulting bases for N = 2 are the same as those designed in [24] and [25]. For the quadratic spline-wavelet bases, i.e. N = 3, the condition of our bases is comparable to the condition of the bases from [24] and [25]. In [3], it was shown that for any spline wavelet basis of order ./V on the real line, the condition is bounded below by 2N~1. This result readily carries over to the case of spline wavelet bases on the interval. Now, the constructions from [24], [25] yields the wavelet bases whose Riesz bounds are nearly optimal, i.e. cond f=s 2N~l for N = 2 and N = 3. Unfortunately, the L2-stability gets considerably worse for N > 4. As can 21 be seen in Table 2, the column "f^", the presented construction seems to yield the optimal Instability also for N = A. Note that the cases N = A, N = A and N = 5, N < 9 are not included in Table 2. It was shown in [9] that the corresponding scaling functions and wavelets do not belong to the space L2. Table 2 The condition of single-scale scaling and wavelet bases N N &y xpN ÜlN 2 2 10 2.00 1.73 2.30 1.97 2.00 2.00 2.02 2.00 2 4 10 2.00 1.73 2.09 1.80 2.00 2.00 2.04 2.00 2 6 10 2.00 1.73 2.26 2.03 2.00 2.00 2.30 2.26 2 8 10 2.00 1.73 2.90 2.78 2.34 2.22 3.14 3.81 3 3 10 3.25 2.76 7.58 6.37 4.49 4.00 7.07 4.27 3 5 10 3.25 2.76 3.93 3.49 4.63 4.00 5.55 4.05 3 7 10 3.25 2.76 3.53 3.11 4.55 4.00 5.13 4.01 3 9 10 3.25 2.76 3.75 3.32 4.44 4.00 5.51 4.23 4 6 10 5.18 4.42 10.88 9.07 14.02 8.00 24.36 9.23 4 8 10 5.18 4.42 6.69 5.88 13.96 8.00 16.98 8.20 4 10 10 5.18 4.42 5.83 5.16 13.82 8.00 15.27 8.00 5 9 10 8.32 7.13 29.87 25.23 67.74 27.44 169.76 68.90 5 11 10 8.32 7.13 12.10 11.74 16.00 16.00 45.12 21.65 5 13 10 8.32 7.13 28.49 45.60 16.00 16.00 22.64 22.23 In Table 3 the condition of the multiscale wavelet bases y^0)iS = 104 > 104 54.08 401.23 3004.08 > 104 > 104 The condition of the single-scale bases adapted to complementary boundary condition of the first order are listed in Table 5. We improve the condition of the constructed bases by the L2-normalization. For N = A the condition number of the bases constructed in this paper is again significantly better than the condition of the bases from [24]. 22 The other criteria for the effectiveness of a wavelet basis is the condition number of the corresponding stiffness matrix. Here, let us consider the stiffness matrix for the Poisson equation: \ \ \ / \ /II comp comp \yComp where Xi/Cj°™p = &Cj°mp U \jf=f0 1 Wj°mp denotes the multiscale basis adapted to complementary boundary conditions. It is well-known that the condition number of Aj0:S increases quadratically with the matrix size. To remedy this, we use the diagonal matrix for preconditioning '>,;aa;,d/;'s. d,, = d1ag(((v,;;r)\(cr)/)1/1 • (132) / ,,,«™py,comp To improve further the condition number of A^c we apply the orthogonal transformation to the scaling basis on the coarsest level as in [7] and then we use the diagonal matrix for preconditioning. We denote the obtained matrix by A°^. The condition numbers of the resulting matrices are listed in Table 6. 10 Adaptive wavelet methods In recent years adaptive wavelet methods have been successfully used for solving partial differential as well as integral equations, both linear and nonlinear. It has been shown that these methods converge and that they are asymptotically optimal in the sense that a storage and a number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined. Thus, the computational complexity for all steps of the algorithm is controlled. The effectiveness of adaptive wavelet methods is strongly influenced by the choice of a wavelet basis, in particular by the condition of the basis. In this section, our intention is to compare the quantitative behaviour of the adaptive wavelet method for the cubic spline wavelet bases constructed in this paper and the cubic spline wavelet bases from [24]. Table 4 The condition number of our multiscale wavelet bases and "PF. and multiscale wavelet bases from [9] and [24] JO.5 Jo,5 L 1 L 1 N N jo s mCDF "o,5 XT/Primbs "o,5 wN "o,5 mCDF "o,5 iuPrimbs "o,5 "o,5 3 3 3 5 6.68 6.25 8.50 8.52 8.17 10.48 3 5 4 5 4.36 5.31 5.14 4.37 5.36 5.16 3 7 4 5 4.04 8.57 4.48 4.04 8.63 4.49 3 9 5 5 4.00 25.40 4.65 4.00 25.76 4.66 4 6 4 5 9.89 141.95 12.90 10.43 160.54 13.58 4 8 5 5 8.27 257.41 8.76 8.27 258.56 8.81 4 10 5 5 8.04 917.10 8.13 8.04 935.38 8.14 4 12 5 5 8.01 3971.65 8.44 8.01 3992.29 8.45 5 9 5 5 17.64 > 104 84.81 18.01 > 104 91.27 Table 5 The condition of scaling bases and single-scale wavelet bases satisfying complementary boundary conditions of the first order N JV J 0> £ 10 o E o 10" -e-CF -♦-■Primbs ■•■■CFort 5 10 15 number of iterations Fig. 9 Convergence history for Id example, comparison of our wavelet bases with and without orthogonalization and wavelet bases from [24]. Q -e-CF -+-Primbs \ *%. ■■»■CFort • V» o >*. \ « V "■»• ^••^ 10 10 degrees of freedom ■g 8000 0 6000 1 4000 -e-CF ■-♦-■Primbs ■■■•■■ CFort m »' .¥ • „e~ 5 10 15 20 25 number of iterations -e-CF -+-Primbs ■•■■CFort . '■•■•■•* 5 10 15 20 number of iterations Fig. 10 Convergence history for 2d example, comparison of our wavelet bases with and without orthogonalization and wavelet bases from [24]. 25 10. Cohen, A.; Daubechies, I.; Vial, P.: Wavelets on the Interval and Fast Wavelet Transforms, Appl. Comp. Harm. Anal. 1, (1993), pp. 54-81. 11. Černá, D.: Biorthogonal Wavelets, Ph.D. thesis, Charles University, Prague, 2008. 12. Dahlke, S.; Fornasier, M.; Primbs, M.; Raasch, T.; Werner, M.: Nonlinear and Adaptive Frame Approximation Schemes for Elliptic PDEs: Theory and Numerical Experiments, preprint, Philipps-Universität Marburg, 2007. 13. Dahmen, W.: Stability of Multiscale Transformations, J. Fourier Anal. Appl. 4, (1996), pp. 341-362. 14. Dahmen, W.: Multiscale Analysis, Approximation, and Interpolation Spaces, in: Approximation Theory VIII, (Chui, C.K.; Schu-maker, L.L, eds.), World Scientific Publishing Co., 1995, pp. 47-88. 15. Dahmen, W.; Han, B.; Jia R.Q.; Kunoth A.: Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines, Constr. Approx. 16, (2000), no. 2, pp. 221-259. 16. Dahmen, W.; Kunoth, A.; Urban, K.: Biorthogonal Spline Wavelets on the Interval - Stability and Moment Conditions, Appl. Comp. Harm. Anal. 6, (1999), pp. 132-196. 17. Dahmen, W.; Kunoth, A.; Urban, K.: Wavelets in Numerical Analysis and Their Quantitative Properties, in: Surface fitting and multiresolution methods 2 (Le Méhauté, A.; Rabut, C; Schumaker, L., ed.), 1997, pp. 93-130. 18. Dahmen, W.; Miccheli, C.A.: Banded Matrices with Banded Inverses, II: Locally Finite Decomposition of Spline Spaces, Constr. Appr. 9, (1993), pp. 263-281. 19. Jia, R.Q.: Stable Bases of Spline Wavelets on the Interval, in: Wavelets and Splines, Athens 2005, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2006, pp. 120-135. 20. Jia, R.Q.: Spline Wavelets on the Interval with Homogeneous Boundary Conditions, Adv. Comput. Math. 30, (2009), no. 2, pp. 177-200. 21. Jia, R.Q.; Jiang, Q.T.: Spectral Analysis of the Transition Operator and its Applications to Smoothness Analysis of Wavelets, SIAM J. Matrix Anal. Appl. 24, (2003), no. 4, pp. 1071-1109. 22. Jia, R.Q.; Liu, S.T.: Wavelet Bases of Hermite Cubic Splines on the Interval, Adv. Comput. Math. 25, (2006), no. 1-3, pp. 23-39. 23. Pabel, R.: Wavelet Methods for PDE Constrained Elliptic Control Problems with Dirichlet Boundary Control, thesis, Universität Bonn, 2005. 24. Primbs, M.: Stabile biorthogonale Spline-Waveletbasen auf dem Intervall, dissertation, Universität Duisburg-Essen, 2006. 25. Primbs, M.: New Stable Biorthogonal Spline Wavelets on the Interval, preprint, Universität Duisburg-Essen, 2007. 26. Primbs, M.: Technical Report for the Paper: 'New Stable Biorthogonal Spline Wavelets on the Interval', Universität Duisburg-Essen, 2007. 27. Raasch, T: Adaptive Wavelet and Frame Schemes for Elliptic and Parabolic Equations, Ph.D. thesis, Marburg, 2007. 28. Schneider, A.: Biorthogonal Cubic Hermite Spline Multiwavelets on the Interval with Complementary Boundary Conditions, preprint, Philipps-Universität Marburg, 2007. 29. Schumaker, L.L.: Spline Functions: Basic Theory, Wiley-Interscience, 1981. 30. Stevenson, R.: Adaptive Solution of Operator Equations Using Wavelet Frames, SIAM J. Numer. Anal. 41, (2003), pp. 1074-1100. 31. Stevenson, R.: On the Compressibility of Operators in Wavelet Coordinates, SIAM J. Math. Anal. 35, (2004), no. 5, pp. 1110-1132. 32. Stevenson, R.: Composite Wavelet Bases with Extended Stability and Cancellation Properties, SIAM J. Numer. Anal. 45, (2007), no. 1, pp. 133-162. 33. Grivet Talocia, S.; Tabacco, A.: Wavelets on the Interval with Optimal Localization, in: Math. Models Meth. Appl. Sei. 10, 2000, pp. 441-462. 34. Turcajova, R.: Numerical Condition of Discrete Wavelet Transforms. SIAM J. Matrix Anal. Appl. 18, (1997), pp. 981-999. 35. Urban, K.: Wavelet Methods for Elliptic Partial Differential Equations. Oxford University Press, 2009. 36. Weyrich, N: Spline Wavelets on an Interval, in: Wavelets and Allied Topics, 2001, pp. 117-189. Postprint of the article published in Applied Mathematics and Computation 219(4), 2012, pp. 1853-1865, https://doi.Org/10.1016/j.amc.2012.08.027 Copyright 2012, Elsevier Inc.,https://www.journals.elsevier.com/applied-mathematics-and-computation Cubic spline wavelets with complementary boundary conditions Dana Černáa, Václav Finěka aDepartment of Mathematics and Didactics of Mathematics, Technical University in Liberec, Studentská 2, 4-61 17 Liberec, Czech Republic Abstract We propose a new construction of a stable cubic spline-wavelet basis on the interval satisfying complementary boundary conditions of the second order. It means that the primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions of the second order, while the dual wavelet basis preserves the full degree of polynomial exactness. We present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator. Keywords: wavelet, cubic spline, complementary boundary conditions, homogeneous Dirichlet boundary conditions, condition number 2000 MSC: 46B15, 65N12, 65T60 1. Introduction In recent years wavelets have been successfully used for solving partial differential equations [2, 11, 12, 16, 27] as well as integral equations [22, 24, 25], both linear and nonlinear. Wavelet bases are useful in the numerical treatment of operator equations, because they are stable, enable high order-approximation, functions from Besov spaces have sparse representation in wavelet bases, condition numbers of stiffness matrices are uniformly bounded and matrices representing operators are typically sparse or quasi-sparse. The quantitative properties of wavelet methods strongly depend on the choice of a wavelet basis, in particular on its condition number. Therefore, a construction of a wavelet basis is always an important issue. Wavelet bases on a bounded domain are usually constructed in the following way: Wavelets on the real line are adapted to the interval and then by tensor product technique to the n-dimensional cube. Finally by splitting the domain into overlapping or non-overlapping subdomains which are images of a unit cube under appropriate parametric mappings one can obtain a wavelet basis or a wavelet frame on a fairly general domain. Thus, the properties of the employed wavelet basis on the interval are crucial for the properties of the resulting bases or frames on a general domain. In this paper, we propose a construction of cubic spline wavelet basis on the interval that is adapted to homogeneous Dirichlet boundary conditions of the second order on Email addresses: dana.cerna@tul.cz (Dana Černá), vaclav.finek@tul.cz (Václav Finěk) the primal side and preserves the full degree of polynomial exactness on the dual side. Such boundary conditions are called complementary boundary conditions [18]. We compare properties of wavelet bases such as the condition number of the basis and the condition number of the corresponding stiffness matrix. Finally, quantitative behaviour of adaptive wavelet method for several boundary-adapted cubic spline wavelet bases is studied. First of all, we summarize the desired properties of a constructed basis: - Polymial exactness. Since the primal basis functions are cubic B-splines, the primal multiresolution analysis has polynomial exactness of order four. The dual multiresolution analysis has polynomial exactness of order six. As a consequence, the primal wavelets have six vanishing moments. - Riesz basis property. The functions form a Riesz basis of the space 1? ([0,1]) and if scaled properly they form a Riesz basis of the space Hq ([0,1]). - Locality. The primal and dual basis functions are local, see definition of locality below. Then the corresponding decomposition and reconstruction algorithms are simple and fast. - Biorthogonality. The primal and dual wavelet bases form a biorthogonal pair. - Smoothness. The smoothness of primal and dual wavelet bases is another desired property. It ensures the validity of norm equivalences. - Closed form. The primal scaling functions and wavelets are known in the closed form. It is a desirable property for the fast computation of integrals involving primal scaling functions and wavelets. - Complementary boundary conditions. Our wavelet basis satisfy complementary boundary conditions of the second order. - Well-conditioned bases. Our objective is to construct a well conditioned wavelet basis. Many constructions of cubic spline wavelet or multiwavelet bases on the interval have been proposed in recent years. In [5, 17, 26] cubic spline wavelets on the interval were constructed. In [14] cubic spline multiwavelet bases were designed and they were adapted to complementary boundary conditions of the second order in [28]. In this case dual functions are known and are local. Cubic spline wavelet bases were also constructed in [1, 9, 20, 21]. A construction of cubic spline multiwavelet basis was proposed in [19] and this basis was already used for solving differential equations in [8, 23]. However, in these cases duals are not known or are not local. Locality of duals are important in some methods and theory, let us mention construction of wavelet bases on general domain [18], adaptive wavelet methods especially for nonlinear equations, data analysis, signal and image processing. A general method of adaptation of biorthogonal wavelet bases to complementary boundary conditions was presented in [18], but this method often leads to very badly conditioned bases. This paper is organized as follows: In Section 2 we briefly review the concept of wavelet bases. In Section 3 we propose a construction of primal and dual scaling bases. The refinement matrices are computed in Section 4 and in Section 5 primal and dual wavelets are constructed. Quantitative properties of constructed bases and other known cubic spline wavelet and multiwavelet bases are studied in Section 6. In Section 7 we compare the number of basis functions and the number of iterations needed to resolve 2 the problem with desired accuracy for our bases and bases from [28]. A numerical example is presented for an equation with the biharmonic operator in two dimensions. 2. Wavelet bases This section provides a short introduction to the concept of wavelet bases in Sobolev spaces. We consider the domain Q C IRd and the Sobolev space or its subspace H C Hs (Q) for nonnegative integer s with an inner product (-,-)H, a norm \\-\\H and a seminorm \-\H. In case s = 0 we consider the space L2 (Q) and we denote by (•, •) and ||-|| the L2-inner product and the L2-norm, respectively. Let J be some index set and let each index A G J take the form A = (j, k), where |A| := j G Z is a scale or a level. Let I2 (J):=lv.J^R,Y, vA|2. (1) A family \l/ := A G J} is called a wavelet basis of H, if i) ^ is a Riesz basis for H, i.e. the closure of the span of ^ is H and there exist constants c, C G (0, oo) such that c ||b||Z2(i7) < j,ij = $i,j$k,h for all (i,k)ej, (j, I) G J. (4) This family is also a Riesz basis for H. The basis ^ is called a primal wavelet basis, while is called a efataZ wavelet basis. In many cases, the wavelet system ^ is constructed with the aid of a multiresolution analysis. A sequence V = \Yj}j>j0i °f closed linear subspaces Vj C H is called a multiresolution or multiscale analysis, if V^C Vjo+iC.C^ C^+iC.fl- (5) and Uj>j0V^- is complete in iJ. The nestedness and the closedness of the multiresolution analysis implies the existence of the complement spaces Wj such that Vj+i = Vj © Wj. 3 We now assume that Vj and Wj are spanned by sets of basis functions $j := {(f)j:k, kelj}, Vj:= {tpj:k, kejj}, (6) where Ij and Jj are finite or at most countable index sets. We refer to 0^ as scaling functions and ipj^ as wavelets. The multiscale basis is given by ^j0,s = &j0 u lj£5o_1 ^i and the wavelet basis of H is obtained by \l/ = $j0 u Uj>j0 dual wavelet system \l/ generates a dual multiresolution analysis V with a dual scaling basis $,0. Polynomial exactness of order JVeN for the primal scaling basis and of order JVeN for the dual scaling basis is another desired property of wavelet bases. It means that Pjv-i (n) c y;-, p^jnjc^, j>j0, (7) where Pm (f2) is the space of all algebraic polynomials on Q of degree less or equal to m. By Taylor theorem, the polynomial exactness of order N on the dual side is equivalent to N vanishing wavelet moments on the primal side, i.e. / P (x) 0A (x) dx = 0, Pe P7v_1, G M *y (8) 3. Construction of Scaling Functions We propose a new cubic spline wavelet basis with six vanishing wavelet moments satisfying homogeneous Dirichlet boundary conditions of order two. Six vanishing wavelet moments on the primal side is equivalent to the polynomial exactness of order six on the dual side. We choose polynomial exactness of this order, because the dual scaling function of order four does not belong to 1? (M.) and the polynomial exactness of order greater than six leads to a larger support of primal wavelets which makes the computation more expensive. The first step is the construction of primal scaling functions on the unit interval. Primal scaling basis is formed by cubic B-splines on the knots tJk defined by '\. I'.'- i--=^, *i:=|, k = 1,.. .2j — 1, (9) + 1 _ 1 t> . ■= -_- t3 =t3 ■= 1 23 • 2J+1 ' 23+l 23+2 • The corresponding cubic B-splines are defined by K (x) := 0i+4 - t{) [ti,..., t{+A] t (t - xf+ , x G [0,1] , where (x)+ := max {0, x} and [ti,... t]y]t f is the iV-th divided difference of /. The set $j := {4>jtk, k = —2,..., 2J— 2} of primal scaling functions is simply given by ^!ll>i- k = -2,...,23 -2, j>0. (10) Thus there are 23 — 5 inner scaling functions and 3 boundary functions at each edge. The inner functions are translations and dilations of a function (2x — k), 0 (x) = hk(j) (2x — k) , x G k=0 The parameters hk and hk are called scaling coefficients. In the sequel, we assume that j > j0 := 4. We define inner scaling functions as translations and dilations of 0: 7j,k 2j/24> (2j ■ -k) , k = 5,...,2j -9. (12) There will be two types of basis functions at each boundary. In the following, it will be convenient to abbreviate the boundary and inner index sets by X X L,l 'j ■R,2 {-2,...,3}, Xf = {4}, Zj = {5,...,2*-9} {2J-8}, Xfx = {2J-7,...,2J-2}, (13) and X XK X- xi u xi xi u xi {-2,...,4}, {2*-8,. ..,2*-2}, XL,1 u XL,2 u jO u XÄ,2 u XÄ,1 = |_2) . . . ; 2J' - 2} . (14) Basis functions of the first type are defined to preserve polynomial exactness and the nestedness of multiresolution spaces by the same way as in [17]: (*) = 2'/2 ^(pfc+2,0(•-/))0(2^-/), kex^\ xe[0,l] (15) = -8 5 where {po, ■ ■ ■ ,p&} is a monomial basis of P5 ([0,1]), i.e. Pi{x) = x\ x G [0,1], i = 0,...,5. The definition of basis functions of the second type is a delicate task, because the low condition number and the nestedness of the multiresolution spaces have to be preserved. This means that the relation 9jA G Vj C Vj+i should hold. Therefore we define 9jA as linear combinations of functions that are already in V}+i- To obtain well-conditioned basis, we define a function 9jA which is close to (pf4 := 2?l2<\) (2J • —4), because (f)fA is biorthogonal to the inner primal scaling functions and the condition of |$f4, k G if'2 U X°| is close to the condition of the set of inner dual basis functions. For this reason, we define the basis function of the second type by 9 9jA (x) = 2j/2 V - 8 - /) , x G [0,1], (16) Z=-3 where hi are the scaling coefficients corresponding to the scaling function JA(x) = V'2 (23+lx~8"0 > xe[o,i]. (17) Z=-5 Figure 2 shows the functions #4i4 and 044. 401-,-,-,-,-1 30- OqI-,-,-,-,- 0 0.2 0.4 0.6 0.8 1 Figure 2: The functions f 4 and 64,4. The boundary functions at the right boundary are defined to be symmetric with the left boundary functions: 0j,k (x) = Ojv-i-k (1 " x), x G [0,1] , k G if. (18) It is easy to see that 9j+1,k (x) = V2 9Jjk (2x), x G [0,1] , k G if, (19) for left boundary functions and 8j+lik(l-x) = V28jik(l-2x), xe[0,l], keif, (20) for right boundary functions. 6 Since the set 0^ := k G Ij} is not biorthogonal to we derive a new set $j:= {fe^el,} (21) from Qj by biorthogonalization. Let Qi = «te^»fc,/ei,- (22) We verify numerically that is invertible. Viewing $j and Qj as column vectors we define % := Q,'0,, (23) Then $^ is biorthogonal to because = <$,, QjTQj) = QjQj1 = I#x3, (24) where the symbol # denotes the cardinality of the set and Im denotes the identity matrix of the size m x m. Remark 1. General approach of adapting wavelet bases to the unit interval was proposed in [18]. The idea is to remove certain boundary scaling functions to achieve homogeneous boundary conditions on the primal side. Then it is necessary to have the same number of basis functions on the dual side. Therefore an appropriate number of inner dual functions is used for the definition of boundary dual generators in formula (15). Applying this approach to cubic spline basis constructed in [5] and basis constructed in [26] we obtain the same resulting basis, because these constructions differs in the definition of some functions which are discarded during adaptation to complementary boundary conditions of the second order. Unfortunately, this basis has large condition number, although the starting basis in [5] is well conditioned. Its quantitative properties are presented in Section 6. 4. Refinement matrices From the nestedness and the closedness of multiresolution spaces it follows that there exist refinement matrices M^o and M^i such that = M£0$i+1, , = QjTS3 = QjT (M%f 0,+1 = QjT (M%)T Qj+1$j+1, (31) the refinement matrix M^o corresponding to the dual scaling basis $^ is given by e r\ 1 M,.n = Q; M^,Q; (32) 8 5. Construction of wavelets Our next goal is to determine the corresponding single-scale wavelet bases tyj. It is directly connected to the task of determining an appropriate matrices M^i such that i+i- (33) We follow a general principle called stable completion which was proposed in [3]. This approach was already used in [5, 17, 26]. In our case, however, the initial stable completion can not be found by the same way, because it leads to singular matrices. Definition 1. Any M^i : I2 (Jj) —> I2 (Ij+i) is called a stable completion of M^o, if where Mj := (Mj-0, MjVL). The idea is to determine first an initial stable completion and then to project it to the desired complement space Wj. This is summarized in the following theorem [3]. Theorem 2. Let $j and $j be a primal and a dual scaling basis, respectively. Let M^o and M^o be refinement matrices corresponding to these bases. Suppose that M^i is some stable completion o/M^o and Gj = Mj1. Then M..;.. : (I M;,,NIj,,) M,.. (35) a/so a stable completion and Gj = M •1 /zas £/ze /orm G, Go-1 (36) Moreover, the collections (37) form biorthogonal systems 0. (38) To find the initial stable completion we use a factorization M^o = HjCj, where Hr.= (39) 9 H L -z ( 0.25 0 0.875 1 0.25 6 0 4.8 1 0 0 0 0 1 0\ 0 0 0 0 1.2 0 1.8125 2 0 0 0 1.25 1 \ 0 0 0 0.3125 0 ) Matrix (Hj) has the size (2j+1 - 7) x (2-?+1 - 9). Its elements are given by: H R H t Li 1, 1 < n < 2i+1 - 9, n odd, m = n + 1 2,m-n+2> 1 < n < 2J+1 - 9, n even, -1 < m - n < 3, 0, otherwise, k (40) (41) where h li h1 0.25, /i{2 14 1 ft7 x? u13 1.5, and c R c l' cj== 1 f° 0 0 0 6 0 0 0 0 b 0 ' b V (1 0 0 \ 0 1 8 0 0 0 0 0 b := -. (42) (43) The factorization corresponding to inner and boundary blocks is not the same as the factorization in [15]. Therefore by our approach we obtain new inner and boundary wavelets. We define B, B, / 0 0 b-1 0 0 0 0 0 0 b-1 10 0 0 0 8 0 0 0 0 0 | B R B Li (44) 0 0 o \ 0 0 0 (45) 10 and (46) /O 0\ i o 0 0 ' Fi? The above findings can be summarized as follows. Lemma 3. The following relations hold: ?i - (\ 0 0 0 0 1 : 0 V (47) V 0, FjC, 0. (48) Now we are able to define the initial stable completions of the refinement matrices M 3,0- Lemma 4. Under the above assumptions, the matrices M,.. := H F. j > Jo, are uniformly stable completions of the matrices M^0- Moreover, the inverse (49) (50) ofMj = (M,n.M,..) /- given by G,n = IJH, . G,. = FjIL'. The proof of this lemma is straightforward and similar to the proof in [17]. Then using the initial stable completion Mj,i we are already able to contruct wavelets according to the Theorem 2. Left boundary wavelets are displayed at the Figure 5. 5.1. Decomposition of a scaling basis on a coarse scale In the previous sections we assumed that the supports of the left and right boundary functions do not overlap and therefore the coarsest level was four. It might be too restrictive, especially in higher dimensions, because then there are many scaling functions. Here we decompose scaling basis $4 into two parts $3 and ^3. It also improves the condition number of the basis. We construct wavelets on the level three to have four vanishing moments. Note that wavelets on other levels have six vanishing moments, but there the vanishing moments guaranties the smoothness of dual functions [10], and four vanishing moments for wavelets are sufficient in the most of the applications. Scaling functions in $3 are defined by (10) for j = 3. Functions in ^3 are defined by (Bf ^ (*) W4)l k x e [0,1] (51) 11 4,0 -J 4,4 V IT -J 1: ^1 It Figure 3: Left boundary wavelets for the scale j = 4. ?^fe is a B-spline of order eight on the sequence of knots tk and (4) denotes the fourth derivative. The sequences of knots tk are given by: (52) ti = [0,0,1/32,1/16,1/8,2/8,3/8,4/8,5/8]; t2 = [0,1/32,1/16,1/8,3/16,2/8,3/8,4/8,5/8]; h = [1/32,1/16,1/8,2/8,5/16,3/8,4/8,5/8,6/8]; U = [1/16,1/8,2/8,3/8,7/16,4/8,5/8,6/8,7/8]; h = [1/8,2/8,3/8,4/8,9/16,5/8,6/8,7/8,15/16]; t6 = [2/8,3/8,4/8,5/8,11/16,6/8,7/8,15/16,31/32]; t7 = [3/8,4/8,5/8,6/8,13/16,7/8,15/16,31/32,1]; t8 = [3/8,4/8,5/8,6/8,7/8,15/16,31/32,1,1]; Functions from the set $3 U ^3 generate the same space as functions from i.e. span $3 U ^3 = span $4. Functions ipsk, k = 1,...,8; have four Lemma 5. the set $4; vanishing wavelet moments. Proof. Since $4 is a basis of the space of all cubic splines on the knots t4 = [0, 0,1/32,1/16, 2/16,..., 15/16, 31/32,1,1]. (53) Functions in $3 are cubic splines on the subsets of these knots. Functions in ^3 are also cubic splines, because they are fourth derivative of the spline of order eight, and they are defined on the subsets of knots t4. Therefore $3 U $3 C span $4. Functions in $3 are linearly independent. Function ip^^ cannot be written as linear combination of functions from $3 U $3\ {03,«}, because it is a cubic spline on sequence of the knots ti containing an additional knot. Hence, ^3 U $3 is a linearly independent subset of span $4, which proves the first assertion. To prove that the functions ipstk, k = 1,..., 8, have four vanishing moments, we use the integration by parts. We obtain for n = 0,..., 3: xn(Bip {x)dx X K)(3)(-) nxn-x (5t8J(3) [x)dx. (54) 12 Since (Bf ) ^ is the spline of order 8 — n on the knots of multiplicity at most two in points 0 and 1, we have (5fJ(n)(0) = (5?J(n)(l) and thus 0, n = 0,... 4, {Blp (x)dx = 0 and xn(B*h)W (x)dx n*-1 (Blf} (x)dx, n ,3. Using (55) and the integration by parts three times, we obtain: )<4> (x)dx = (-1)"n! (Blf-n) (1) - {B^ (0) ?8 \ (4-™) for n = 1,..., 3, which proves the assertion. (55) (56) (57) (58) □ Remark 2. In some constructions, the condition number of the wavelet basis is improved by orthogonalization of boundary wavelets or by the orthogonalization of scaling functions on the coarsest level. In our case, the improvement was insignificant. 5.2. Norm equivalences It remains to prove that \l/ and \l/ are Riesz bases for the space L? ([0,1]) and that properly normalized basis ^ is a Riesz basis for Sobolev space Hs ([0,1]) for some s specified below. The proofs are based on the theory developed in [13] and [17]. For a function / defined on the real line a Sobolev exponent of smoothness is defined as sup {s : f G Hs (M.)}. It is known that primal scaling functions extended to the real line by zero have the Sobolev regularity at least 7 = § and that dual scaling functions extended to the real line by zero have the Sobolev regularity at least 7 = 0.344. Theorem 6. i) The sets := {&j}j>jo and j^i j := 30 cIHIz2(x3) < < C II h\k(x3) for all b = {bk}keXj G I2 (I,-), j > j0. (59) ii) For all j > jo, the Jackson inequalities hold, i.e. and inf \\v-Vj\\< 2 33 |Mlffs([o 1]) for all v e Hs ([0,1]) and s < N, inf \\v-Vj\\< 2 33 |Mlffs([o 1]) for all v e Hs ([0,1]) and s < N. Hi) For all j > j0, the Bernstein inequalities hold, i.e. Wvj\\Ha([o 1]) ~ 2SJ ||^'|| for all Vj G Sj and s < 7, and JJ\\H!>([0,1]) < 2s3 \\vj\\ for all Vj G Sj and s < 7. (60) (61) (62) (63) 13 Proof, i) Due to Lemma 2.1 in [17], the collections := {&j}j>j0 and {^i} := j , are uniformly stable, if $j and $j are biorthogonal, < 1, 0i < l, ke lj, j > j0, and and are locally finite, i.e. and # {k' G X, : n njjk ^ 0} < 1, for all fc G X„ j > j0, Cljik/ n flj-fc ^ 0 I < 1, for all k G X„ j > j0, (64) (65) (66) where Qjtk := supp 0^ and Qjjk '■= supp 0^. By (24) the sets $^ and $^ are biorthogonal. The properties (64), (65), and (66) follow from (10), (12), and (19). ii) By Lemma 2.1 in [17], the Jackson inequalities are the consequences of i) and the polynomial exactness of primal and dual multiresolution analyses. iii) The Bernstein inequalities follow from i) and the regularity of basis functions, for details see [17]. □ The following fact follows from [13]. Corollary 1. IL^e have the norm equivalences \v\\2Hs~ 22sj0 \v^jo,k). °jo,k hex. J2 22sj 3=30 Yl (V^3,k)^. 'J3,k (67) where v G Hs ([0,1]) and s G (—7,7). The norm equivalence for s = 0, Theorem 2, and Lemma 4, imply that * ■ 'I',, U |J M>., and M' := -I-,, U Q ^ (68) 3=30 3=30 are biorthogonal Riesz bases of the space L2 ([0,1]). Let us define D=(DAjA)A^, DAjA:=^A2lAl, \,\ej. (69) The relation (67) implies that D sty is a Riesz basis of the Sobolev space Hs ([0,1]) for s G (-7,7)- 6. Quantitative properties of constructed bases In this section, we compare quantitative properties of bases constructed in this paper, cubic spline-wavelet basis from [26] and cubic spline multiwavelet basis recently adapted to homogeneous boundary conditions in [28]. The condition of multi-scale wavelet bases is shown in Table 1. Our wavelet basis is denoted by CF, a basis from 14 [28] is denoted by Schneider and a basis from [26] adapted to complementary boundary conditions by method from [18] is denoted by Primbs. The last basis is the same as the basis from [5] adapted to complementary boundary conditions by method from [18], see Remark 1. Other criteria for the effectiveness of wavelet bases is the condition number of a corresponding stiffness matrix. Here, let us consider the stiffness matrix: A*, = «*?,,■■ (70) It is well-known that the condition number of Aj0)S increases quadratically with the matrix size. To remedy this, we use a diagonal matrix for preconditioning a prec _ -pv-1 a -pv-1 x where D,0,s = diag(<^fc,^fc>1/2) . (72) In [7] the anisotropic wavelet basis were used for solving fourth-order problems. Here, we use isotropic wavelet basis, i.e. we define multiscale wavelet basis on the unit square by *I5 = *l° U (J *f > (73) i=3 where $2° = $3 ® $3) \\>-" . M/;i.vl/; ,..M', (74) The symbol ® denotes the tensor product. The preconditioned stiffness matrix 2DA^esc for the biharmonic equation defined on the unit square is similar to the one dimensional case. Condition numbers of the stiffness matrices are listed in Table 1 and Table 2. The condition number of the stiffness matrix corresponding to wavelet basis by Primbs exceeds 104 already for number of levels j = 3. Wavelet basis from [17] adapted to complementary boundary conditions by method from [18] is very badly conditioned, its quantitative properties can be found in [28]. 7. Numerical example Now, we compare the quantitative behaviour of the adaptive wavelet method with our bases and bases from [28]. Both bases are formed by cubic splines and have local Table 1: The condition numbers of wavelet bases and stiffness matrices, jo = 3 for CF and Schneider, jo = 4 for Primbs. j CF \T/. . ^30,3 Schneider Primbs CF a prec A30,3 Schneider Primbs 1 8.3 1.9 14.9 64.8 472.0 1111.0 3 12.5 2.4 45.9 66.5 569.5 1116.9 5 15.3 2.6 69.8 66.6 640.8 1117.0 7 18.0 2.7 85.8 66.7 693.0 1117.0 15 Reprinted by permission from Springer Nature: Springer Nature, Results in Mathematics 66(3-4), 2014, pp. 525-540, https://doi.org/10.1007/s00025-014-0402-6 Copyright 2014, https://link.springer.com/journal/25 Quadratic spline wavelets with short support for fourth-order problems Dana Černá • Václav Finěk Abstract In the paper, we propose constructions of new quadratic spline-wavelet bases on the interval and the unit square satisfying homogeneous Dirichlet boundary conditions of the second order. The basis functions have small supports and wavelets have one vanishing moment. We show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small. Keywords Wavelet • Quadratic spline • Homogeneous Dirichlet boundary conditions • Condition number • Biharmonic equation Mathematics Subject Classification (2000) 46B15 • 65N12 • 65T60 1 Introduction In this paper, we propose a construction of quadratic spline wavelet bases on the interval that are well-conditioned, adapted to homogeneous Dirichlet boundary conditions of the second order, the wavelets have one vanishing moment and the shortest possible support. The wavelet basis of the space Hq ^(0, l)2^ is then obtained by an isotropic tensor product. Wavelet bases are useful for solving the fourth-order problems. In [11], a construction of cubic spline wavelet basis was proposed and it was shown that the Galerkin method based on this wavelet basis is very efficient even in comparison with multigrid methods. We show that our wavelet basis is even better conditioned than basis in [11]. Moreover, since our wavelets have vanishing moments, they can be used in adaptive wavelet methods. First of all, we summarize the desired properties of a constructed basis: - Riesz basis property. We construct Riesz bases of the space Hq (0,1) and Hq ^(0, l)2^. - Polynomial exactness. Since the primal basis functions are quadratic B-splines, the primal multiresolution analysis has polynomial exactness of order three. - Vanishing moments. The inner wavelets have one vanishing moment, the wavelets near the boundary do not need to have vanishing moments. The authors have been supported by the ESF project " Strengthening international cooperation of the KLIMATEXT research team" No. CZ.1.07/2.3.00/20.0086. D. Černá Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic Tel.: +420-48-535-2407 Fax: +420-48-535-2332 E-mail: dana.cerna@tul.cz V. Finěk Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic 2 Dana Černá, Václav Finěk - Short support. The wavelets have the shortest possible support among quadratic spline wavelets with one vanishing moment. - Locality. The primal basis functions are local. - Closed form. The primal scaling functions and wavelets are known in the closed form. - Homogeneous Dirichlet boundary conditions. Our wavelet bases satisfy homogeneous Dirichlet boundary conditions of second order. - Well-conditioned bases. Our objective is to construct a well conditioned wavelet basis. Moreover, in a comparison with constructions in [2], [8], [12], [13] that are quite long and technical, the construction in this paper is very simple. Many constructions of spline wavelet or multiwavelet bases on the interval have been proposed in recent years. In [1], [2], [8], [12] cubic spline wavelets on the interval were constructed. In [7] cubic spline multiwavelet bases were designed and they were adapted to complementary boundary conditions of second order in [13]. In these cases dual functions are known and are local. Spline wavelet or multiwavelet bases whose duals are not local were constructed in [4], [9], [10], [11]. Some of these bases were already adapted to boundary conditions. The advantage of our construction is the shortest possible support for a given number of required vanishing moments. Vanishing moments are necessary in some applications such as adaptive wavelet methods [5], [6]. Originally, these methods were designed for wavelet bases with local duals. However, it was shown in [14] that wavelet bases without local dual basis can be used if the solved equation is linear. 2 Wavelet bases This section provides a short introduction to the concept of wavelet bases in Sobolev spaces. In this paper, we consider the domain Q = (0,1) or Q = (0, l)2. We consider the Sobolev space or its subspace by H C Hs (Q) for nonnegative integer s and the corresponding inner product by {-,-)H, a norm by \\-\\H and a seminorm by \-\H. In case s = 0 we consider the space L2 (Q) and we denote by (•, •) and ||-|| the L2-inner product and the L2-norm, respectively. Let J be some index set and let each index A 6 J take the form A = (j, k), where A := j 6 Z is a scale or a level. Let for v = {vx}XeJ , vx e 1 and I2 (J) = {v : v = {vx}XeJ ,vx(ER, ||v||2 < oo} . (2) A family & := {vjx, A 6 is called a (primal) wavelet basis of H, if i) & is a Riesz basis for H, i.e. the closure of the span of & is H and there exist constants c, C 6 (0, oo) such that |b||2< E xej ) < C2 'A' for all A 6 J, where Qx is the support of and at a given level j the supports of only finitely many wavelets overlap at any point x 6 Q. By the Riesz representation theorem, to any basis of the space H there exists a unique family & = j^A) A 6 j C H biorthogonal to i.e. H for all (i, k) e J, (j, l) e J, (4) Quadratic spline wavelets with short support for fourth-order problems where 5ij denotes the Kronecker delta, i.e. 5ij = 1 for i = j and 5ij = 0 for i ^ j. This family is a Riesz basis for H if and only if the primal basis is a Riesz basis for H. The functions do not need to be local, therefore SP do not need to be a wavelet basis in the sense of the above definition. The basis SP is called a dual basis. Wavelets are usually constructed using a function ifi called a mother-wavelet by ipjjk = 2j/2ip (2jx - k + n) , n e N. Also the inner wavelets in this paper are constructed by this way. This does not implicate that the dual basis has a mother-wavelet. In many cases, the wavelet system *P is constructed with the aid of a multiresolution analysis. A sequence V = {Vj}j>j0i °f closed linear subspaces Vj C H is called a multiresolution or multiscale analysis, if VjocVjo+1C...CVjcVj+1C...H (5) and Uj>j0Vj is complete in H. The nestedness and the closedness of the multiresolution analysis implies the existence of the complement spaces Wj such that Vj+\ = Vj © Wj. We now assume that Vj and Wj are spanned by sets of basis functions $j = {j,k, kelj}, ^ = #»UU^ (7) 3=30 3>30 Let us denote j,k, k E Xj } , #j- = fc, fc e Jj} , (8) and Vj = span #j, Wj = span ^ The spaces Vj are also nested: VjCVj+1, j>j0. (10) Most common way of construction of wavelet bases is using dual functions. In our paper, we use a different approach and construct scaling functions 0^ as quadratic splines and we derive wavelets directly as linear combinations of functions (j)j+\^, where the coefficients of the linear combinations are chosen such that wavelets have vanishing moments. Polynomial exactness of order N £ N for the primal scaling basis and of order JVeN for the dual scaling basis is another desired property of wavelet bases. It means that TN^{Q)dVj, f^ifycVj, j>jQ, (11) where Pm (Q) is the space of all algebraic polynomials on Q of degree less or equal to m. The polynomial exactness of order N on the dual side is equivalent to N vanishing wavelet moments on the primal side, i.e. / P (x) ipx (x) dx = 0, for any P e f^_v ^eN 30 (9) 4 Dana Černá, Václav Finěk 3 Primal scaling basis A primal scaling basis is generated from function 0. Let 0 be a quadratic B-spline defined on knots [0,1, 2, 3]. It can be written explicitly as: T> x e [0,1], ~x ~\~ 3i£ ~2 5 x £ ^1 j 2j . ~2~ —H 3> ^ [2, 3j. 0, otherwise, (13) The function 0 satisfies a scaling equation [8]: , (j>(2x) 30(2x-l) 30 (2a:-2) 0 (2a; - 3) ^ = ^1 +-4-+ -+ - • ^14) For j > 2 and rr £ [0,1] we set 0j-fc (a:) = 2J'/20(2J'a: - k + 1), fc = 1,... 2j - 2. (15) The graphs of the functions 0^ on the coarsest level j = 2 are displayed in Figure 1. 0 0.5 1 Fig. 1 Primal scaling basis for j = 2. We define a wavelet -0 as ^) = -^(2^-l) + ^0(2a:-2). (16) Then supp-0 = [0.5,2.5] and -0 has one vanishing wavelet moments, i.e. /oo i/)(x)dx = 0. (17) -oo The graph of -0 is shown in Figure 2. We define a boundary wavelet -05 by: 0>6(a:) = a(2x) +6 0(2a: - 1), (18) where a and b are real parameters. Since we want to have wavelets with the shortest possible support for a given number of vanishing moments, we will consider two choices of the parameters: a) a = 2 5 b = ~2 5 b) a = 1, 6 = 0. The properties of these wavelets are summarized in the following lemma. Quadratic spline wavelets with short support for fourth-order problems 5 '0.5 1 1.5 2 2.5 Fig. 2 Wavelet ip. Lemma 1 i) The function ifib(x) defined by (18) with the choice of parameters a) satisfies supp-05 = [0, 2] and /oo ipb(x)dx = 0. (19) -oo ii) The function ifib(x) defined by (18) with the choice of parameters b) satisfies supp-05 = Proof The length of the support of the function ifib is derived from the lengths of the supports of the functions (f)(2x) and (f)(2x — 1). By (13) we have supp (f>(2x) = [0,1.5] and supp (f>(2x - 1) = [0.5, 2] . (20) Since the functions (j)(2x) and (j)(2x — 1) are given in the closed form, the formula (19) can be verified easily. Thus, we can choose boundary wavelet with one vanishing moment and larger support or boundary wavelets with shorter supports but without vanishing moments. If / G Hq (0,1) and / is constant at the interval [0, e], 0 < e < 1, then / has to be zero at [0, e]. The same holds for the interval [1 — e, 1]. Hence / G Hq (0,1) can not be nonzero constant near the boundary and therefore in some applications such as adaptive wavelet methods the vanishing moment does not play the significant role for boundary wavelets. The graphs of boundary wavelets ifib are displayed in Figure 3. All the following lemmas and theorems are valid for both choices of parameters. Fig. 3 Boundary wavelet ipt, for a) and b), respectively. For j > 2 and x G [0,1] we define xj)j,k{x) = 2j/2i;(2jx - k + 2), k = 2,2j - 1, (21) 0i,iW = 2J/2j,k/ 10^1^(0,1) , k = 1,..., V - 2} , (22) *3 = { k = 1, • • •, 2?} . In Section 5 we show that the sets l+s 00 &s = 2 U |J #j (23) 3=2 3=2 are a multiscale wavelet basis and a wavelet basis of the space Hq (0,1), respectively. We use h ® j; to denote the tensor product of functions u and v, i.e. (u 0 u) (2:1,2:2) = ■u (rri) u (2:2). We set = ® I\(j)j,k ® 0j,dff2(fi) , fc, / = 1,..., 2j - 2j = fc ® / fc 0 i>j,i\Hi(n) , fc = 1, • • •, 2J' - 2, Z = 1,... 2J'} G2 = ^ 0jV / ^ jAh2{n) , fc = 1, • • •, 2j, I = 1,... 2j - 2} Gi = where Q = (0, l)2. We show that the sets defined by l+s 00 V™ = f2 U j (G) U G2 U G3) , <^2D = F2 U J (G) U G2 U G3) (24) are a wavelet basis and a multiscale wavelet basis of the space Hq (Q). 4 Refinement matrices From the nestedness and the closedness of multiresolution spaces it follows that refinement matrices M^o and M^i such that By (14), the entries of the refinement matrix M^o satisfy: n = 1,..., 2' - 2, 1 < m + 2 - 2n < 4, otherwise, where h= [^1,^2,^3,^4] (Mj-,0 13 3 1 _4' 4' 4' 4_ is a vector of coefficients from scaling equation (14). It follows from the equations (16) and (18) that the matrix M^i is of the size 23 and has the structure T there exist (25) (26) (27) 2J+1 - 2) fa b 0 0 0 0 M 1 71 0-ii 0 00... 00 0 0 0 -iiO 0 0\ 0 0 0 0 0...0 0 0 0 -± ± 0 0...0 0 00 0 b a (2Í Quadratic spline wavelets with short support for fourth-order problems 7 There also exist refinement matrices M^o and M^i corresponding to dual spaces that satisfy: ^■ = Mjo^+i. *j = ti%A+i- (29) The structure of the matrix M^o is derived in the proof of Lemma 2. We do not need to know the structure of the matrix M^i in this paper. The Euclidean norm of a vector v is denoted by ||v||2 and the spectral norm of the matrix M is denoted as ||M||2. The following lemma is crucial for the proof of a Riesz basis property. Lemma 2 The norm of the matrix M,- o satisfies M ■3,0 < 2P,p hi 6 hi 4' Proof We prove the lemma for the choice a) of parameters for the boundary wavelet, for the choice b) the proof is similar. We denote the entries of the matrix M^o as Mk,i, k = 1,.. .2i+1 - 2, I = 1,.. .,2i - 2. Due to the biorthogonality of the sets ^ U 2J-6 ' (7-a)2 (-3-2a/2)^ where the constants an and j3n are given by (37) (38) (39) (40) (41) (42) OLr. (7-a)2 (-3-2a/2) 2n-6 (43) Quadratic spline wavelets with short support for fourth-order problems 9 and (7-a)2 (-3-2^/2) n—5 ' (44) 3 3 Note that an f=s 1 and j3n s=s 0. Since the matrices Cj and Dj are invertible, we can define Bj = A71 = D^C:1 Substituting this into (31) we obtain the entries of the matrix Mj,o: d\ dr- M 1,1 -3-2V2)11-11 (-3-2V2)1 Mipi-i =M2:2i-i = M323-i = M2j+i_4::i = M2j+i_3j = M2j+i_2j, and for k = 1,..., 2' - 2, I = 1,..., 2' - 2, we have 1 dfc , dn+i-k + \n-l\ ' (45) M2ki = Bkl 3 - 2v^) (-3 - 2V2) (-3 - 2V2) + \i-i\ + ,\n-l\ (46) The entries M.2k-\,i are given by (33). It is well-known that for any matrix M of the size m x n with entries Mk 1: M||2<,/||M||1||M||oo, where m n ||M||1 = max V|MM|, HM^ = max V|MM|. l=l,...,n*—' k=l,...,m£—' k=l 1=1 In our case, from (45), (48), and a formula for a sum of a geometric sequence we obtain: (47) (48) M 3,0 M 3,0 Thus M ■3,0 < 3a/2 and 2, k = 1,... 2'} (52) is a Riesz basis of i7g (0,1). First we define suitable projections Pj from Vj+\ onto and show that these projections satisfies (51). Then we show that 0/ which differs from (52) only by scaling is also a Riesz basis of Hq (0,1). We denote lj = {1,2,..., 2* -2} and Jj = {l, 2,..., 2^'} (53) and for j > 2 we define ^ = *}fcez, U and F, = (r,,^) . (54) Let a set 10 Dana Černá, Václav Finěk be given by Since obviously p. — f p. J Fj ) — Ij , (56) (57) functions from tj are duals to functions from Fj in the space Vj+\. Since f ■ 1 is not a sparse matrix, these duals are not local. We define a projection Pj from Vj+\ onto Vj by pif=Y,(f>hk)i,k. (58) Lemma 3 Let f £ Vj+Í, aJk = (f, (j)j,k), &j lis .11 < TP n — isJ ajll2 — Z ' P — ln4- BI fcex,- , j > 2, and S,- : a,-+i h-> a,-. T/ien JJll2 Proof We have (59) E E al+1 \j+l,hhk)j,k- keXj leXj+i Therefore 4*3 + 1,1/ 1 aj Let us denote then we can write and By Lemma 2 the assertion is proved. Lemma 4 ^4 projection Pj satisfies \\PmPm+l...Pn-l\\(n-m\ p-- leXj+\,keXj M ■3,0- (60) (61) (62) (63) In 6 Íři4' (64) for all 2 < m < n and a constant C independent on m and n. Proof Let fn e Vn and fm = PmPm+i... Pn-ifn- We represent fj by fj = Y.kex, a{4j for j = m,n and we set a.j = jafc j • It is known [1] that {4>j,k, k £ Fj) is a Riesz basis of Vj = spanj and there exist constants C\ and C2 independent of j such that: d \\aj\\2 < fcex,- < ^2 ||aj||2 By Lemma 3 we have for p — — • In 4" II f m I) í; C*2 II 3-m II2 — C2 II Sjtj 12 I) Sm_|_i J] 2 • • • j) Sn_i J] 2 I a^ 12 < C22p{n-m) ||an||2 < Cj1C22p{n-m) ||/n|| . Thus (64) is proved. (65) (66) Quadratic spline wavelets with short support for fourth-order problems 11 Theorem 1 The set {2-402,fc, k = 1, 2} U {2-2^jtk,j > 2, k = 1,... V } (67) is a Riesz basis of H% (0,1) for < // < 2.5 Proof By Lemma 4 and Theorem 5.3. from [11], the set {2-402,fc, fc = 1, 2} U {2-2*i/>j,k,j > 2, fc = 1,... V} (68) is a Riesz basis of the space H% (0,1) for jjif < < v, where y is the Sobolev exponent of smoothness of the basis, i.e. v = 2.5. Theorem 2 The set & is a Riesz basis of Hq (0,1). Proof From (21) there exist nonzero constants C\ and C2 such that Ci22j < \ipj,k\H^n) < C222j', forj > 2, fc = 1,..., 2\ and Ci24 < |02,fc|H2(r?) < forfc = 1,2. Let b = {a2)k, k £ X2} U J > 2, fee J7,| be such that = &h + 5^ blk < oo- fcex2 kejj,j>2 We define 24a2,fc I^2.*Ih§(o,i) , kel2, bjfk 22ib and b = {a2,k, k £ 22} U {bj,k,j > 2, fc £ J,-}. Then b J >2, fc£ ^, |b||2 < Ci < oo. Theorem 1 implies that there exist constants C3 and C4 such that C.3 ||b||2 < ^ a2,fc2-402,fc + ^ /',./■2 -'''•;./, fcex2 kejj,j>2 C4 ||b||2 > 52 °2,fc2 42,k + 5] 6j>2 2j,^'2 ff2(0,l) and similarly C2 < E E «2,fc 02,fc -4>2,k + 5] ^2,fc ^r'(O.l) /,• J ,, -2 l^'.fclff02(0,l) ff2(0,l) °2,fc J2,k -ij.k H20(0,1) (69) (70) (71) (72) (73) (74) (75) kex2 l^2>fck2(o,i) ' kejdj>2 l^'i,k,4>j,i^ = 5i,j5k,i, for all (i,k)ej, (j,/) G J, (5) where 5jj denotes the Kronecker delta, i.e. Sij = 1 for i = j and Sij = 0 for i ^ j. This family is also a Riesz basis for H, but the functions ifjjj need not be local. The basis V is called a c/na/ wavelet basis. In many cases, the wavelet system V is constructed with the aid of a multiresolution analysis. A sequence V = \Yj}j>j0i °f closed linear subspaces Vj C H is called a multiresolution or multiscale analysis, if V^C Vjb+1C...C^-C^-+1C...tf (6) and Uj>j0Vj is complete in iJ. The nestedness and the closedness of the multiresolution analysis implies the existence of the complement spaces Wj such that Vj+i = Vj © Wj. We now assume that Vj and Wj are spanned by sets of basis functions $j := {(f)j,k, k G Z,-} , Vj := {ijjjik, k G Jj} , (7) 3 where Xj and Jj are finite or at most countable index sets. We refer to 0^ as scaling functions and -0^ as wavelets. The multiscale basis and the wavelet basis of H are given by VI'„.- = 'I',, U U V"r V" = <1*- U U Xl'r (8) i=io j>jo The dual wavelet system \l/ generates a dual multiresolution analysis V with a dual scaling basis $,0. Polynomial exactness of order JVeN for the primal scaling basis and of order n G N for the dual scaling basis is another desired property of wavelet bases. It means that Ptv_i (Q) C Vj and P^f_1 (fl) C V}, j > jo? where Pm (f2) is the space of all algebraic polynomials on Q of degree less or equal to m. The polynomial exactness of order n on the dual side is equivalent to n vanishing wavelet moments on the primal side, i.e. / P (x) ip\ (x) dx = 0, for any P G P^, ^eM ^, (9) 3. Primal scaling basis A primal scaling basis is the same as the basis constructed in [4, 16]. This basis is generated from functions 0 and 0{>. Let 0 be a cubic B-spline defined on knots [0,1, 2, 3,4]. It can be written explicitly as: J:k(x) = 23^2(f){23x — k), k = 2,.. .2j — 2, (14) 0,-xOr) = 2j/2b(2jx), 0^_1(x) = 2^206(2J(l-x)). 4 The graphs of the functions b(^x) satisfies suppipb = [0,1.5]. b) The function ipb(x) = 4>b(2x) — 0.75b(^x) — 0.45 b(^x) — 1.35 0(2rr) + O.6 0(2x— 1) satisfies supp^ = [0,2.5], ipb(x)dx = 0, and xipb(x)dx = 0. (20) Proof. The length of the support of the function ipb is derived from the lengths of the supports of functions 0f,(2rr), (j)(2x), and h(23(\-x)). We denote ®i = {^/l^l^(o,i)^ = ^ (22) ^ = { 0^10^1^(0,1)^ = 1>--->2J}. Then the sets l+s ^s = $2 u (J ^ and # = $2 U (J ^ (23) i=2 i=2 are a multiscale wavelet basis and a wavelet basis of the space Hq (0,1), respectively. We use m®d to denote the tensor product of functions u and v, i.e. (u (rri,^) = ■u (rri) i> {X2). We set ^ = G) = G2 = G) = j,l \h20(u) » k, I = 1,. ••,2J- -1) 4>3,l lff02(Sl) ,k = l,... ,2J- 1,/ = l,...2i jl 'lH02(n) ,k = l,... ,2V = l,...2j - 1 i>3, 'Iff2(n) ,k,l = 1,. ••,2J } where f2 = (0,1) . A wavelet basis and a multiscale wavelet basis of the space Hq (Q) are defined as l + S OO ^2D = F2 U |J (G) U G2 U G3) , V2D = F2 U |J (G] U G2 U G3) . (24) i=2 i=2 Remark 1. Wavelet basis of the space H2 (Q) can be constructed in a similar way. We add two boundary functions 05i and 052 that are B-splines on sequences of knots [0, 0, 0, 0,1] and [0, 0, 0,1,2], respectively. Then scaling basis is generated from the functions 052, 06 and 0 as in (14), see also [2], and boundary wavelets are constructed as appropriate linear combinations of 051? 052 and 05 in a similar way as above. 4. Refinement matrices From the nestedness and the closedness of multiresolution spaces it follows that there exist refinement matrices M^o and M^i such that 'h Mj„,..- Mj.., .• (25) In these formulas we view the sets of functions $^ and tyj as column vectors with entries 0^, k = 1,..., 2J — 1, and ipj,k, k = 1,..., 2\ respectively. Due to the length of 7 the support of primal scaling functions, the refinement matrix M^0 has the following structure: where Mj0 is a (2J Mí,o), +i M 3,0 \ MR 3) x (2J — 3) matrix given by 0, otherwise n = 1,... ,23 - 3, 0 < m (26) 2n < 4, (27) where [h0,h1,h2,h3,h4 113 11 8' 2'4' 2' 8 (28) is a vector of coefficients from scaling equation (11). We denote a vector of coefficients from scaling equation (13) by h6= [/^UUa] 1 11 1 1 4'16'2'8 (29) Then M^ = -^h^ and the matrix is obtained from a matrix M^ by reversing the ordering of rows. It follows from the equations (15) and (17) that the matrix M^i is of the size (2J+1 - 1) x 2J and has the structure / 1 m n 0 M,-,i v2 0 0 ... 0 \ T 0 -i 1 2 0 0 0 ■\ o 0 4 i -\ 0 0 V 0 0 0 0 _I i _i o 2 x 2 u 0 n m 1 (30) There also exist refinement matrices M^o and M^i corresponding to dual spaces that satisfy: (31) where the sets $^ and are viewed as column vectors. The Euclidean norm of a vector v is denoted by ||v||2 and the spectral norm of the matrix M is denoted as ||M||2. The following lemma is crucial for the proof of a Riesz basis property. Lemma 2. The norm of the matrix M,- o satisfies M 3,0 <2*,p = l + £f. 8 Proof. We prove the lemma for the choice c) of parameters for boundary wavelet, for choices a), b), and d) the proof is similar. We denote the entries of the matrix Mj]0 as M3k'?, k = l,...y+1 1, / = 1,...,2-? - 1. Due to biorthogonality of the sets U and U we have (32) and M^Mj-o = 0 3i (33) where Ij denotes the identity matrix and Oj denotes the zero matrix of the appropriate size. From (30) and (33) we have M'f = 0.45 M3», M^°+1_11 = 0.45 M*°+1_ ■2,Z> (34) and for A; odd, k = 3,..., 23 +i 3. (35) We substitute these relations into (32) and we obtain a new system of equations Ij, where / 21 / 20 A, 1 7! 21 3 20 8 3 5 8 4 o I 0 V o 3 8 0 0 \ 0 3 8 21 / 20 ' (36) and B, contains MJf, for k even, i.e. the entries B{, of the matrix B, satisfy: B3 m2kp kj = 1,...2J - 1. (37) We factorize the matrix Aj as Aj CjDj, where /I | 0 0 °3~ v2 3 5 8 4 0 2 u 8 0 0 0 0 o \ 0 3 8 - I 8 / (38) 9 and / 3Z ' 40 40 -1 40-3 0 0 . . 0 0 1 0 0 0 0 1 0 0 1 40(- _3)23-5 1 40(- -3)23-4 1 0 0 1 0 0 0 0 1 0 0 . . 0 0 40(- _3)23-3 1 40(- 1 40(- _3)23-5 -1 \ 40-3 J_ 40 37 40 \ 40(-3)23-3 40 y More precisely, the entries D3k l of the matrix are given by: D3 1,1 D3 UkA D3 D3 D3 -^23-1,23-1 37 40' D3 23+l-/«,23-l (-1)' 40 •3fc-2' 1, for*; = 2, ...,23-2, 0, otherwise. for A; = 2, It is easy to verify that C3- = C •1 has entries: and the matrix D.1 has the structure: / d{ 0 4 i V 4 o |fc-Z| D -l 0 | Mfc l,...,m ^—' (48) (49) k=l 1=1 In our case, from (45), (49), and a formula for a sum of a geometric sequence we obtain: M 3,0 Thus M 3,0 < 3\/2 and M,-0 < 2y/2. In 3 <2V3 = 2P for p=l +-. 2~ In 4 (50) (51) □ The consequence of the proof of Lemma 2 is that the matrix Mj = (M^o, M^i) representing the discrete wavelet transform is invertible. Lemma 3. The matrix Mj = (M^o, M^i) is invertible. Proof. We prove the lemma for the choice c) of parameters for boundary wavelet, for other choices the proof is similar. The matrix Mj is invertible if and only if the matrix Mj = (M^o, M^i) satisfying MjMj = Ly exists and is unique. The existence and uniqueness of the matrix Mjj0 is already shown in the proof of Lemma 2. The entries Ml'} of the matrix M,-1 satisfy for / = 1,..., 2j+1: and M(j = Sltl - 0.45 MQ, Mg+1_hl = Sltl - 0.45 Mg+1 -2,Z> A; odd, k = 3,..., 23 +i 3. (52) (53) Using these relations we obtain a system of equations with the matrix Aj defined by (36). From the proof of Lemma 2 follows that Aj is invertible. Therefore the matrix M,-1 exists and is unique. □ 11 5. Riesz basis on Sobolev spaces For j > 2 we define a column vector ^j,fc-2i-i, k = 2J,... 2J+1 — 1. The symbol (•, •) denotes the standard L2 (f2) inner product. If u and v are two vectors of functions of the length n, then (u, v) denotes matrix with entries (uk,vi), k,l = l,...,n. We set Fj = (Tj, Tj), where tj = F^Tj. We denote Xj = {1,2,...,2i - 1} and J3 = {l, 2,..., V } (55) and the entries of tj as h,k = (tj) , k e lj, i>jtk = (tj) , k e Jj. (56) Since obviously V (57) functions from tj are duals to functions from 1^ in the space Vj+\. Since F •1 is not a sparse matrix, these duals are not local. We define a projection Pj from Vj+i onto Vj by Lemma 4. Let f G V,-+i, a?k = ^f,4>j,kj, slj = {ak}keij> - 2> and SJ : ai+1 ^ ai Then ||SJ-||2<2P,p = l + gf. Proof. We have = E E ai+1 (h+V'fai^fak- Therefore Let us denote Y ai+1 \3+l,h3,k) ■ (6°) lex. ■3 + 1 s3i,k = (hk, 3+i,i), = {sik}leXj+iMX_ (61) then we can write a^ = S^a^i, and = •? = m, n. Since $^ is a Riesz basis of Vj [2, 16], there exist constants C\ and C*2 independent of j such that: Ci ||aj||2 < By Lemma 4 we have for p = 1 fcex,- < C2 Ha ■j 112 • In 3. In 4' 11 fm 11 ^ C*2 11 am 112 — C*2 11 Sm 112 II 112 • • < C22p(n-m) ||an||2 < 9l2p{n-m) ||/„ 'n—1II2 II an II2 Thus (63) is proved. Theorem 6. TTie set ^ is a Riesz basis of Hq (0,1). Proof. By Lemma 5 and Theorem 5.3. from [16], the set {2-2 2, fc = 1,... 2} U {2 2V,,../ > 2, fc = 1,... 2^'} is a Riesz basis of the space Hq (0,1) for 1 + < u < 2.5. Since obviously c22j < |^,fc|Ho2(n) < C22\ for j > 2, fc = 1 the set \l/ defined by (23) is a Riesz basis of the space Hq (0,1 (64) (65) □ (66) 21. (67) □ Theorem 7. TTie set ^ is a ffies^ fraszs of H2 ((0,1)2). Proof. The theorem is a consequence of Lemma 6, (67), and Theorem 5.3. from [16]. □ 6. Quantitative properties of constructed bases In this section, we compare the condition numbers of the stiffness matrices for the biharmonic problem in two dimensions for different wavelet bases. For Q = (0, l)2 we consider the biharmonic equation A2u = f on Q, u du dn 0 on dQ, (68) where A is the Laplace operator and n is the outer unit normal vector. The variational formulation is Au = f, where A = (A\I>2D, A\I>2D), u = uT^2D, and f = (/, \I>2D). It is known that then condA < C < 00. Since As = (A\[>2D, A\[>2D) is a part of the matrix A that is symmetric and positive definite, we have also condAs < C. The 13 s N a) b) c) d) JZ11 CF12 S09 1 49 14.6 13.6 8.5 50.7 34.0 128.1 484.4 2 225 21.4 18.1 14.3 72.3 34.9 141.3 583.4 3 961 23.7 20.2 17.5 84.4 35.1 212.0 626.9 4 3969 24.5 21.4 18.2 91.3 35.3 257.6 653.5 5 16129 24.8 22.2 18.4 95.3 35.5 281.2 673.2 6 65025 25.2 22.6 18.6 98.0 35.8 297.2 689.4 Table 1: The condition numbers of the stiffness matrices As of the size N x N corresponding to multiscale wavelet bases with s levels of wavelets. condition numbers of the stiffness matrices As are shown in Table 1. A construction by Jia and Zhao from [16] is denoted as JZ11, a construction from [4] is denoted as CF12, a construction of multiwavelet basis from [22] is denoted as S09 and wavelet bases constructed in this paper are denoted as a), b), c), and d) according to the choice of parameters for the boundary wavelet. The size of the stifness matrix is N x N for wavelet bases constructed in this paper, it differs for other bases. The condition number for our wavelet bases is comparable to the wavelet basis from [16], but the difference is that wavelets from [16] have not vanishing moments and therefore can not be used in some applications such as adaptive wavelet methods. Wavelet bases from [4, 22] have significantly larger condition number. Remark 2. We can also treat the fourth-order problem subject to nonhomogeneous Dirichlet boundary conditions: Ou A2u = f on Q, u = g on dQ, —— = h on dVL. (69) on Let w G H2 (Q) be a function such that dw w = g on dQ, —— = h on dQ. (70) On Then the solution u of the problem (69) can be computed as u = w + u, where u solves the problem (jii A2u = f - A2w on O, u = 0 on d£l, — = 0 on d£l. (71) On If Q = (0,1), we can simply set w to be a Hermite cubic polynomial: w (x) = Ax3 + Bx2 + Cx + D (72) with A = 2g (0)-h(0)-2g(l) + h(l), B = -3g (0) + 2h (0) + 3g (l)-h (1), C = -h(0), D = g(0). The case Q = (0,1) can be treated in a similar way. Since in formulation (69), the values of normal derivative of u are not well defined at corners, we will consider more 14 precise formulation: dw dw w = g on dQ, — (x,0) = hx (x), — (l,y) = h2 (y), (73) dw dw — (x, 1) = h3(x), — (0,y) = h4(y), x,y e [0,1] . dy ox If w G C2 (Q) then MO) = ^(0,0) = | (0,0), (74) dhi d2w d2w dh4 dx dxdy ' dydx ' dy ' and similarly at other corners. Therefore, we assume that h (0) = || (0, 0), h (1) = |£ (1, 0), h3 (0) = g (0,1), (75) do , , dh\ , , d/u . , d/ii . , dho , , '"W = af-d7<°>=*7(0)- d7<1)= *7(0>- d/i-? . , dh2 , \ dh-x , dh4 , , djW = *7(1)- d7<0)=d^(1)- We first construct a function u1 that satisfies boundary conditions at the part of the boundary {[0,y],y G [0,1]} U {[1, y] , y G [0,1]}. We set m1 (x, y) = A (y) + B (y) x2 + C{y)x + D (y) (76) with A (y) = 2g (0,y) + h4 (y)-2g (l,y) + h2 (y), B (y) = -3g (0,y)-2h4 (y)+3g (l,y)-h2 (y), C (y) =Ji4 (y), D (y) = g (0, y). We define g = g - u1 on dQ, hx (x) = ht (x) -(x,0) and h3 (x) = h% (x) — (x, 1). We construct a function u2 that satisfies u2 = g on dQ and Sif^ ~ <9"u2 ~ du2 du2 — (x,0) = h1(x), —(x,l) = h3(x), _(0,j,) = —(l,j,) = 0. (77) We set w2 (x, y) = A (x) y3 + B (x) y2 + C(x)y + D (x) (78) with A (x)_ = 2g (x, 0) + hi (x) - 2g (x, 1) + h3 (x), B (x) = -3g (x, 0) - 2hi (x) + 3g (x, 1) — h3 (x), C (x) = hi (x), D (x) = g (x, 0). Due to (75) it can be simply verified that w = u1 + u2 satisfies (73). 7. Numerical example In this section, we compare the quantitative behaviour of the adaptive wavelet method with a basis constructed in this paper and bases from [4, 22]. All these bases are formed by cubic splines. We first briefly review an adaptive wavelet method. The method was proposed by Cohen, Dahmen and DeVore in [8]. We use a slightly modified version from [3] with an adaptive matrix-vector multiplication from [5]. 15 For Q = (0, l)2 we consider the fourth-order problem (68). Let \E'2D be a wavelet basis constructed in this paper. As mentioned above, the original equation (68) can be reformulated as an equivalent biinfinite matrix equation Au = f, where A = (A^2D,A^2D), u = uT^2D, and f = (f,^2D). Thus the original problem is equivalent to the well-posed problem in I2. While the classical adaptive methods uses refining and derefining a given mesh according to a-posteriori local error estimates, the wavelet approach is different. Instead of turning to a finite dimensional approximation, we try to devise a convergent iteration for the /2-problem. Then all infinite-dimensional quantities have to be replaced by finitely supported ones and the routine for the application of the biinfinite-dimensional matrix A approximately have to be designed. The simplest convergent iteration for the /2-problem is a Richardson iteration which has the following form: u0 := 0, un+i :=un + w(f-Aun), n = 0,l,.... (79) For the convergence, the relaxation parameter u has to satisfy p : = ||I- uA\\2 < 1, (80) where ||-||2 is a spectral norm. Then the iteration (79) convergences with an error reduction per step ||un+i - u||2 < p ||un - u||2 , (81) where II-IL is the Euclidean norm. Condition (80) is satisfied if 0 < u < t-2— , where Xmax is the largest eigenvalue of A. It is known that the optimal relaxation parameter Cj and the corresponding error reduction can be computed as 2 / " \ Xmax Xmin ^ (A) 1 /r><-»\ u = ~\—n;—' 9^= ~\—n;— = ^ur (82) where \min is the smallest eigenvalue of A. Hence the estimate of the number of iterations needed to resolve the problem with desired accuracy depends on the condition number of the matrix A that can be estimated by Ajmax, where jmax is the maximal level used in the computations. In the algorithm the sparse representation of the vector f is needed. It can be found due to the relation: (f,^,k},) < C 2~'A' for all A G J and at a given level j the supports of only finitely many wavelets overlap at any point x G Q. Remark 2.1. A Riesz basis for H is actually a (Schauder) basis for H. The condition that the closure of the span of is H implies that for any / G H there exists {a\} I1 (J) such that XGJ X a\ip\. XGJ If {b\}XGj G I1 (J) is such that / = X ax^x = X bx^x XGJ XGJ then due to I1 (J) C I2 (J) and (2.3) we have XGJ X axipx ~ X bx^> XGJ XGJ 0. (2.4) (2.5) (2.6) H Hence, a\ = b\ and the expansion (2.4) is unique. For the two countable sets of functions r, V C H , the symbol ^r,ly denotes the matrix (r,f)H:={(7,7>HWef (2-7) Remark 2.2. It is known that the constants and from Definition 2.1 satisfy C* = Y/Amin((^,^)H), = ^\max(.(y,y)H), (2.8) where Xmin ((^, ^)H) and \max ((^, ^)H) are the smallest and the largest eigenvalues of the matrix H, respectively. Let M be a Lebesgue measurable subset of Rd. The space L2 (M) is the space of all Lebesgue measurable functions on M such that the norm 1/2 \f(x)\ dx . M is finite. The space L2 (M) is a Hilbert space with the inner product (f,9) = / f(x)g(x)dx, f,geL2(M) JM (2.9) (2.10) 4 Dana černá and Václav Finěk The Sobolev space Hs (Rd) for s > 0 is defined as the space of all functions / £ L2 (Mr) such that the seminorm (2.11) is finite. The symbol / denotes the Fourier transform of the function / defined by 1(0 = I /•:•<•:•< ';-'sW<) = 7r^r / f(OW)U + \C\2s)dC, f,geH°(Rd), (2.13) (27TJ jrd v 7 and the norm ll/ll.ff3(Rd) = \j (/' f)n'(VLd)- (2-14) For an open set M C Md, -£F (M) is the set of restrictions of functions from Hs (Md) to M equipped with the norm ll/llff>(M) = mf {||s||H.(Kd) : 9 S H8 (m) and ff|M = /j . (2.15) The space H~s (m) is defined as the dual space to Hs (M). Let (M) be the space of all continuous functions with the support in M such that they have continuous derivatives of order r for any r £ R. The space Hq (m) is defined as the closure of (m) in Hs (Rd). It is known that II/IIhi(m) = I/Ihi(m) +11/11. (2-16) where \f\m{M) = \/(V/,V/) (2.17) is the seminorm in H1 (m) and V/ denotes the gradient of /. 3. Construction of scaling functions A primal scaling basis is the same as a scaling basis in Ref. 2, 18. It is generated from functions cj), (f>bi and (f>b2 as follows. Let 0 be a cubic B-spline defined on knots {0,1,2, 3,4}. It can be written explicitly as cf>(a 6 ' + 2a;2 - 2x ■ Ax2 + Wx - (4-x)3 6 ' 0, x £ [0,1], f ,x £ [2,3], x £ [3,4], otherwise. Then

(x) 8 2) 0 (2x - 3) 0 (2x - 4) + 2 + 8 ' (3.1) (3.2) Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions 5 Let (f>bi be a cubic B-spline defined on knots {0, 0,0,1,2} and 062 be a cubic B-spline defined on knots {0,0,1,2, 3}, i.e., 0&i(a;) Zf- - 2f- + 3a;, x G [0,1] 4 o, and 062 (z) 12 llxJ I 3x2 12 2 „2 I 2 (3-x)a 3a;2 6 o, a;G [1,2], otherwise, a;G [0,1], a;G [1,2], zG [2,3], otherwise. Then 06i and 062 satisfy the scaling equations2'18 061 (2a;) 3062 (2a;) 30 (2a;) 061 (x) 062 (x) 2 062 (2a;) 4 110 (2a;) 16 0(2a; - 0(2a 4 16 2 8 For j G N, j > 3 and x G [0,1] we set 0jjfe (a;) = 2j/2(f>(2jx - k), k = 3,... 2j - 1, 0,,! (a;) = 2J/206i(2Ja;), 0,- 2i+1 (a;) = 2^2(f>bl(^(l - x)), 0,-2 (a;) = 2^2>62(2'a;), ^ (*) = 2j/2062(2J(l - a;)). Furthermore, we define $j = {4j,k/Hj,k\\,k= l,...,2j +1} and V}= span .,■ (3.3) (3.4) (3.5) (3.6) (3.7) It was proved in Ref. 2 that the sets j are uniform Riesz bases of the space Vj. This means that the sets j are Riesz bases of the space Vj with Riesz bounds independent on j. The graphs of the functions 0^ on the coarsest level j = 3 are displayed in Figure 1. Fig. 1. Functions 3 k> k = 1,. . . ,9. 6 Dana černá and Václav Finěk 4. Construction of wavelets In some applications such as adaptive wavelet methods,7'8 vanishing moments of wavelets are needed. In our case, we construct wavelets with two vanishing moments, i.e. xkip(x)dx = 0, A: = 0,1. (4.1) supp For k > 3 we set Vj as the space of continuous piecewise linear function: Vj = \ v G C(0,1) : Hfi S Pi \ i ~~7T~~~ I for A; = 0,...,2J' - ll , (4.2) where Pi (a, b) is the space of all algebraic polynomials on (a, b) of degree less than or equal to 1. Clearly, with this choice the dimension of Vj is 2J + 1 that is the same as the dimension of Vj. We construct wavelets ipj,k, k = 1,..., 2J', such that ipj^ G Vj+i fl Vj , where Vj is the orthogonal complement of Vj with respect to the L2-norm. Then ^j,k,4>) = 0 (4.3) for all functions cf> G Vj and (4.1) is satisfied. Since we want ipj^ G Vj+i, we define a generator wavelet ip as M ^{x) = Y,9k^x-k). (4.4) k=0 Then supp V> = [0, 4f + 2]. Let V = G C (o, y + 2^ : «|(M+i) 6 Pi (fc, H 1), /c = 0,..., y + l| . (4.5) The dimension of V is + 3 and ip can be found as the solution of the system M (ip,fi) = 0, i = l,...y + 3, (4.6) — +3 where {/i}i!=1 is a basis of V. For M < 5, the system (4.6) has only trivial solution. Therefore, we choose M = 6 and compute Then for -1 7 -119 -119 7 -1 184'46' 184 ' ' 184 '46'184 (4.7) ^•,fc(a;)=2J'/V(2J'a;-A; + 2)|[0il], k = 3,2J - 2, j G N, j > 3, (4.8) the condition (4.3) is satisfied and the functions ip and ^j & have two vanishing moments. The support of the wavelet ip is [0,5]. The graph of ip is shown in Figure 2. We define boundary wavelets i\)b\ and ip^ by: il>bx{x) = gtffapx) + g?b2(2x) + ^SfcV(2a; - Ä + 2), (4.9) 6 ^2^) = gfM^x) + Si62f 0(2^ - Ä + 2), fe=2 Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions 7 Fig. 2. Wavelets ip, 4>bi and 4>b2- where 939 -393 6233 70 ' 20 ' 560 ' 2770661 256057 -4,1 , -493633 20761777 (4.10) -76369591 ,7,-3 1828560 457140 76992 '1828560 7314240 Then supp^i = [0, 3], supp?/^ = [0,4] and both boundary wavelets have two vanishing moments. For j G N, j > 3 and x G [0,1] we define ^{x)=2*'^bl(2?x), ^(x) =2*'^bl(2*(\ - x)), (4.11) i/jj,2(x) = 2j'2i/,b2(2jx), tl)j,2i-i{x) = 2^2^b2(23(l - x)). and We denote {VW || , A: = 1,..., 2j } , IF,- = span ^. 2+s (4.12) (4.13) $3 U (J i&j and ^ = $3 U (J ^. j=3 j=3 In the following, we prove that ^ is a Riesz basis of the space L2 (0,1). The set is a finite dimensional subset of 5'. Theorem 4.1. TYie seis ^j, j > 3, are uniform Riesz bases ofWj. Proof. We compute the matrix (4.14) using (4.4) and (4.9). For example, for j = 3 we obtain /1.000 0.128 0.103 0.003 0 0 0 0\ 0.128 1.000 0.432 -0.145 -0.014 0 0 0 0.103 0.432 1.000 -0.029 -0.077 0.001 0 0 0.003 -0.145 -0.029 1.000 -0.029 -0.077 -0.014 0 0 -0.014 -0.077 -0.029 1.000 -0.029 -0.145 0.003 0 0 0.001 -0.077 -0.029 1.000 0.432 0.103 0 0 0 -0.014 -0.145 0.432 1.000 0.128 V o 0 0 0 0.003 0.103 0.128 1.000 / (4.15) 8 Dana černá and Václav Finěk where the numbers are rounded to three decimal digits. The matrix Fj for j > 3 has a similar structure. The first two rows and columns and the last two rows and columns corresponds to boundary wavelets and for k, I = 3,.. . 2J' — 2: 1, k = l, -0.029, \k - l\ = 1, (Fj)kj = { -0-077, \k-l\ = 2, (4.16) -0.001, j A; - l\ = 3, 0, otherwise. It is easy to see that Fj is banded and diagonally dominant. Estimates for the smallest eigenvalue XJmin and the largest eigenvalue XJmax of the matrix Fj can be computed using the Gershgorin circle theorem: ymin > min X3max < max Fj I - I Fj ii / j ik fc = l n k=l > 0.2, < 1.8, (4.17) (4.18) (4.19) where F-k are the entries of the matrix Fj. With the help of Remark 2.2 the assertion is proven. □ The proof that ^ is a Riesz basis is based on the following theorem.9'13 Theorem 4.2. Let JgN and let Vj and Vj, j > J, be subspaces of L2 (0,1) such that Vj C Vj+i, Vj C Vj+1, dimVj = dimVj < oo, j > J. (4.20) Let j be uniform Riesz bases ofVj, $j be uniform Riesz bases of Vj, "J/j be uniform Riesz bases of V^ fl Vj+i, where V^ is the orthogonal complement of Vj with respect to the L2-inner product, and let $ = {^,AgJ} = $jU (J 3=j Furthermore, let the matrix ^3 ' ^3 (4.21) (4.22) be invertible and the spectral norm of Gj 1 is bounded independently on j. In addition, for some positive constants C, 7 and d, 7 < d, let Tdm i\ (423) inf \\v - vj j) < C2-id \\v\\Hd(0 n , v S Hg (0,1), VjGVj ' and for 0 < s < 7 let vj\\iid(o,i) - CZ33 \\vj\\ > vj £ Vj, (4.24) and let similar estimates (4-33) and (4-^4) hold for 7 and d on the dual side. Then there exist constants k and K, 0 < k < K < 00, such that k ||b||2 < I>2 -\\U xej <Ä"||b||2, b {bx}XGjel2(J) (4.25) Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions 9 holds for s £ (—7,7). Now we are ready to prove the Riesz basis property (2.3) for ^. Theorem 4.3. The set ^ is a wavelet basis of the space L2 (0,1). Proof. For j > 3 we consider the set = {j,k,k = 1,...,2' +1} that is a Riesz basis of the space Vj. Recall that Vj is defined by (4.2). Let (x + l,xe [-1,0], 4>{x) = I 1 - x, x e [0,1], [ 0, otherwise, and for x £ [0,1] we define 4>j:k (x) = 2j/24> (2jx - k + l) , k = 2,..., 2j, 4>jtk (x) = 2(j+1)/20 (2jx - k + 1) , k = 1, 2j + 1. It was proved in Ref. 2 that j,k,k= l,...,2j +1} are uniform Riesz bases of Vj. (4.26) (4.27) (4.28) (4.29) (4.30) The matrix G,- $j,$j> is G, II 11 11 o o 40 40 80 u u 19 _9_ 17 JJ_ n 120 20 80 120 U J_ 13 11 13 _±_ 60 60 20 60 120 n 1 13 11 13 0 0 0 0 0 0 1 120 60 20 60 120 0 0 0 V 0 0 0 0 0 1 13 11 13 _1_ n 120 ^n nn ^n 1 nn 13 60 1 120 11 13 1 20 60 120 13 11 60 20 11 17 120 80 11 80 13 60 9 60 _ JJL 20 120 I! I! , 40 40 / (4.31) It is easy to verify that the matrix Gj is banded and strictly diagonally dominant. Therefore, it is invertible and the spectral norm of Gj1 is bounded independently on j. It is known9 that when 7 is the Sobolev exponent of smoothness of the basis functions and d is the polynomial exactness of Vj than (4.23) and (4.24) are satisfied. In our case, the Sobolev exponent of smoothness is 7 = 3.5 and the polynomial exactness of Vj is d = 4. On the dual side, 7 = 1.5 and d = 2. Therefore, due to Theorem 4.2, relation (4.25) is satisfied for s £ (—1.5,3.5). Since we proved that (4.25) holds for s = 0, the set ^ is indeed a wavelet basis of the space L2 (0,1). □ It remains to prove that when the wavelet basis ^ is normalized in the i^-seminorm, then it is a wavelet basis of the space Hq (0,1). We denote X3 := {0,1,..., 8} and J, : j 1.....2') . (4.32) 10 Dana černá and Václav Finěk Theorem 4.4. The set {4>3,k/ l3,*|ffi(o,i) >fc eX3} U {%fc/|^j,fc|Hi(o,i) , j > 3,fc g J}} zs a wavelet basis of the space H] (0,1). (4.33) Proof. We follow the Proof of Theorem 2 in Ref. 4. From the proof of Theorem 4.3, we know that relation (4.25) holds for s = 1. Therefore the set {2-3^k,k g I3} U j2 'r,.;...., > 3,fc g ^} (4.34) is a wavelet basis of the space Hr] (0,1). From (3.6), (4.8) and (4.11) there exist nonzero constants C\ and C2 such that Ci2J < hMtfifn) ^ ^ for j > 3, A: g Jj, and Ci23 < | 3, fc g Jj j be such that a-gz3 k<=J.j,j>3 We define a3,fc 23a3ií: /c g X3, 6j;fc 1^3,^1^1(0,1) ^ l^'.fc 1^1(0,1) and b = {a3ifc, k g X3} u {bJik,j > 3, fc g J}}. Then b < 00. j > 3, kej;, (4.35) (4.36) (4.37) (4.38) < Ci < 00. (4.39) Since the set in (4.34) is a Riesz basis of Hr] (0,1) there exist constants C3 and C4 such that C3 ||b|L < Therefore k£l3 kejj,j>3 C4 ||b||2 > (4.41) a3,fc b- feei3 ^3'frl^(o,i) fce^}>3 l^'fcl"ž(o,i) and similarly C2 < £>3,fc Hi(0,l) (0,1) V--^-3fe + V feei3 I^Ihžío.D ' fceji,j>3 1^(0,1) i/ I ^>fe Ho1 (0,1) (4.42) □ Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions 11 It is known2'17 that an orthogonalization of the scaling functions on the coarsest level can lead to improved quantitative properties of the resulting wavelet basis. Therefore, we define the set nrt = {4Tl kel3} (4.43) by ^-K-1^, K=($3,$3>. (4.44) Then the set of scaling functions 3rt is orthonormal and oo is a wavelet basis of the space 1? (0,1) and its appropriate rescaling is a wavelet basis of the space H\ (0,1). 5. Multivariate wavelets We present two well-known constructions of multivariate wavelet bases on the unit hypercube tt = (0,1) .23 They are both based on tensorizing univariate wavelet bases and preserve Riesz basis property, locality of wavelets, vanishing moments and polynomial exactness. 5.1. Anisotropic construction For notational simplicity, we denote ^2,k :=%% k£j2:=l3 (5.1) and J:={(j,k), j >2, kejj}. (5.2) Then we can write ^°rt = fe, 3 > 2, k G Jj} = {^x, A G J} . (5.3) Recall that for A = (j, k) we denote |A| = j. We use u v to denote the tensor product of functions u and v, i.e. (u <8> v) (x\, x2) = u (x\) v {x2). We define multivariate basis functions as: ^x = f=1^xi, A = (Ai,...,Ad) G J, 3 = Jd = J®...®J. (5.4) Since xS>ort is a Riesz basis of L2 (0,1) and xS>ort normalized with respect to iJ1-seminorm is a Riesz basis of Hq (0,1), the set *™:={^,AeJ} (5.5) is a Riesz basis of L2 (fi) and its normalization j—-,AGj) (5.6) is a Riesz basis of Hq (fl). The set ^ni:={^,A = (Ai,...,Ad), |A,|<2 + s} (5.7) is a finite-dimensional approximation of \J)am. 12 Dana černá and Václav Finěk 5.2. Isotropic construction We define for j > 3 and k = (&!,... /c^) multivariate scaling functions: JM, (5-8) and $jfl0 == {^,k, k = (fci,... kd), kt e X,, i = 1,...,d} . (5.9) For e G {0,1} we define We denote the index set: j>'-{j;.l~l: (5-n) For k = (hi,... kd) and e = (ei,..., e^) we define multivariate functions ^j,k,e = ®di=1^j,ki,ei (5.12) and the set of wavelets on the level j as = (Ak,e, fci e J,- ei, ee£}, where E = {0, l}d \ {0} . (5.13) It is known that then the set oo tfifl0 = $ifl° U (J *f (5-14) is a wavelet basis of L2 (fi) and its normalization with respect to the iJ1 (J7)-seminorm is a Riesz basis of Hq (fl). The set 2+s ¥f° = ¥3so U (J $>f° (5.15) is a finite dimensional subset of fylso. 6. Quantitative properties In this section, we present the condition numbers of the stiffness matrices for the following elliptic problem: — eAu + au = f on fl, u = 0ondO., (6-1) where A is the Laplace operator, e and a are positive constants. The variational formulation for an anisotropic wavelet basis is Aaniuani = fani^ (g^ where Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions 13 An advantage of discretization of elliptic equation (6.1) using a wavelet basis is that the system (6.2) can be simply preconditioned by a diagonal preconditioner.11 Let D be a matrix of diagonal elements of the matrix A, i.e. Daj|u = Ax^dx^, where 5\:IM denotes Kronecker delta. Setting \ani\ —1/2 (6.4) /j-jemz\l/2 ^ani /■|-janz\ 1/2 ^ani we obtain the preconditioned system Aarauara = fani. It is known11 that there exist constants C\, C2, and C such that 0 < Ci < and thus condA™ < C < oo. Let 1 by ('HY^l;;:::!.. <7-2) where kj = (2j+2 + if. The method for anisotropic wavelet basis is similar. A criterion ||rj|| < ej, where Tj := AljSOUj — ijSO, is used for terminating iterations of the conjugate gradient (CG) method at level j. It is possible to choose smaller ej on coarser levels,14 but in our case we choose €j constant for all levels, because other choices of ej did not lead to significantly smaller number of iterations in our experiments. Namely, for the given number of levels s we set €j = 10~52~3s, j = 0,..., s, for the isotropic case and €j = 10~42~3s, j = 0,..., s, for the anisotropic case. We explain the choice of ej. Let u be the exact solution of (6.1) and < = (Drr1/2(u:)T*fl, (7.3) where u* is the exact solution of the discrete problem (6.10). It is known23 that IF us\\Hm{n) We have < C2-(4-mK (7.4) vs := A*fl°ufl - f;so = A*fl°ufl - A^0us*. (7.5) where us is the approximate solution of (6.10) by the conjugate gradient method. The relation (6.11) implies 1...... tll 1 Due to Theorem 4.3 we have |rfl||2 < ||úfl - u*||2 < — ||rfl||2 (7.6) C3||ús-ú*||2 < \\us-us\\HHn) < C4 \\us - ú*||2 . (7.7) Hence, by (7.4), (7.6) and (7.7) IK " «lljfi(íí) = 11^ " U*s + U*s - u\\hi(SI) (7-8) < \\us ~ 3+s there exists a matrix T*so such that ^f0 = T*so3+s. This matrix represents the discrete wavelet transform and the multiplication of the matrix T*so with a vector requires O (N) work, where N x N is the size of the matrix T*so. For details see e.g. Ref. 17. Thus Ar = (Dr)_1/2TrAf+3(Tr)T(Dr)_1/2, (7.11) where A*+3 is the stiffness matrix with respect to the basis $3+s. Since A*+3 is banded and D*so is diagonal, the multiplication of the matrix A*so with a vector requires O (N) floating-point operations. We conclude that one CG iteration requires O (N) floatingpoint operations. We denote the number of iterations on the level j as Mj. The number of operations needed to compute one CG iteration on the level j requires about one quarter of operations needed to compute one CG iteration on the level j + 1. Thus Mj iterations at level j is equivalent to Mj 4P~S iterations at level s. Therefore, we define the total number of equivalent iterations by j=0 The results are listed in Table 7 and Table 8. The residuum is denoted rs, u is the exact solution of the given problem and us is an approximate solution obtained by multilevel Galerkin method with s levels of wavelets. It can be seen that the number of conjugate gradient iterations is quite small and that \\us - liHrc _ ||iia - u\\ ^ J_ IK+i -^lloo IK+i - ^11 16' i.e. that order of convergence is 4. It confirms the theory. (7.13) Table 7. Number of iterations and error estimates for multilevel conjugate gradient method for isotropic wavelet basis. s NM ||rJ| ||tí= — íí|I ||ti= — ti|| lit/.* — t/. 11___ Hw* — u _II ^ II_II ^_II 00_II ^_M_II s_II 00_II s_ 1 289 17.00 1.00e-6 1.026-5 2.95e-6 1.026-5 2.95e-6 2 1 089 17.06 1.516-7 6.95e-7 2.49e-7 6.96e-7 2.49e-7 3 4 225 16.75 1.29e-8 4.83e-8 1.61e-8 4.82e-8 1.616-8 4 16 641 15.31 1.786-9 2.87e-9 9.92e-10 2.86e-9 9.92e-10 5 66 049 14.48 1.596-10 1.796-10 6.18e-ll 1.776-10 6.186-11 6 263 169 12.77 3.216-11 1.126-11 3.77e-12 1.10e-ll 3.75e-12 Acknowledgments The authors have been supported by the project "Modern numerical methods II" financed by Technical University in Liberec. The authors would like to thank T. Šimková for her help with numerical experiments. 18 Dana černá and Václav Finěk Table 8. Number of iterations and error estimates for multilevel conjugate gradient method for anisotropic wavelet basis. s NM lir., II \\us — -zx 11___ — till \\u* — u\\^ \\u* — u _II ^ II_II ^_II oo_II ^_M_II s_lipo_II s_ 1 289 9.25 8.156-6 1.03e-5 2.97e-6 1.02e-5 2.95e-6 2 1 089 11.13 1.16e-6 7.106-7 2.49e-7 6.96e-7 2.49e-7 3 4 225 11.42 1.33e-7 4.91e-8 1.62e-8 4.82e-8 1.61e-8 4 16 641 12.05 1.32e-8 2.90e-9 9.93e-10 2.86e-9 9.92e-10 5 66 049 12.14 1.31e-9 1.76e-10 6.20e-ll 1.77e-10 6.18e-ll 6 263 169 11.95 1.32e-10 1.14e-ll 3.78e-12 l.lOe-11 3.75e-12 References 1. J. M. Carnicer, W. Dahmen, and J. M. Peňa, Local decomposition of refinable spaces,Appl. Comput. Harmon. Anal. 3 (1996) 127-153. 2. D. Černá and V. Finěk, Construction of optimally conditioned cubic spline wavelets on the interval, Adv. Comput. Math. 34 (2011) 219-252. 3. D. Černá and V. Finěk, Cubic spline wavelets with complementary boundary conditions, Appl. Math. Comput. 219 (2012) 1853-1865. 4. D. Černá and V. 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Article Sparse wavelet representation of differential operators with piecewise polynomial coefficients Dana Černá and Václav Finěk1 1 Department of Mathematics and Didactics of Mathematics, Technical University in Liberec, Studentská 2, Liberec 46117, Czech Republic; * Correspondence: dana.cerna@tul.cz; Tel.: +420-48-535-2837 Abstract: We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore the wavelets have eight vanishing moments and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving one-dimensional elliptic equation and the two-dimensional Black-Scholes equation with a quadratic volatility. Keywords: Riesz basis; wavelet; spline; interval; differential equation; sparse matrix, Black-Scholes equation 1. Introduction Wavelets are a powerful and useful tool for analysing signals, detection of singularities, data compression and numerical solution of partial differential and integral equations. One of the most important properties of wavelets is that they have vanishing moments. Vanishing wavelet moments ensure so called compression property of wavelets. It means that a function / that is smooth, except at some isolated singularities, typically has a sparse representation in a wavelet basis, i.e. only a small number of wavelet coefficients carry most of the information on /. Similarly as functions also certain differential and integral operators have sparse or quasi-sparse representation in a wavelet basis. This compression property of wavelets leads to design of many multiscale wavelet-based methods for the solution of differential equations. First wavelet methods used orthogonal wavelets, e.g. Daubechies wavelets or coiflets [1,34]. Their disadvantage is that the most orthogonal wavelets are usually not known in a closed form and that their smoothness is typically dependent on the length of the support. The orthogonal wavelets that are known in a closed form are Haar wavelets. They were succesfully used for solving differential equations e.g. in [21,31,32]. Another useful tool is the short Haar wavelet transform that was derived and used for solving differential equations in [3-5]. Since spline wavelets are known in a closed form and they are smoother and have more vanishing moments than orthogonal wavelets of the same length of support, many wavelet methods using spline wavelets were proposed [27,28,30]. For a review of wavelet methods for solving differential equations see also [17,29]. It is known that spectral methods can be used to study singularity formation for PDE solution [2,20,39]. Due to their compression property wavelets can also be used to study singularity formation for PDE solutions. The wavelet approach simply insists in analyzing wavelet coefficients that are large in regions where the singularity occurs and very small in regions where the function is smooth and derivatives are relatively small. Many adaptive wavelet methods are based on this property [13,14]. We focus on an adaptive wavelet method that was originally designed in [13,14] and later modified in many papers [24,25,37], because it has the following advantages: www.mdpi. com/j ournal / axioms Axioms 2 of 21 • Optimality. For a large class of differential equations, both linear and nonlinear, it was shown that this method converges and is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, depend linearly on the number of parameters representing the solution and the number of these parameters is small. Thus, the computational complexity for all steps of the algorithm is controlled. • High order-approximation. The method enables high order approximation. The order of approximation depends on the order of the spline wavelet basis. • Sparsity. The solution and the right-hand side of the equation have sparse representation in a wavelet basis, i.e. they are represented by a small number of numerically significant parameters. In the beginning iterations start for a small vector of parameters and the size of the vector increases successively until the required tolerance is reached. The differential operator is represented by a sparse or quasi-sparse matrix and a procedure for computing the product of this matrix with a finite-length vector with linear complexity is known. • Preconditioning. For a large class of problems the matrices arising from a discretization using wavelet bases can be simply preconditioned by a diagonal preconditioner and the condition numbers of these preconditioned matrices are uniformly bounded. It is important that the preconditioner is simple such as the diagonal preconditioner, because in some implementations only nonzero elements in columns of matrices corresponding to significant coefficients of solutions are stored and used. It should be noted that also other spline wavelet methods utilize some of these features, but up to our knowledge there are not other wavelet methods than adaptive wavelet methods based on ideas from [13,14] that have all these properties. For more details about adaptive wavelet methods see Section 6 and [13,14,19,24,25,37,38]. In this paper, we are concerned with the wavelet discretization of the partial differential equation We assume that qk (x) > Q > 0, the functions pkj, qk, po and / are sufficiently smooth and bounded on O, and that pk / satisfy the uniform ellipticity condition where C > 0 is independent on x. The discretization matrix for wavelet bases is typically not sparse, but only quasi-sparse, i.e. the matrix of the size N x N has O (N x log N) nonzero entries. For multiplication of this matrix with a vector a routine called APPLY have to be used [6,14,24]. However, it was observed in several papers, e.g. in [23] that "quantitatively the application of the APPLY routine is very demanding, where this routine is also not easy to implement". Therefore, in [23] a wavelet basis was constructed with respect to which the discretization matrix is sparse, i.e. it has O (N) nonzero entries, for equation (1) if the coefficients are constant. The construction from [23] was modified in [10,11] with the aim to improve the condition number of the discretization matrices. Some numerical experiments with these bases can be found in [9,15]. In this paper, our aim is to construct a wavelet basis such that the discretization matrix corresponding to (1) is sparse if the coefficients pk /, qk and po are piecewise polynomial functions of degree at most n on the uniform grid, where n = 6 for Pki, n = 5 for qk and n = 4 for pg- Our construction is based on Hermite cubic splines. Let us mention that cubic Hermite wavelets were constructed also in [10,11,18,23,26,33,35,36,40]. (1) d d d EY^PKi(x)xkxi>cYLxl' x = (xi,...,xd), (2) k=l1=1 k=l Axioms 3 of 21 Example 1. We have recently implemented adaptive wavelet method for solving the Black-Scholes equation W ^ Pk,l Q Q dZV £ c W 37 " L -^-a^lSkSi^^ - r £ Sk—+rV = 0, (3) dt ky^t 2 dSkdSi £-[ dSk where (Slr...,Sd,t) E (0,Sfax) x ... x (0,S%ax) x (0,T). We used the 0-scheme for time discretization and tested the performance of the adaptive method with respect to the choice of a wavelet basis for d = 1,2,3. Some results can be found in [9]. In the case of cubic spline wavelets, the smallest number of iteration was required for the wavelet basis from [8]. The discretization matrix for most spline wavelet bases is not sparse, but only quasi-sparse and thus the above mentioned routine APPLY have to be used. For wavelet bases from [10,11,23] the discretization matrix corresponding to the Black-Scholes operator is sparse if volatilities c, are constant. However, in more realistic models, volatilities are represented by non-constant functions, e.g. piecewise polynomial functions [41]. For the basis that will be constructed in this paper the discretization matrix is sparse also for the Black-Scholes equation with volatilities c, that are piecewise quadratic. 2. Wavelet bases In this section, we briefly review the concept of a wavelet basis in Sobolev spaces and introduce notations, for more details see e.g. [38]. Let H be a Hilbert space with the inner product (■, ■}H and the norm | ■ \\H and let (■, ■} denote the L2-inner product. Let J be an index set and let each index A 6 J take the form A = (j,k), where |A| := j 6 Z is a level. For v = {v\}Aej, ^a 6 ^ /we define H2:= ( E K|2>) , Z2(J):={v: ||v||2 < oo}. (4) Vej ) Our aim is to construct a wavelet basis in the sense of the following definition. Definition 1. A family Y := {ip\, A 6 J} is called a wavelet basis of H, if i) T is a Riesz basis for H, i.e. the closure of the span of Y is H and there exist constants c,C E (0, oo) such that c ||b||2 < 2 , H 0 is defined as the space of all functions / £ L2 (m.^ such that the seminorm \f\H><#) = hrriL/tf)2lei28^ do) ■ - v(2ti)^» , is finite. The symbol / denotes the Fourier transform of the function / defined by /(£)= / f(x)e-'^dx. (11) The space Hs [RdJ is a Hilbert space with the inner product II/IIhs(r**) - \J(/'/)ff(e^)- (13) (271) and the norm For an open set M C M.d, Hs (M) is the set of restrictions of functions from Hs (Kd j to M equipped with the norm II/IIh»(m) =inf {IIsIIh^r") :£e hs(m) andg|M = /}. (14) Let (M) be the space of all continuous functions with the support in M such that they have continuous derivatives of order r for any r £ R. The space Hq (M) is defined as the closure of (M) in Hs (Kd) . It is known that II/IIhi(m) = I/Ihi(m) +ll/ll' (15) where _ \f\HHM) = \/(Vf,Vf) (16) is the seminorm in H1 (M) and V/ denotes the gradient of /. 3. Construction of scaling functions We start with the same scaling functions as in [10,11/18,23,26,33,35,36]. Let (x + l)2(l-2x), a: 6 [-1,0], ( (x + l)2x, x£[-l,0], 3 and x G [0,1] we define '/202 (2>> - 1)) , and <$>j={,■. (19) Then the spaces Vj form a multiresolution analysis. We choose dual space Vj as the set of all functions Figure 1. Scaling functions on the level j = 3. v G L2 (0,1) such that v restricted to the interval (j^ir 27=2 ) *s a polynomial of degree less than 8 for any k = 1,..., 2-'~2, i.e. ^■ = {^eL2(o,i):p|^^ eng (^,-£2) forfc = 0.....2^2|, (20) where n§ (a, b) denotes the set of all polynomials on (a, b) of degree less than 8. Let Wj = Vj1- n vj+1, (21) where is the orthogonal complement of Vj with respect to the L2-inner product. If a function g is a piecewise polynomial of degree n we write deg g = n. Lemma 2. Let the spaces Wj, j > 3, are defined as above. Then all functions g G W,- and /z G W^ i,j > 3, |z — j\ > 2, satisfy (ag,h)=0, (bg',h)=0, (cg',h')=0, (22) where a, b, c are piecewise polynomial functions such that a,b,c G Vp, p < max (i, j), deg a < 4, deg b < 5, and deg c < 6. Proof of Lemma 2. Let us assume that j > i + 2. We have g 6 W; c Vj+i C Vj_2 C V,, degg < 3, fl G t/j, deg a < 4, and thus ag G Vj. Since /z G Wj and is orthogonal to Vj we obtain (a g,h) = 0. Axioms 6 of 21 Similarly, the relation (b g',h) = 0 is the consequence of the fact that bg' E Vj and h E Wj. Using integration by parts we obtain {cg',h') = -{d g' + cg",h). (23) Since d g' + c g" E Vj and h E Wj we have (c g'r h') = 0. The situation for j < i + 2 is similar. □ Therefore the discretization matrix for the equation (1) is sparse. Let Yj be a basis of Wj. The proof that CO T = {fcAe J} = ®3u\jYj. (24) 7=3 is a Riesz basis of the space L2 (0,1) and that T is a Riesz basis of the space Hq (0,1) when normalized with respect to the H1-norm is based on the following theorem [16,23]. Theorem 3. Let } E N and let Vj and Vj, j > }, be subspaces ofL2 (0,1) such that Vj c Vj+1, Vj c Vj+1, dim Vj = dim Vj < oo, j> J. (25) Let <&j be bases ofVj, &j be bases ofVj, Yj be bases ofV^ n Vj+\, such that Riesz bounds with respect to the L2-norm of<&j, &j, and Yj are uniformly bounded, and let Y be given by (24). Furthermore, let the matrix Gj := (jrj) (26) be invertible and the spectral norm of G • z's bounded independently on j. In addition, for some positive constants C, 7 and d, such that 7 < d, let mf\\v-Vj\\ j,j) (30) is invertible and the spectral norm ofGj1 is bounded independently on j. Proof of Theorem 4. Let Oy, Vj and Vj be defined as above. For i = 0,..., 7 we define Pi (x) { (31) 0, otherwise, Axioms 7 of 21 and 0j#k+i+i=2{i~2)/2Pi(2h2x-k)> keZ' i = 0,...,7. (32) Then the set @;- = jfy^,k = 1,...,2>+1} is a basis of V,- and the matrix A;- = 0;), / > 3, has the structure: Ai = V A (33) where A is the matrix of the size 10 x 8. Our aim is to apply several transforms on and &j such that new bases y of Vj and y of Vj are local and the matrix Gj defined by (30) and its transpose Gj are both strictly diagonally dominant. First, we replace functions Qj^ by functions gj^ in such a way that the matrix of L2-inner products of (pj^ and gjj is tridiagonal. Therefore we define gj#k+i+l - Yl c'j,l6j,8k+l' 1=1 where the coefficients cl-j are chosen such that i = 0,...,7, k = 0,...,2>- 1, (34) {hvSm) = °' Ip - fl > L #,P> = l p =1.....2>+1 ~ 1- (35) For m = 8k + z + 1, z SUPP 0,..., 7, A: = 1,...,2'-2 -2, we substitute (34) into (35) and using = 0 for Z 7^ 0,..., 9, we obtain systems of 8 linear algebraic equations with 8 unknown coefficents: A'c' = e, for£: = 2/.../2^2 2' c, = WvJI=1 (36) the system matrices A' that are submatrices of A containing all rows of A except z'-th and (z + 2)-th rows, and e' are unit vectors such that (e'); = Sy. The symbol denotes the Kronecker delta. We computed all of the system matrices precisely using symbolic computations and verified that they are regular. Thus the coefficients cl-j exist and are unique. The matrix Bj defined by (B^)fc; = \[gj,v structure k, I = 1,... ,2>+1, is tridiagonal and has the \ (37) Axioms 8 of 21 where 13.199 0 0 0 0 0 0 0 1.000 0.098 0 0 0 0 0 0 -2.185 1.000 -24.781 0 0 0 0 0 0 -0.138 1.000 0.104 0 0 0 0 0 0 13.887 1.000 -6.026 0 0 0 0 0 0 - -0.074 1.000 0.041 0 0 0 0 0 0 34.953 1.000 8.824 0 0 0 0 0 0 -0.018 1.000 0.023 0 0 0 0 0 0 -9.423 1.000 0 0 0 0 0 0 0 -0.092 bL = B{2„ .,10}' bR = B{1„ .,8,10}- (38) and (39) The symbol denotes the submatrix of the matrix B containing rows from B with indices from M. In (38) the numbers are rounded to three decimal digits. We apply several transforms on (pj^ and denote the new functions by hl,l = l,...2i+1| and <3>;- = structure as Ay and By, i.e. l-3g;,2+8fc for k = 0,... ,2^2 and t^y = gjrt for / ^ 2 + 8fc. Let 4>;- = - {+1 j. The matrix Gy defined by (30) has the same +1 -8 + k,2'+1 -8 + /" fc=0,...,8,/=l,.. = G, z' = 2,...,2>'" = GL, = GR, (43) Axioms 9 of 21 where ( 0 -1.6863 0 0 0 0 0 0 1.0000 0.1278 0 0 0 0 0 0 0 -2.8555 0 2.5773 0 0 0 0 0 -0.1790 1.0000 0.1040 0 0 0 0 0 0 0 2.8443 0 0.5229 0 0 0 0 0 -0.0737 1.0000 0.0413 0 0 0 0 0 0 0 -0.7599 0 -0.2011 0 0 0 0 0 -0.0179 1.0000 0.0228 0 0 0 0 0 -0.1689 0 2.4295 0 0 0 0 0 0 0 -0.0920 0 0 0 0 0 0 0 -0.2011 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.3675 0 0 0 0 0 0 0 0 V o 0 0 0 0 0 0 0.3330 (44) and '{2,...,15}' G{1,...,8} .. 0 -0.920 (45) Thus the matrices Gj and G J are diagonally dominant and invertible and due to the Johnson's lower bound for the smallest singular value [22], we have Vmin (Gj) > 0.117, G7 (Gj) < 8.527. (46) It remains to prove that <&j are uniform Riesz bases of Vj and j are uniform Riesz bases of Vj. Since (pjjc are locally supported and there exists M independent of / and k such that <^ have L2(0,1) and similarly for (pjk we have = E E Cj,kCj,l f 4>jjc (x) j and €>y are uniform Riesz bases of their spans. □ 4. Construction of wavelets Now we construct a basis Yj of the space Wj = n Vj+\ such that cond Yj < C, where C is a constant independent on j, and functions from Yj are translations and dilations of some generators. We propose one boundary generator ipb and functions ip',i = 1,..., 8, generating inner wavelets such that the sets Yj= {iPj,k/k=l.....2>+1}, j>3, (49) Axioms 10 of 21 contain functions rpj^ defined for x g [0,1] by ipjA(x) = 2!/2ipb(2!xy iph2j+1(x)=2!/2ipb(2!(l-x)y (50) ,',8fc+/+i (*) = 2'/2ipl \2>x-4k), I = 1,... 8, 1< 8k + I + 1< 2'+1. We denote the scaling functions on the level / = 1 by ~2 — 1, and I = 1,..., 6. Substituting (52) into (53) we obtain the system of linear algebraic equations with the solution h' = {hi^}].^ of the form fl/,lul + fl/,2u2 + fl/,3U3, I = 1,2,3, and where «/ ^ and b\ k are chosen real parameters and [u1,u2,u3] = 29 120 5716 95361 31787 31787 0 0 1 592 13477 56300 95361 95361 95361 0 1 0 13456 95361 39892 95361 49671 31787 1 0 0 0 0 0 11708 4541 26022 4541 116428 4541 13456 95361 39892 95361 49671 31787 1 0 0 592 13477 56300 95361 95361 95361 0 1 0 29 120 5716 95361 31787 31787 0 0 1 [v1,v2,v3] = 4,5,6, / n 17 1727 \ 10500 2625 10500 0 0 -1 13 293 1123 875 2625 2625 0 -1 0 1 5 53 12 21 84 -1 0 0 34 6 386 175 25 525 0 0 0 1 5 53 12 21 84 1 0 0 13 293 1123 875 2625 2625 0 1 0 11 17 1727 10500 2625 10500 V 0 0 1 / (54) (55) (56) Axioms 11 of 21 For I £ {7,8} let the functions xjr have the form 28 ¥ = E hkln := K_103, K = (03,03), (64) and we redefine O3 as O3 := Ogrt. Furthermore we wrote a program that computes the condition number of the wavelet basis containing all wavelets up to the level 7 with respect to the both L2-norm and H1-norm for given Axioms 12 of 21 parameters «£/, b^i and d, and performed extensive numerical experiments. In the following we consider the parameters that lead to good results: ai = («14, «1,2, «i/3 ) = (4.62,4.43,0.67), (65) a2 = (fl2/i, «2,2, «2,3) = (7.196227729728021, -4.658487033189625, -2.279869518963229), a3 = («3,1, «3,2, «3,3) = (-0.775021413514386,0.613425421561151,0.151825757948663), bi = (&i,i, b1/2, bh3) = (0.24,-3.92,4.17), b2 = (^2,1,^2,2,^2,3) = (4.214132381596882,-2.612399654970785,-1.411579368326525), t»3 = (^3,1,^3,2,^3,3) = (-0.601286696663076,-0.778487053796787,-0.180033928710130), d = (di,...,d7) = (-0.075,-0.363,-0.616,-0.134,0.344,0.580,0.099), and after computing ipb and ip',i = 1,..., 8, using these parameters, we normalize them with respect to the L2-norm, i.e. we redefine ipb := ipb / ipb and ip1 := ip1 / \\ip'\\. The wavelets 1/3,1,1/3,9 that are dilations of ipb, ip1,..., ips are displayed in Figure 2. 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 •J 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 2. Wavelets ^31,..., ^3,9. Theorem 5. The sets Yj with the parameters given by (65) are uniform Riesz bases ofWjfor j > 3. Proof of Theorem 5. Since we constructed wavelets such that many of them are orthogonal, there is only small number of nonzero entries in Nj. Since wavelets are normalized with respect to the L2-norm, we have (N,)yc = 1- (66) Axioms 13 of 21 Direct computation yields that (N;)k=l,I=2.....9 = (N;)fc=2,V=2/-l.....2/-8 = Z' <67) (Ni)fc=2.....9,=1 = (N;)» 'fc=2,...,9,/=l \ //it=2)-l,..,2)-8,l=2 where z = (0.0022, -0.0927, -0.0166, -0.0339, -0.0075,0.0045, -0.2652,0.2439), (68) and for i = 1,..., 2>~2 — 2 we have (N;0jt=8r,8r+l,/=8r+8,8r+9 ~~ N ^ (N) = NT \ Uk=8i+8JSi+9,l=8i,8i+l ' where , -0.2048 0.1885 \ N = . (70) -0.1885 0.1734J The numbers in (68) and (70) are rounded to four decimal digits. All other entries of Ny are zero. The structure of the Gram matrix Ny = (Yy,Yy) is displayed in Figure 3. Using Gershgorin theorem the 0 Figure 3. The structure of the matrix Ny. smallest eigenvalue Am,„ (Ny) > 0.21 and the largest eigenvalue Amax (Ny) < 1.79. Therefore Ty are uniform Riesz bases of their spans. □ Theorem 6. The set Y is a Riesz basis ofL2 (0,1) and when normalized with respect to the H^-norm it is a Riesz basis ofH^ (0,1). Proof of Theorem 6. Due to the Theorem 3, Theorem 4 and Theorem 5, the relation (29) holds both for s = 0 and s = 1. Hence, T is a Riesz basis of L2 (0,1) and {2-5cp3rk,k = 1.....16} U {2->>;>; > 3,k = 1.....(71) Axioms 14 of 21 is a Riesz basis of H1, (0,1). To show that also I fefc I fe* II Hi (0,1) -,Jt = 1,...,16 S U *Pj,k -,j > 3,Jt= 1,...,2'+1 h!(0,1) (72) is a Riesz basis of H1, (0,1), we follow the Proof of Theorem 2 in [7]. From (18) and (50) there exist nonzero constants C\ and C2 such that Ct2> < Hi(n) 3, k = l,...,2i+1, and Ci23 < || 3, Jfc = 1,..., 2^+1 J be such that 16 00 2-h = E älk + E E £j* < °°- k=l j=3 k=\ We define a3,k life- 3,fc||fjl(0,l) -, Jt = 1,...,16, = 2>b Hj(0,l) andb = {fl3/fc/ Jfc = 1,...,16} U {b;>j > 3, Jc = 1,... ,2i+1}. Then |b||2 < Ci < 00. Since the set (71) is a Riesz basis of H1, (0,1) there exist constants C3 and C4 such that C3IN2 < 16 00 2i+1 £ au2-5 C4||b||2> 16 E k=l 211 1 E a3,k2~3fo,k + E E h]Xl '>;,* ;=3 fc=l Hi (0,1) 16 ä 00 2/+1 £ E „„.T fe + E E , v »m fc=l II fell Hi (0, (0,1) ;=3fc=l HS (0,1) Hj (0,1) and similarly C2 < 2i+1 \k t=l 11 fee 11H* (0,1) ;=3fc=l ^ Hj (0,1) Hj (0,1) (73) (74) (75) -,;> 3, *=!,...,2>'+1, (76) (77) (78) (79) (80) □ Axioms 15 of 21 The condition number of the resulting wavelet basis with wavelets up to the level 10 with respect to the L2-norm is 17.2 and the condition number of this basis normalized with respect to the H1-norm is 6.0. The sparsity patterns of the matrix arising from a discretization using a wavelet basis constructed in this paper and a wavelet basis from [23] for the one-dimensional Black-Scholes equation with quadratic volatilities from Example 1 is displayed in Figure 4. 0 200 400 600 800 1000 0 200 400 600 800 1000 Figure 4. The sparsity pattern of the matrices arising from a discretization using a wavelet basis constructed in this paper (left) and a wavelet basis from [23] (right) for the Black-Scholes equation with quadratic volatilities. 5. Wavelets on the hypercube We present a well-known construction of a multivariate wavelet basis on the unit hypercube fi = (0,1 f, for more details see e.g. [38]. It is based on tensorizing univariate wavelet bases and preserves Riesz basis property, locality of wavelets, vanishing moments and polynomial exactness. This approach is known as an anisotropic approach. For notational simplicity, we denote Jj = {!,..., 2>+1} for j > 3, and 1>2,k ■= kEj2:=J3, J := {(j,k), ; > 2, E J}} . (81) Then we can write Y = {ipjrh j>2,kejj} = {ipx, a 6 J} . (82) We use u ® v to denote the tensor product of functions u and v, i.e. (u eg) v) (x\,x2) = u(x\)v (^2). We define multivariate basis functions as: Va = ®?=iVv a = (ai.....Ad)ej, j = Jd = J®...® J. (83) Since T is a Riesz basis of L2 (0,1) and T normalized with respect to H1-norm is a Riesz basis of Hi (0,1), the set Yani := {tp\,\ 6 j} (84) is a Riesz basis of L2 (Cl) and its normalization with respect to the H1-norm is a Riesz basis of Hq (fi). Using the same argument as in the Proof of Lemma 2 we conclude that for this basis the discretization matrix is sparse for the equation (1) with piecewise polynomial coefficients on uniform meshes such that deg pjy < 6, deg qk < 5, and deg a0 < 4. 6. Numerical examples In this section, we solve the elliptic equation (1) and the equation with the Black-Scholes operator from Example 1 by an adaptive wavelet method with the basis constructed in this paper. We briefly Axioms 16 of 21 describe the algorithm. While the classical adaptive methods typically uses refining a mesh according to a-posteriori local error estimates, the wavelet approach is different and it comprises the following steps [13,14,17]: 1. One starts with a variational formulation for a suitable wavelet basis but instead of turning to a finite dimensional approximation, the continuous problem is transformed into an infinite-dimensional /2-problem. 2. Then one proposes a convergent iteration for the /2-problem. 3. Finally, one derives an implementable version of this idealized iteration, where all infinite-dimensional quantities are replaced by finitely supported ones. To the left-hand side of the equation (1) we associate the following bilinear form , , f A dv dw JL dv a(v,w):= 2^ Pk,l^-^-+ l^qk^-w+ipQVwdx. (85) "^íc/=l k axl k=\ k The weak formulation of (1) reads as folows: Find u £ fíj (fi) such that a (u, v) = (f, v) for all v e Hq (fi). (86) Instead of turning to a finite dimensional approximation, the equation (86) is reformulated as an equivalent biinfinite matrix equation Au = f, where (A)a^ = «(Va,^), (f)a=(/^a>, (87) for 6 Y, and Y is a wavelet basis of Hq (CI). We use the standard Jacobi diagonal preconditioner D for preconditioning this equation, i.e. D\rJi = 'Oxji^Xji- If the coefficients are constant one can also use an efficient diagonal preconditioner from [12]. The algorithm for solving the /2-problem is the following: 1. Compute sparse representation ij of the right-hand side f such that 11 f — f j 11 is smaller than a given tolerance ej. The computation of a sparse representation insists in thresholding the smallest coefficients and working only with the largest ones. We denote the routine as ij := RHS[i,ej}. 2. Compute K steps of GMRES for solving the system Av = ij with the initial vector \j. Each iteration of GMRES requires multiplication of the infinite-dimensional matrix with a finitely supported vector. Since for the wavelet basis constructed in this paper, the matrix is sparse, it can be computed exactly. Otherwise, it is computed approximately with the given tolerance ej by the method from [6]. We denote the routine z = GMRES [A, íj,Vj, K]. 3. Compute sparse representation Vj+i of z with the error smaller than ej. We denote the routine Vj+i := COARSE[z,e2]. It insists in thresholding the coefficients. We repeat the steps 1., 2., and 3. until the norm of the residual ij = ||f — Avy|| is not smaller than the required tolerance S. Since we work with the sparse representation of the right-hand side and the sparse representation of the vector representing the solution, the method is adaptive. It is known that the coefficients in the wavelet basis are small in regions where the function is smooth and large in regions where the function has some singularity. Therefore, by this method the singularities are automatically detected. Axioms 17 of 21 We use the following algorithm that is modified version of the original algorithm from [13,14]: Algorithm 7. u: =SOLVE [ A, f, e ] 1. Choose /c0, klrk2 6 (0,1), K e N. 2. Set; := 0, vq := 0 and e := |f ||2. 3. While e > e ;:=; + i, e := /cge, ej := /qe, e? := k2e, fj := RHS[f,ej], z := GMRES[A, ij, Vj-i, K] Vj := COARSE[z,ej], Estimate rj = f — Avj and set e := ||2. end while, 4. u := v;-, 5. Compute approximate solution u = E«agu uA1Pa- For an appropriate choice of parameters kg, k\, k2 and K and more details about the routines RHS and COARSE we refer to [13,14,38]. Example 2. We solve the equation -eu" + x2u' + u = f on (0,1), u (0) = u (1) = 0, (88) where e = 0.001 and the right-hand side / is corresponding to the solution u{x) = x(\- e50*-50) for x e [0,1]. (89) We solve this equation using the adaptive wavelet method described above with the wavelet basis constructed in this paper. The approximate solution and the derivative of the approximate solution that were computed using only 79 coefficients are displayed in Figure 5. The significant coefficients were located near the point 1, because the solution has a large derivative near this point. 0.2 0.4 0.6 0.8 1 Figure 5. The approximate solution (left) and the derivative of the approximate solution (right) for Example 1. The sparsity patterns of the matrices arising from discretization of equation (88) using wavelets constructed in this paper and wavelets from [23] are the same as the sparsity patterns of matrices for Example 1 that are displayed in Figure 4. Convergence history is displayed in Figure 6. The number of iterations equals the parameter j from Algorithm 7, the number of basis functions determining the Axioms 18 of 21 approximate solution in j-th iteration is the same as the number of nonzero entries of the vector vy and the L°°-norm of the error is given by \\u ~ "lloo = max \u(x) — u(x)\. (90) *e[o,i] number of iterations number of iterations number of basis functions Figure 6. Convergence history for Example 1. The number of basis functions and the L°°-norm of the error are in logarithmic scaling. Example 3. We consider the equation 2 dSkdSt dt L^lSkSl^i-rYJSk^+rV=f, (91) for (S\,S2) 6 fi := (0, l)2 and t 6 (0,1). We choose parameters of the Black-Scholes operator as Pi,i = 92,2 = 1, Pi,2 = P2,i = 0.88, <7i (x) = O.lx2 - O.lx + 0.66, a2 (x) = O.lx2 - O.lx + 0.97, r = 0.02, and we set the right-hand side /, the initial and boundary conditions such that the solution V is given by V(SlrS2,t) = e-rtS1S2(l-eZ0S^Z0) (l - ezos^zo) (92) for (Si,S2,t) E fi x (0,1). We use the Crank-Nicolson scheme for the semidiscretization of the equation (91) in time. Let M 6 N, t = Mr1, tt = It, I = 0,...,M, and denote V/(Si,S2) = V (Si, S2, ti) and // (Si, S2) = / (Si, S2, t{). The Crank-Nicolson scheme has the form: Vl+i ~ Vl f fa,/ c c d2(Vl+i + Vl) r 2 djVi+t + Vt) rjV^ + Vt) fl+1+fi -z--I-i -rak^km—ag -tg--bk-^7--1--o-= —o—■ (yj) t ^ 4 dSkdSi 2 dSfc 2 2 In this scheme, the function V/ is known from the equation on the previous time level and the function V/+l is an unknown solution. Thus, for the given time level t\ the equation (93) is of the form (1) and we can use the adaptive wavelet method for solving it. The approximate solution V\ for t = 1 / 365 that was computed using 731 coefficients is displayed in Figure 7. It can be seen that the gradient of the solution Vj has largest values near the point [1,1]. Therefore the largest wavelet coefficients correspond to wavelets with supports in regions near this point and wavelet coefficients are small for wavelets that are not located in these regions. Thus many wavelet coefficients are ommited and the representation of the solution is sparse. Convergence history is shown in Figure 8. 7. Conclusions In this paper, we constructed a new cubic spline multiwavelet basis on the unit interval and unit cube. The basis is adapted to homogeneous Dirichlet boundary conditions and wavelets have eight vanishing moments. The main advantage of this basis is that the matrices arising from a discretization Axioms 19 of 21 Figure 8. Convergence history for Example 2. The number of basis functions and the L°°-norm of the error are in logarithmic scaling. of a differential equation (1) with piecewise polynomial coefficients on uniform meshes such that deg /fy z < 6, deg qk < 5, and deg ag < 4, are sparse and not only quasi-sparse. We proved that the constructed basis is indeed a wavelet basis, i.e. Riesz basis property (5) is satisfied. We performed extensive numerical experiments and present the construction that leads to the wavelet basis that is well-conditioned with respect to the L2-norm as well as the H1-norm. Acknowledgments: This work was supported by grant GA16-09541S of the Czech Science Foundation. Author Contributions: The authors contributed equally to this work. Conflicts of Interest: The authors declare no conflict of interest. References 1. Beylkin, G. Wavelets and fast numerical algorithms. 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The pricing of derivatives on assets with quadratic volatility. Appl. Math. Financ. 2001, 8, 235-262. © by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/). Postprint of an article published in Electronic Transactions on Numerical Analysis 48, 2019, pp. 15-39. Copyright 2018, Kent State University Electronic TVansactions on Numerical Analysis. Copyright © 2017, Kent State University. ISSN 1068-9613. ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT SATISFYING HOMOGENEOUS BOUNDARY CONDITIONS * DANA CERNA t AND VACLAV FINEK * Abstract. In the paper, we construct a new quadratic spline-wavelet basis on the interval and on a unit square satisfying homogeneous Dirichlet boundary conditions of the first order. The wavelets have one vanishing moment and the shortest support among quadratic spline wavelets with at least one vanishing moments adapted to the same type of boundary conditions. The stiffness matrices arising from a discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers and the condition numbers are small. We present some quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis require smaller number of iterations than these methods with other quadratic spline wavelet bases. Moreover, due to the short support of the wavelets, one iteration requires smaller number of floating point operations. Key words, wavelet, quadratic spline, homogeneous Dirichlet boundary conditions, condition number, elliptic problem AMS subject classifications. 46B15, 65N12, 65T60 1. Introduction. Wavelets are a powerful tool in signal analysis, image processing, and engineering applications. They are also used for the numerical solution of various types of equations. Wavelet methods are used especially for preconditioning of systems of linear algebraic equations arising from the discretization of elliptic problems [9], adaptive solving of operator equations [6, 7], solving of certain type of partial differential equations with a dimension independent convergence rate [12], and a sparse representation of operators [2]. The quantitative properties of any wavelet method strongly depend on the used wavelet basis, namely on its condition number, the length of the support of wavelets, the number of vanishing wavelet moments and the smoothness of basis functions. Therefore, a construction of appropriate wavelet bases is an important issue. In this paper, we construct a quadratic spline wavelet basis on the interval and on the unit square that is well-conditioned and adapted to homogeneous Dirichlet boundary conditions of the first order. The wavelets have one vanishing moment and we show that the support is the shortest among all quadratic spline wavelets with one vanishing moment. The condition numbers of the stiffness matrices arising from the discretization of elliptic problems using the constructed basis are uniformly bounded and small. Let f2tfWee- Remark 2.2. The constants := sup {c : c satisfies (2.1)} and := inf {C : C satisfies (2.1)} are called Riesz bounds and the number cond 'F = /is called the condition number of 'F. It is known that the constants and satisfy: where \min ((W, and Amaa; ((W, are the smallest and the largest eigenvalues of the matrix ('F, ^) jj, respectively. We define a scaling basis as a basis of quadratic B-splines in the same way as in [22, 5, 20]. Let be a quadratic B-spline defined on knots [0,1,2,3]. It can be written explicitly as: 4, xe[0,l], -x2 + 3x-|, xe[l,2], 0, otherwise. The function satisfies a scaling equation [5] >(2x) 30(2x-l) 30(2x-2) 0 (2x - 3) (2.3) 0(x) 4 4 4 4 Let 0b be a multiple of the quadratic B-spline defined on knots [0, 0,1,2] such that H^bH^i 9a^ " 4 ■3x, ie[0, l], 4 - 3x+ 3, x e [1,2], (2.4) 0b(x) = ^ 3|! 0, otherwise. The function b satisfies a scaling equation [5] 4>b (2x) 9 (2x) 3 (2x - 1) (2.5) 4>b (x) = The graphs of the functions b and are displayed in Figure 2.1. i, i.e. 0 0.5 1 FIG. 2.1. The scaling functions and i, and the wavelets ip and ?/>(,. For j > 2 and x e [0,1] we set (2.6) 4>j,k(x) = 2j/2(p(2jx-k + 2), k = 2,2j — 1, ^-,i(a;) = 2>l*b(Vx), '/206(2'(l - *))■ ETNA Kent State University and Johann Radon Institute (RICAM) 4 D. CERNA AND V. FINEK We define a wavefet tp and a boundary wavefet ipt, as (2.7) ^) = -^(2x-l) + ^(2x-2) and ^b(x) = + tip.. Due to the normafization of (2 • — k) , k G Z} and ip satisfies (2.8), then the length of the support of ip is at least 2. Proof. Since ip G span {

[2x - k) . feez for some coefficients G R. Let us suppose that the fength of the support of ip is at most 2. Then supp-0 C [j/2, (j + 4)/2] for some j G Z. Since ip (x) = 0 for aff x G [k/2, (k + l)/2], where k G Z\ {j, j + 1, j + 2, j + 3}, the coefficients = 0 for aff k G Z\ {j, j + 1}. Due to (2.8) we have Oj + aj+i = 0- Thus up to a muftipfication by a constant and shifting by k/2, k G Z, there is onfy one wavefet that has the fength of support at most 2 and this wavefet is a wavefet defined by (2.7). □ Using the simifar argument as in the proof of Lemma 1 it is easy to see that afso the boundary wavefet tpi, has the shortest possibfe support among aff boundary wavefets with one vanishing moment corresponding to scafing functions defined by (2.6). For j > 2 and x G [0,1] we define (2.9) TpjAx) = 2j/2ip(2jx -k + 2),k = 2, ...,2j - 1, yj^ix) = 2^2^x), ^ (x) = -2^2^(\ - x)). We denote the index sets by Xj = {k G Z : 1 < k < 2j} . We define * j = {i,k ,k^X3}, ^3 = fe, k G Xj } , and oo j'o+s—1 (2.10) * = *2U|J*J-, *fl = *j0U (J *jt 3o=2. J=2 3=30 fn Section 4 we prove that 'F, when normafized with respect to the H1-seminorm, forms a wavefet basis of the Sobofev space Hq (0,1). A basis on 51^ = (0, l)d is buift from the univariate wavefet basis by a tensor product [21]. Let j > 2, k = (fci,..., kd), k G Xj := Xj x ... x Xj, and x = (x\,..., xj) G fid- We define the muftivariate scafing functions by d jM (x)=n^fe' ' i=i ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT and for any e = (ei,..., ej,kn ei = l, The basis on the unit cube f2e eEd,ke lf,j > 2}. This approach is called an isotropic approach. It preserves the regularity and polynomial exactness. Another approach is an anisotropic approach. The anisotropic basis on the unit square is W ,-i- M'vi',1- In these formulas we view the sets $j and Wj as column vectors with entries jtk and tpj^ such that for all j,l > 2,1 > j, k E lj, m E li. Let us denote and view these sets as column vectors. Then C span 2 and the entries M^, k G Ij+i, I G Ij, of the matrix M^o be given by: dl 1-' a|i-'r al«-'l: o 1V12J + 1,l — m2> + 1-l,l — a\n-l\ + ß|l-i| : where n = 2f a = —3 — 2v2, (3.4) 4 = 6a„ 3 +v2' dl = -366 an a 2—n 11 + 6v2 36 \r a 2 n4-2n ll + 6v2 13-9v2 and for k = 2, ... , n — 1 and I G Ij let d{ di M2k,l-M2k-U - a|fe_i|+ a|i_i| + a|„_i| : where (3.5) Then (3.6) j —6bana2 k k~ 3 + v2 366 an a fc+3-2n ll + 6v2 Mj0Mjt0 = Ij, and MLMj,o = Oj w/zere Ij denotes the identity matrix and Oj denotes the zero matrix of the appropriate size. Proof. By similar approach as in [26, 25] we derive the explicit form of the entries M^, k G Ij+i, I G of the matrix M,- o sucn mat (3-6) is satisfied. From (3.3) we obtain (3.7) M2k-u M- 2k,l, for k = 1, ,2>. ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT We substitute (3.7) into (3.6) and we obtain a new system AjBj = L,-, where where 1 vi is 8 1 4 0 1 4 is 8 v2 13 12 1 4 (M) = (i,i), (M) = (2J>") (HA.i=^l. k = l,k^l,k^23, otherwise, i 4 13 12 and B, is the 23 x2J matrix with entries B3k t = M^0,, k,l el,-. We factorize the matrix A,- as A,- = H,C,D, L2k,l where 3+2^ 4 1 4 0 0 0 0 1 4 3+2^2 4 and D, 6 b 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 6 More precisely, the entries D3k t of the matrix Dj are given by: n3 1,1 D3 V2 n3 n3 uk,k n3 uk,l n3 -^n+l-fe.n ,fc-2 ' fork = 2, otherwise. fork - 1, It is easy to verify that Cj = C ■ 1 has entries C3kl = a 'fe 'I, and the matrix D 1 has the structure: d{ d3 d3 an-l di 0 d3n 0 di_, d32 d{ ETNA Kent State University and Johann Radon Institute (RICAM) D. CERNA AND V. FINEK with di given by (3.4) and (3.5). Since the matrices Cj, Dj and Hj are invertible, we can define (3.8) ^ V l)řl(,řl|,řl Substituting (3.8) into (3.7) the lemma is proved. □ LEMMA 3. There exist unique matrices Mji, j > 2, such that (3.9) M.J0Mjtl = Oj, and MLMja = Ij Proof. For I G T3+1 and k G X,- the entries M-jf \ of the matrix Mj,i satisfy Using these relations we obtain a system of equations with the matrix Aj defined in the proof of Lemma 5. Since the matrix Aj is invertible, the matrix Mj,i exists and is unique. □ LEMMA 4. We have = Mj,o*j + Mj^jfor all j > 2. Proof. Due to (3.2) we have from the left-hand side and using (3.6) and (3.9) the HMvIL Mio" MIi_ r Multiplying this equation by the matrix lemma is proved. □ For any matrix M of the size m x n we set M,-,o,M,-i i Ml 2 = sup ^-t v£¥^,v^éO ||v| and IMI max 1=1,...,n IMII It is well-known that (3.10) fe=i |M||2< max k—l,...,m i=i IMIUIMIL. LEMMA 5. The matrices Mj 0, j > 2, have uniformly bounded norms, i.e., there exists Cel independent of j such that M 3,0 < C for all j > 2. Proof. Since the matrices Mjq are known in the explicit form, they have a regular structure and the values in each column and row are exponentially decreasing, we compute upper bounds for the 1-norm and oo-norm such that we compute several largest values in each row and column and estimate the sum of the remaining entries. We obtain and due to (3.10) we have M ■3,0 < 2.04 < 1.42, □ M ■3,0 < 2.91, For comparison we computed the norms of the matrices Mjq numerically and we found that M 3,0 < 2 for j 12, and M12j0 1.9999997. LEMMA 6. Let Sj = Mj0Mj+1 0, j > 3, and Sj be the matrix given by Sj) t = (Si)2fe-i,i + (Si)2fe,i ' k e ^j-i^ e Zj+2- ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT Then there exists a constant C independent of j such that Proof. Let Kj be a 2J x 2J+1 matrix with entries < C < 2v2. (3.11) and let Lj = MT0 — Kj. We know the explicit expression of the matrix Lj, because the explicit expressions of both Mjo and Kj are known. We have S, = Mj0Mj+lj0 = K,K , i + K ,1.,. i + I.řKř., + I..I.. i • Let us denote N,- = KjKj+i, Oj = K,L and Nj, 6j, Pj, and Qj be derived from Nj, Oj, Pj and Qj by similar way as Sj from Sj. Then Sj = Nj 6,+p, ,j -rx j -rQj.From (3.11) we have for k E Xj,l E Ij+i (^,21-1 = 0*^,21=^1, where Ufe = v; = 1 1 111111 1 1 n = 2J. Due to the structure of the vector Ufe we can write a + 1 T -ufev;, where Ufe 1 1 I 1 I a a i—fe „!-2 , „!-4 , ■ ■ ■ , a2 , a, a3 ■ ■ ■ , „2»-l-l J , __I I J_ J_ _1_1T l-2i nl-4 i ■ ■ ■ i a1 1 a2' a4 ' ' ' ' „2ri-z-i I > La1 ^ 7 a / even, I odd. For k > ^,1 E l~j+i, I even, we have (N,)fe,2i = -(E«' I 771 — 1 3m —k—I E 777=^ + 1 i + 1 — fc — 777 E a' i+fc+l-3m 777—fc + 1 a+1 /..i-fel-(^)5 , ,.i-fel-a)' f-2-2fc^ (a3) n—k l - l - l - Similarly for k > / E Ij+i, I odd, we obtain a+1 ,3m — fc—i E a ! + l /+! —fe—m 777—fc+1 / + /C+1-3777 ETNA Kent State University and Johann Radon Institute (RICAM) 10 D. CERNA AND V. FINEK If k < |, I g I even, then we have (NA,2i = — I Eß L 771—1 3m-fc-( Ea m-\-k—l J2 al+k+1- 3777 If < Z g ' odd, then we have (N,)fe,2*— (E i m—l 3m-):-! _|_ ßm+fc+2-i fe+1 E ßi+fe í+fc+l-3m m=í±i+l --1 a2k 1-VV +a 2 2---ha 2 2 1 - 1 - 1 - To compute an upper bound for the norm of the matrix Sj, we compute bounds for the sums of the absolute values of the entries in rows and columns for matrices Nj, Oj, Pj, and Qj. Since the values in the columns of the matrix Nj are exponentially decreasing, we can compute several largest values in each column and estimate the sum of absolute values of the remaining entries. We denote Ij+2 = {1,2,3,4, V+2 - 3, V+2 - 2, V+2 - 1, V+2 } , Ij+2 = T3+2\X3+2 and we set N. k,l = 0, for k £ Tj-i. For I such that I mod 8 g {0,1, 6, 7} and I g ij+2 we obtain E fe=i < V JJk,i N. liJ-i.' N, UJ.' UJ-2 E fe=i N. k,l E fe=L|J+2 N. N, k,l < 0.018 + 0.727 + 0.239 + 0.007 + 0.001 < 1. For I such that I mod 8 g {2, 3, 4, 5} and I g Ij+2 we obtain E fe=i < V JJk,i N, LiK' N. UJ-2 E fe=i k,l E fe=L|J+2 N. k,l LiJ+i.' < 0.101 + 0.566 + 0.037 + 0.002 + 0.004 < 1. For I g Ij+2 we have ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT 11 We use the similar approach for computing the sums of absolute values of the entries in rows. We obtain E fe=i Similarly, we obtain 2J- N. E fe=i o k,l < k,l < 0.73, I e Tj+2, 1.00, I e ij+2 0.13, I e lj+2, 0.04, I e lj+2, E 3+2, l=l E i=i N. k,l < O k,l < 5.95, k = l,2i-6.80, otherwise. 0.30, k=l,2i-0.02, otherwise, E fe=i k,l < 0.15, I e ij+2, 0.05, I e ij+2 E j+2> l=1 k,l < 0.68, /c = l,2J-] 0.04, otherwise, y-1 E fe=i Q: k,l < 0.03, Z e ij+2, 0.01, Z e ij+2 E J+2, i=1 Q.5 k,l < 0.06, /c=l,2J'-1, 0.01, otherwise. Therefore using (3.10) we have < vT.l • 7 < 2V2. For comparison we computed the norms of matrices Sj numerically and we found that < 2.27 for 12, 2.2623 and it seems that this value does not further increase with increasing j. LEMMA 7. Let to, n > 2, to < n, then there exists a constant C < 2 such that M^0M^+lj0...M^0M^+lj0 < C m£,<Ä+1j0 ... MT_lß Proof. For to and n fixed such that to, n > 2, to < n, we use the notation: R = m£,<Ä+i,0 ■ Mi S = MTnML 1n-l,0> ° ~~ lvj-n,0lv-ln+l,0- Due to the structure of the matrices Mj,o given in Lemma 5 we have Rfc,2i = R-fe,2i-l, k £lm,l £ln-i. Therefore, we can write RS = RS, where the matrix R is 2m x 2™_1 matrix containing the even columns of the matrix R, i.e. R^ = Rk,2i, and the matrix S is given by We have 1/2 R sup Rx E E R-jm* sup x£R,x^0 2 x£l,x^0 Ilxll2 xGK.x^U 11^-112 Let x be a vector of the length q = 2n such that X2j-i = x2j = Xj and let i = {xel«: x2j-i = x2j, x ^ 0} . ETNA Kent State University and Johann Radon Institute (RICAM) 12 Then llxIL = \/2 llxIL and we have D. CERNA AND V. FINEK 1/2 R = sup E E 2-1Rfejíxí keim yei„ 2-1/2||i|l 1/2 E E R*,i*« keim \iein Using Lemma 6 we obtain < sup xeR«,x#0 IRSII 2-V2 11x11 |R||2 V2 ŘŠ < Ř Š 2 2 < CIIRII with C < 2. □ LEMMA 8. There exist constants C G M and p < 0.5 swc/z that for all m, n > 2, m < n, we have (3.12) ivrT ivrT M ■n-1,0 < C 2p(™_m) Proof. The assertion of the lemma is a direct consequence of Lemma 5 and Lemma 7. □ 4. Riesz basis on Sobolev space. In this section, we prove that W is a Riesz basis of Hq (Six) and ^2D is a Riesz basis of Hq (SI2). The proof is based on the lemmas from Section 3 and on the theory developed in [19] that is summarized in the following theorem. THEOREM 9. Let H be a Hilbert space and let Vj, j > J, be the closed subspaces of L2 such that Vj C Vj+i and U^jVj is dense in H. Let Hq for fixed q > 0 be a linear subspace of H that is itself a normed linear space and assume that there exist positive constants A\ and A2 such that a) If f G Hq has decomposition f = Ej>j fj> fj ^ V then (4.1) l/llH,<^i£2m-HH 3>j b) For each f G Hq there exists a decomposition f = Ej> j fj> fj ^ Vj> mcn that (4.2) E2«||/J||^j Furthermore, suppose that Pj is a linear projection from Vj+i onto Vj, Wj is the kernel space of Pj, $j = {4>j,k, k G Tj} are Riesz bases ofVj with respect to the L2-norm with uniformly bounded condition numbers and = {ipj,k, k el,} are Riesz bases ofWj with uniformly bounded condition numbers. If there exist constants C and p such that 0 < p < q and (4.3) then (4.4) ||PmPm+i...P„-i|| J,keIj) is a Riesz basis of Hq. Now we define suitable projections Pj from Vj+i onto Vj and show that these projections satisfy (4.3). Then we show that W which differs from (4.4) only by scaling is also a Riesz basis of Hq (0,1). For j > 2 we define t3 = {^j,k}kelj U {ipj,k}kex. and Fj = (Tj,Tj) ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT 13 Let a set (4.5) be given by (4.6) Since obviously u (,,.,} r ■ — ft ■ 3 j 3' ^3 ' r j / — Ij , keJi functions from fj are duals to functions from Tj in the space Vj+\. Since f ■ 1 is not a sparse matrix, these duafs are not focaf. We define a projection Pj from Vj+i onto Vj by pjf= E (f>hk)h,k. LEMMA 10. There exist p < 0.5 such that a projection Pj satisfies (4.7) \\PmPm+1...P„_!|| j,k^j, &j = |afe} > J > 2, and Sj : aJ+i slj. Then feel, feel, Therefore Let us denote al= E ai+1 ■i \, si = 1 Sfk r then we can write &j = Sj-aj+i and due to Lemma 4 we have ■3,0- Now, fet us consider fn E Vn and /m = PmPm+i ■ ■ ■ Pn-ifn- Then fj can be represented by fj = 2~^feei afe^i f°r J = m,n and we set a^ = |ai| ^ ■ Since 2, with the condition numbers bounded independently on j, namely cond Wj < 2. Proof. The matrix Uj = (Wj, Wj) is tridiagonal with entries 27 1 12' (Uj)^,. -, = —, k = 2,... ,2^ 2, 47 1920' (uA,fe+1 2,...,2'-l 1 3>k+l,k 0, otherwise. Thus, Uj is strictly diagonally dominant and using Gershgorin circle theorem we obtain \min (Uj) > 55 ~ 0.0333, Amaa; (Uj) < ^ « 0.1333, and cond Wj < 2. □ We also computed the eigenvalues of the matrix Uj numerically and the numerical values \min ~ 0.0333 and Xmax ~ 0.1333 correspond to the values computed using Gershgorin theorem. Thus the inequality in Theorem 11 seems to be sharp. Theorem 12. The set {2-24>2,k, k e T2) U {2-ji/)jtk,j >2,keT3) is a Riesz basis of Hq (0,1). Proof. Using the same argument as in [19] we conclude that (4.1) and (4.2) follows from the polynomial exactness of the scaling basis and the smoothness of basis functions and are satisfied for H = L2 (0,1) and Hq = Hq (0,1), 0 < q < 1.5. Due to Lemma 10 the condition (4.3) is fulfilled. Therefore by Theorem 9 the assertion of Theorem 12 is proved. □ Theorem 13. The set [ J > 2>k e Ij] , where \-\Hi^0 ^ denotes the Hq (0, \)—seminorm, is a Riesz basis of Hq (0, 1). Proof. We follow the proof of Lemma 2 in [25]. From (2.9) there exist constants C\ and C2 such that (4.8) and (4.9) C7122<|02jfe|Hol(Q)2 < CVIIb 4 ll"ll2 : ifi(0,l) for any b = {a2jfe, k e T2} U {bjtk, j > 2, k e Jj}. Using (4.8), (4.9), and (4.10) we obtain |b||2<^ E feei2 a2k-. o2,k »2,fe m E ifi(0,l) and IblL > Ci 2" C74 E feei2 ß2,fel 'J2,k »2,fe m E (0,1) ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT 15 REMARK 4.1. By Theorem 9 and the proof of Lemma 10 if p satisfies (3.12) then the norm equivalence (2.1) for from Section 2 normalized with respect to the Hs-norm is satisfied for H = Hs, where s e (p, 1.5). Since we proved in Section 3 that there exists p satisfying (3.12) such that p < 0.5 we proved the norm equivalence (2.1) for Hs with s e (0.5,1.5). We computed the norms in (3.12) also numerically and we found that this theoretical estimate of p is not sharp. It seems that (3.12) holds also for any p > 0. THEOREM 14. The set ty2D normalized with respect to the H1-seminorm is a Riesz basis o/Hq I (0,1) Proof. Recall that ipj^ are defined by (4.5) and (4.6). For k = (k\, k2) let us define 2 ^ = j,k± <8> j,k2 Then for k = (ki, k2) and 1 = (li, l2) we have 2 N — Ůk1,l15k2,l2 and P2D defined by P?Df- is a projection from V2+1 onto V2, where V2 = Vj Vj for j > 2. We denote S2D = M^0 M^0. It is well-known that for any matrix B we have ||B (g> B||2 = ||B||2. Using this relation and the same arguments as in the proof of Lemma 10 we obtain for /„ e V2 and fm = P^P^+i ■ ■ ■ Pn-ifn the estimate: ||/m|| < Ci ||am||2 < C2 jjS2^ S^_1 ... S2- ||a„||2 = C2 M m,0 ■ M n-1,0 M m,0 M n-1,0 ^"112 < C7322^"-m) ||a„||2 < c74 22^"-m) ||/„|| with 2p < 1. Hence by Theorem 9 the assertion of the theorem is proved. □ 5. Quantitative properties of constructed bases. In this section, we present the condition numbers of the stiffness matrices for the Helmholtz equation (5.1) -eAu + au = f on fid, u = 0 on dfld, where A is the Laplace operator, e and a are positive constants. We also study the case e = 1 and a = 0, i.e., the Poisson equation, and the case e = 0 and a = 1. The variational formulation is (5.2) where Au A = e(W, W) + a(W,W), u = (uy^, f = (/, . An advantage of discretizing the elliptic equation (5.1) using a wavelet basis is that the system (5.2) can be simply preconditioned by a diagonal preconditioner [9]. Let D be a matrix of diagonal elements of the matrix A, i.e., Da,m = Aa,m5a,m, where S\tfI denotes Kronecker delta. Setting Ä = (Dr1/2A(D)~1/2, u=(D)1/2u, f=(D)~1/2f, we obtain the preconditioned system (5.3) Aü It is known [9] that there exist a constant C such that cond A < C < oo. Let ^s be defined by (2.10) for d = 1 and similarly for d > 1. We define Afl = e (Ws, Ws) + a (^, , us = (usf ^s, ffl = (/, ETNA Kent State University and Johann Radon Institute (RICAM) 16 D. CERNA AND V. FINEK Let Ds be a matrix of diagonal elements of the matrix As, i.e., (DS)A = (AS)A We set Afl = (Dfl)-1/2 Afl (Dfl)-1/2 , ufl = (DS)V2 ufl, 1 = (Dfl)-1/2 ffl and we obtain the preconditioned finite-dimensional system (5.4) Aflufl = ffl. Since As is a part of the matrix A that is symmetric and positive definite, we have also condAs < C. The condition numbers of the stiffness matrices As for e = 1, a = 0, and d = 1, 2, are shown in Table 5.1. By Remark 2.2 these numbers correspond to the squares of the condition numbers of Ws with respect to the H1-seminorm. We computed also the condition numbers of Ws with respect to the H1-norm. The values were very close to the values presented in Table 5.1 (the difference was less than 1%). For comparison, we computed also the condition numbers for other wavelet bases and displayed them in Figure 5.1 and Figure 5.2. The bases CF2 and CF3 refer to the wavelet bases from this paper with the coarsest level 2 and 3, respectively. Dj0 and Pj0 refer to the quadratic spline wavelet basis with 3 vanishing moments and the coarsest level j0 from [11] and [20], respectively. We modified the construction from [3] to homogeneous boundary conditions. The resulting quadratic spline wavelet basis with three vanishing wavelet moments with the coarsest level jo is denoted as Bj0. We found that bases Dj0, Pj0, and Bj0 lead to the same results and we realized that they contain the same wavelets up to a multiplication with a constant factor. Semiorthogonal quadratic spline wavelets with three vanishing moments on the interval were constructed in [5]. In Appendix A we show that the semiorthogonal quadratic spline wavelet basis corresponding to scaling functions that are B-splines on the Schoenberg sequence of knots such that wavelets have three vanishing moments and the basis is adapted to homogeneous boundary conditions do not exist. Therefore, we adapt this basis such that semiorthog-onality is preserved and 2J — 2 wavelets on the level j have three vanishing moments and 2 wavelets on the level j are without vanishing moments. We denote the resulting basis as CQ. We also tested wavelet bases from [11, 20] with 5 vanishing moments, but the condition numbers were larger than for bases with 3 vanishing moments. All wavelets used in numerical experiments are presented in Appendix A. Although it was not proved in this paper that using appropriate tensorising of ID wavelet basis we obtain the wavelet basis in 3D, we listed the condition numbers of the stiffness matrices As for 3D case in Table 5.2. The condition numbers for several constructions of quadratic spline wavelet bases and various values of parameters e and a are compared in Table 5.3. Table 5.1 The condition numbers of the stiffness matrices As of the size N X N corresponding to multiscale wavelet bases with s levels of wavelets for the one-dimensional and the two-dimensional Poisson equation. s N ID ^max condÄs N 2D ^max condÄs 1 8 0.50 1.38 2.77 64 0.25 1.88 7.5 2 16 0.50 1.41 2.83 256 0.19 2.08 11.1 3 32 0.50 1.42 2.83 1 024 0.16 2.17 13.7 4 64 0.50 1.42 2.84 4 096 0.14 2.20 15.4 5 128 0.50 1.42 2.84 16 384 0.13 2.22 16.6 6 256 0.50 1.42 2.84 65 536 0.13 2.23 17.4 7 512 0.50 1.42 2.84 262 144 0.12 2.23 17.9 8 1024 0.50 1.42 2.84 1 048 576 0.12 2.23 18.3 We computed also the condition numbers for the discretization matrices As corresponding to e = 0, a = 1, and d = 1. By Remark 2 these condition numbers represent the squares of the L2 condition numbers of Ws normalized with respect to the L2-norm. The results are displayed in Figure 5.1. In this paper, we proved that the constructed basis is a Riesz basis in Hq (0,1). The condition numbers of matrices As corresponding to e = 0 and a = 1 for the new basis seems to be unbounded and thus it seems that the new basis is not a ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT 17 Table 5.2 The condition numbers of the stiffness matrices As of the size N X N corresponding to multiscale wavelet bases with s levels of wavelets for the three-dimensional Poisson equation. s N ^max condAs 1 512 0.15 3.23 47.4 2 4096 0.04 3.69 85.0 3 32768 0.03 3.83 113.8 4 262144 0.03 3.87 132.9 5 2097152 0.03 3.89 145.3 Table 5.3 The condition numbers of the stiffness matrices As of the size 65536 X 65536/or several choices oft and a for our bases and bases from [11. 20]. e a CF2 CF3 (jport (jport CQ D2 D3 1000 1 17.4 16.3 17.1 16.4 62.0 116.3 98.4 1 0 17.4 16.7 17.1 16.4 62.0 116.3 98.4 1 1 17.4 16.7 17.1 16.4 62.0 116.6 98.5 irr3 1 72.1 35.9 35.6 22.5 61.1 328.1 139.2 irr6 1 746.0 577.0 425.7 287.6 46.3 1878.0 1115.4 0 1 872.6 687.4 511.0 351.5 46.4 2034.6 1251.4 Riesz basis in L2 (0,1), see also Remark 4.1. Since the condition numbers of matrices As for e = 1 and a = 0 corresponding to the anisotropic basis 'J®'I' with respect to the H1-seminorm depend on the condition numbers of *f>s both with respect to the L2-norm and the H1-seminorm, they are also increasing, see Figure 5.2. Thus in our case an isotropic wavelet basis from Section 2 has bounded and significantly smaller condition number than an anisotropic basis. We performed numerical experiments with both types of bases, but since the isotropic system lead to significantly better results we present in Section 6 only the experiments with the isotropic wavelet bases. 25 20 15 1D,e=1,a=0 -*-C-F2 -e-B2,D2,P2 -«-B3.D3.P3 CQ ......B..-er:S , o- -o - o- -o - ©- -o- - e- -o- -e - -o- -© 0000000000 10 12 14 1D,e=0,a=1 FIG. 5.1. The condition numbers of the matrices As, s = J — jo + 1, for the one-dimensional problem (5.1) with parameters e = 1, a = 0, and e = 0, a = 1. The parameter J denotes the finest level and jo denotes the coarsest level. 6. Numerical examples. In this section we use the constructed wavelet basis in the wavelet-Galerkin method and the adaptive wavelet method. 6.1. Multilevel Galerkin method. We consider the problem (5.1) with fl2, e = 1 and a = 0. The right-hand side / is such that the solution u is given by: u (x, y)=v (x) v(y), v (x) = x (l - e50a;-50) ETNA Kent State University and Johann Radon Institute (RICAM) 18 D. ČERNÁ AND V. FINĚK 2D-iso 2D-aniso FIG. 5.2. The condition numbers of the matrices As, s = J — jo + 1, for e = 1, a = 0 and two-dimensional wavelet bases constructed using an isotropic approach and an anisotropic approach. The parameter J denotes the finest level and jo denotes the coarsest level. We discretize the equation using the Galerkin method with wavelet basis constructed in this paper and we obtain discrete problem Asiis = fs. We solve it by conjugate gradient method using a simple multilevel approach similarly as in [27, 19]: 1. Compute As and fs, choose vo of the length 42. 2. For j = 0,..., s find the solution Uj of the system AjUj = f, by conjugate gradient method with initial vector Vj defined for j > 1 by ' j — 1; i 1, ■ ■ ■ , kj , 0, i = kj,. .. , kj-^i, where kj = 22^+1\ Let u be the exact solution of (5.1) and (ü:f(Dsr1/2* where u* is the exact solution of the discrete problem (5.4). It is known [21] that due to the polynomial exactness of the spaces span *f>s there exists a constant C independent of s such that (6.1) ||U-U;|| 3=0 The results are listed in Table 6.1. It can be seen that the number of conjugate gradient iterations is quite small and that K -Mlloo _ \\Us-u\\ _ I \\us+1 -utt ~ \\us+1 -u\\ ~ 8' ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT 19 i.e., that the order of convergence is 3. It corresponds to (6.1). The parameters r2 and rm in Table 6.1 are the experimental rates of convergence, i.e. / >, _ (\\us-l - Ml / \\Us - U\\) _ lQg(K-l -uWoq/WUs-uWoo) [r2)s~ log 2 >(r°°)s- bg2 We presented also the wall clock time in Table 6.1. It includes the computation of the right-hand side, the system matrix, iterations and evaluation of the solution on the grid with the step size 2_J0_S, where jo is the coarsest level. Table 6.1 Number of iterations and error estimates for multilevel conjugate gradient method. CF2 s N M \\Us-u\\oo \\us - u\\ r2 time [s] 1 64 18.50 3.19e-l 4.54e-2 0.04 2 256 21.63 1.32e-l 1.27 1.26e-3 5.17 0.05 3 1 024 23.66 2.60e-2 2.34 2.02e-3 2.64 0.06 4 4 096 23.00 2.91e-3 3.16 2.45e-4 3.04 0.09 5 16 384 20.89 4.06e-4 2.84 2.89e-5 3.08 0.16 6 65 536 18.37 5.35e-5 2.92 3.41e-6 3.08 0.30 7 262 144 15.68 6.82e-6 2.97 4.23e-7 3.01 0.99 8 1 048 576 13.02 8.63e-7 2.98 5.28e-8 3.00 3.89 9 4 194 304 10.35 1.08e-7 3.00 6.59e-9 3.00 14.87 10 16 777 216 8.85 1.41e-8 2.94 8.25e-10 3.00 58.12 D2,P2,B2 s N M K - «II«, \\us - u\\ r2 time [s] 1 64 27.50 3.19e-l 4.54e-2 0.04 2 256 48.88 1.32e-l 1.27 1.26e-3 5.17 0.07 3 1 024 59.22 2.60e-2 2.34 2.02e-3 2.64 0.11 4 4 096 59.38 2.91e-3 3.16 2.45e-4 3.04 0.19 5 16 384 50.76 4.06e-4 2.84 2.89e-5 3.08 0.33 6 65 536 39.44 5.35e-5 2.92 3.41e-6 3.08 0.68 7 262 144 29.92 6.84e-6 2.97 4.23e-7 3.01 2.20 8 1 048 576 21.50 8.64e-7 2.98 5.29e-8 3.00 9.53 9 4 194 304 17.66 1.09e-7 2.99 6.73e-9 2.97 47.39 10 16 777 216 15.79 1.38e-8 2.98 9.43e-10 2.84 248.41 CQ s N M \\Us-u\\oo \\us - u\\ r2 time [s] 0 64 13.00 3.19e-l 4.54e-2 0.03 1 256 30.25 1.32e-l 1.27 1.26e-3 5.17 0.05 3 1 024 35.06 2.60e-2 2.34 2.02e-3 2.64 0.07 4 4 096 33.82 2.91e-3 3.16 2.45e-4 3.04 0.14 5 16 384 30.30 4.06e-4 2.84 2.89e-5 3.08 0.21 6 65 536 25.32 5.35e-5 2.92 3.41e-6 3.08 0.41 7 262 144 20.74 6.84e-6 2.97 4.23e-7 3.01 1.39 8 1 048 576 17.87 8.64e-7 2.98 5.29e-8 3.00 5.55 9 4 194 304 14.82 1.08e-7 3.00 6.73e-9 2.97 21.62 10 16 777 216 12.36 1.36e-8 2.99 8.56e-10 2.97 83.54 6.2. Adaptive wavelet method. We compare the quantitative behavior of the adaptive wavelet method with our wavelet basis, the wavelet basis from [11], and the wavelet basis that is a modification of the basis from [5], see Appendix A. We consider the equation (5.1) with d = 1, e = 1, a = 0, and the solution u (x) = e I"S" iI — e~i + sin37rx, x G [0,1] . ETNA Kent State University and Johann Radon Institute (RICAM) 20 d. cerna and v. finek Note that u is the sum of the infinitely differentiabfe function and the function g (x) = e_l*_s| which has not derivative in the point 0.5. Let g be the Fourier transform of g, i.e. H0= I g(x)e-^dx. Since 64 K| 2ß (16£2 + 1) is finite for u < 3/2 and it is not finite for u > 3/2, the sofution u befongs to the Sobofev space u e Hs (0,1) onfy for s < 3/2. Therefore it is not guarantied that (6.1) holds and that the Gaferkin method converges with the optimaf rate. Since u is continuous and piecewise smooth, it can be shown that u befongs to the Besov space fij! T (0,1) for any s > 0 and t = (s + 1/2) 1. ft is therefore convenient to sofve this probfem with the adaptive wavefet method proposed in [6, 7], because it is proved that this method converges with the optimaf rate for functions from such spaces. More precisefy, fet Uj be the approximate sofution in the jth step and fet pj denote the error in the energy norm which is in this exampfe the same as the iP-seminorm, i.e., Pj = Let Uj be the vector of coefficients corresponding to Uj and fet Nj be the number of nonzero entries of Uj. ft foffows from the theory devefoped in [7] that if the used basis is a quadratic spfine wavefet basis then there exists a constant C independent on j such that (6.2) Pj < CNrr for any r < 2. The method insists in sofving the infinite preconditioned system (5.3) with Richardson iterations. The afgorithm contains the routine COARSE that is based on threshofding the coefficients and the routine RHS that approximate the vector of the right-hand side that is infinite by a finite vector with a prescribed accuracy. For detaifs about these two routines we refer to [7]. ft is possibfe to modify the afgorithm such that the routine COARSE is avoided, see [14]. Furthermore, it is necessary to have a routine that enables to compute a muftipfication of the biinfinite matrix A with a finitefy supported vector. This routine caffed APPLY was proposed in [7] and modified in [24, 12]. We use the version from [24]. We use the similar version of the method and notations that is presented as CDD02SOLVE in [14]. We compute the relaxation parameter uj and the error reduction factor pby and we set 6 = 0.3 and K e N such that 2pK/6 < 0.6. We use the folfowing version of the method: Algorithm 15. SOLVE [A, f, e] ->• ue P = cond A — 1 cond A + 1' 1. Set j := 0, Uo := 0, and eg > 2. While 6j > e do u„ z0 := For I 1, end for, ., K do Zi-l + i RHS[f 2ujK J-APPLYlÄ.z^!,^] Uj := COARSE[zK, (1 - 0) €j], end while, We use the foffowing parameters in the numericaf experiments: ETNA Kent State University and Johann Radon Institute (RICAM) QUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT 21 0 50 100 150 200 250 number of iterations time[s] FIG. 6.1. The convergence history for adaptive wavelet scheme with various wavelet bases. - CF2: uj = 1.04, p = 0.48, K = 4, - D2: uj = 0.89, p = 0.70, K = 7, - CQ3: uj = 0.95, p = 0.87, K = 18. The convergence history is shown in Figure 6.1. Since the entries of the matrix A, the estimates of eigenvalues of A and the parameters uj, p and K were precomputed for every basis, the wall clock time includes the computation of the right-hand side and the computation of iterations. The experimental convergence rate, i.e., the parameter r from (6.2) estimated for the observed values (Nj, pj) by the least square method, for bases CF2, D2, and CQ was r « 1.87, r « 1.95, and r « 1.77, respectively. It can be seen that the number of iterations and the computational time needed to resolve the problem with desired accuracy is significantly smaller for the new wavelet basis. Moreover, due to the shorter support of the wavelets, the stiffness matrix is sparser and thus one iteration requires smaller number of operations. Appendix A. Quadratic spline wavelet bases. In this section we present inner and boundary scaling functions and wavelets that were used in numerical experiments in Section 6. The wavelet bases is generated from these function by the similar way as in (2.6) and (2.9). Let be given by (2.2) and b = 2b/3, where b is given by (2.4). Since diagonal preconditioning (5.4) is similar to the normalization of the basis with respect to the energy norm, the multiplication of b with a constant has no effect on resulting condition numbers presented in Section 5 and numerical results in Section 6. The wavelets are given by $ (x) = £ 9k (2x - k) . k=0 k=0 for i = 1,2. The values of the parameters gk and g\ are presented for several constructions below. A.l. Primbs wavelet basis. The parameters for the construction from [20] are given by [go, [sř-i, [g2-i, ,ff7] = [-3,-9,7,45,-45,-7,9,3] /64, ,g&\ -10 65 _ _9_ 6"'~14' 10 5 65 T'~6' IT' 31 25 n is _5_ ~21'14'14 13 3 1 ~9~' 2' 2 /64, /64. More precisely, in [20] the parameters are multiplications of these parameters, but as we already mentioned different normalization does not play a role, because we use diagonal preconditioning (5.4) in our experiments. A.2. Dijkema wavelet basis. There are several constructions in [11]. We used the parameters that are listed in the file mats.zip attached to [11], but we found that in the case of quadratic spline wavelets with three vanishing moments and homogeneous boundary conditions these parameters are multiples of the parameters from [20] and thus they lead to the same results. A.3. Modification of Chui-Quak wavelet basis. In [5] the semiorthogonal quadratic spline wavelets with three vanishing moments were adapted to the interval. We adapt these wavelets to homogeneous boundary conditions. Since wavelets on the level j are linear combinations of scaling functions on the level j + 1, they ETNA Kent State University and Johann Radon Institute (RICAM) 22 D. CERNA AND V. FINEK are given by 2J+1 parameters. We want to preserve semiorthogonality, therefore we have 2J conditions on orthogonality to scaling functions on the level j. Furthermore, we want to preserve three vanishing moments. We obtain homogeneous system with 2J + 2 independent equations with 2J+1 variables that has only 2J — 2 independent solutions. Therefore there exists only 2J — 2 wavelets with three vanishing moments that are semiorthogonal. We add two wavelets on each level that are semiorthogonal but without vanishing moments. We obtain wavelets with parameters: [do, ■ ■■,97 r i in g-i,- -,9&\ r 2 21 9-1,- ■,9e\ [450, -780 TP -332,148,-29,1,0,0] /480, 1949 3481 3362 1618 11 ' 11 11 ' 11 ,-29,1 /480. A.4. Modification of Bittner wavelet basis. In [3] spline wavelet bases on the interval were constructed. We use the similar approach as in [3], but for quadratic spline wavelets with three vanishing moments satisfying homogeneous boundary conditions. The inner wavelet is the third derivative of the sixth-order B-spline on knots [0,1, 2, 5/2, 3,4, 5]. The boundary wavelets are the third derivatives of the sixth-order B-splines on knots [0, 0,1/2,1,2,3, 4] and [0, 0,1, 3/2, 2, 3,4], respectively. We found that by this approach we again obtain the same wavelets up to a constant factor as in [11, 20]. REFERENCES [1] T. BARSCH, Adaptive Multiskalenverfahren für elliptische partielle Differentialgleichungen - Realisierung, Umsetzung und numerische Ergebnisse, PhD thesis, RWTH Aachen, 2001. [2] G. BEYLKIN, R. COIFMAN, and V. ROKHLIN, Fast wavelet transforms and numerical algorithms i, Comm. Pure Appl. Math., 44 (1991), pp. 141-183. [3] K. BITTNER, Biorthogonal spline wavelets on the interval, in Wavelets and Splines: Athens 2005, G. Chen and M.-J. Lai, eds., Modern methods in mathematics, Brentwood, TN, 2006, Nashboro Press, pp. 141-183. [4] C. BURSTEDDE, Fast Optimized Wavelet Methods for Control Problems Constrained by Elliptic PDEs, PhD thesis, Universität, Bonn, 2005. [5] C. K. CHUI and E. quak, Wavelets on a bounded interval, Birkhäuser Basel, Basel, 1992, pp. 53-75. [6] A. COHEN, W. DAHMEN, and R. DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comput., 70 (2001), pp. 27-75. [7] -, Adaptive wavelet methods ii - beyond the elliptic case, Found. Comput. Math., 2 (2002), pp. 203-245. [8] W. DAHMEN, B. Han, R.-q. Jia, and A. Kunoth, Biorthogonal multiwavelets on the interval: cubic Hermite splines, Constr. Approx., 16 (2000), pp. 221-259. [9] W. DAHMEN and A. KUNOTH, Multilevel preconditioning, Numer. Math., 63 (1992), pp. 315-344. [10] W. DAHMEN, A. KUNOTH, and K. URBAN, Biorthogonal spline wavelets on the intervaläATstability and moment conditions, Appl. Comput. Harmon. Anal, 6 (1999), pp. 132 - 196. [11] T. DlJKEMA, Adaptive tensor product wavelet methods for solving PDEs, PhD thesis, Universiteit Utrecht, 2009. [12] T. J. DlJKEMA, C. schwab, and R. STEVENSON, An adaptive wavelet method for solving high-dimensional elliptic pdes, Constr. Approx., 30 (2009), pp. 423-455. [13] T. J. DlJKEMA and R. STEVENSON, A sparse laplacian in tensor product wavelet coordinates, Numer. Math., 115 (2010), pp. 433-449. [14] T. GANTUMUR, H. HARBRECHT, and R. STEVENSON, An optimal adaptive wavelet method without coarsening of the iterands, Math. Comput., 76 (2007), pp. 615-629. [15] s. GRIVET-TALOCIA and A. TABACCO, Wavelets on the interval with optimal localization, Math. Models Meth. Appl. Sei., 10 (2000), pp. 441-462. [16] N. HlLBER, O. REICHMANN, C. schwab, and C. WINTER, Computational Methods for Quantitative Finance, Springer, Berlin, 2013. [17] R.-q. JIA, Spline wavelets on the interval with homogeneous boundary conditions, Adv. Comput. Math., 30 (2009), pp. 177-200. [18] R.-q. JIA and s.-T. LIU, Wavelet bases of hermite cubic splines on the interval, Adv. Comput. Math., 25 (2006), pp. 23-39. [19] R.-q. JIA and W. ZHAO, Riesz bases of wavelets and applications to numerical solutions of elliptic equations, Math. Comput., 80 (2011), pp. 1525-1556. [20] M. PRIMBS, New stable biorthogonal spline-wavelets on the interval, Result. Math., 57 (2010), pp. 121-162. [21] K. URBAN, Wavelet Methods for Elliptic Partial Differential Equations, Oxford University Press, Oxford, 2009. [22] D. černá and V. FINEK, Construction of optimally conditioned cubic spline wavelets on the interval, Adv. Comput. Math., 34 (2011), pp. 219-252. [23] -, Cubic spline wavelets with complementary boundary conditions, Appl. Math. Comput., 219 (2012), pp. 1853 - 1865. [24] -, Approximate multiplication in adaptive wavelet methods, Cent. Eur. J. Math., 11 (2013), pp. 972-983. [25] -, Cubic spline wavelets with short support for fourth-order problems, Appl. Math. Comput., 243 (2014), pp. 44 - 56. [26] -, Quadratic spline wavelets with short support for fourth-order problems, Result. Math., 66 (2014), pp. 525-540. [27] -, Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions, IJWMIP, 13 (2015), p. 1550014 (21 pages). Preprint of an article published in International Journal of Wavelets, Multiresolution and Information Processing 17(1), 2019, article No. 1850061], https://doi.org/10.1142/S0219691318500613 Copyright World Scientific Publishing, https://www.worldscientific.com/worldscinet/ijwmip International Journal of Wavelets, Multiresolution and Information Processing © World Scientific Publishing Company CUBIC SPLINE WAVELETS WITH FOUR VANISHING MOMENTS ON THE INTERVAL AND THEIR APPLICATIONS TO OPTION PRICING UNDER KOU MODEL DANA ČERNÁ Department of Mathematics and Didactics of Mathematics, Technical University of Liberec Studentská 2, 461 17 Liberec, Czech Republic dana. cerna@tul. cz The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions. The wavelets have four vanishing moments and they have the shortest possible support among all cubic spline wavelets with four vanishing moments corresponding to B-spline scaling functions. We provide a rigorous proof of the stability of the basis in the space L2 (0, 1) or its subspace incorporating boundary conditions. To illustrate the applicability of the constructed bases we apply the wavelet-Galerkin method to option pricing under the double exponential jump-diffusion model and we compare the results with other cubic spline wavelet bases and with other methods. Keywords: Wavelet; cubic spline; short support; Galerkin method; option pricing; Kou model. AMS Subject Classification: 65T60, 65D07, 47G20, 65M60, 91G80 1. Introduction Wavelets have already found applications in numerous fields, including signal analysis, image processing, approximation theory, engineering applications, and numerical simulations. They have been used for the numerical solution of various types of partial differential and integral equations. Wavelet methods are suitable for preconditioning of systems of linear algebraic equations arising from the discretization of elliptic problems,19 adaptive solution of operator equations,16 the numerical solution of certain types of partial differential equations with a dimension independent convergence rate.23 Wavelet methods seem to be superior to classical methods especially for the solution of integral and partial integro-differential equations, because they enable to represent the integral term by sparse or almost sparse matrices while the classical methods suffer from the fact that the matrices arising from discretization are full.3'12 The quantitative properties of any wavelet method strongly depend on the used wavelet basis, namely on the length of the support of basis functions, the number of vanishing wavelet moments, the smoothness of basis functions and the condition number of the basis. Hence, a construction of appropriate wavelet bases is an important issue. In this paper, we construct a cubic spline wavelet basis on the interval and we adapt this basis to homogeneous Dirichlet boundary conditions of the first order. The bases are well-conditioned, the wavelets have four vanishing moments and we show that the support is the shortest among all cubic spline wavelets with four vanishing moment corresponding to B-spline scaling functions. The wavelet basis is composed of scaling l 2 Dana Černá functions and inner and boundary wavelets. The inner wavelets are the same as wavelets constructed in Ref. 11, 28. We provide a rigorous proof of the Riesz basis property. While such proofs are usually based on semiorthogonality of the wavelets,15'34 the local supports of the biorthogonal wavelets,18'20 the estimates of the norms of certain projections7'36 or spectral properties of the matrices of the inner products of primal and dual scaling functions,8'9'24 we use a different approach that is based on analyzing the sets of inner and boundary wavelets separately and verifying the minimal angle condition between spaces generated by inner and boundary wavelets. To illustrate the applicability of the bases we apply the Crank-Nicolson scheme combined with the Galerkin method with the constructed basis for option pricing under the double exponential jump-diffusion model that was proposed by Kou in Ref. 39. This model is represented by a nonstationary partial integro-differential equation. We show the decay of elements of the matrices arising from discretization of the integral term. Due to this decay the discretization matrices can be truncated and represented by quasi-sparse matrices while the most standard methods suffer from the fact that the discretization matrices are full. Since the basis functions are piecewise cubic we obtain a high order convergence and the problem can be resolved with the small number of degrees of freedom. We present numerical examples for European options and we compare the results with other cubic spline wavelet bases and with other methods. For more details about wavelet-Galerkin method and using this method for the numerical solution of various option pricing problems we refer the reader to Ref. 25, 31, 32, 44. First, let us briefly recall constructions of cubic spline wavelet and multiwavelet bases on the interval. Several constructions of biorthogonal B-spline wavelet bases on the interval were proposed in Ref. 20. In these cases both the primal and dual wavelets are local, but the disadvantage of these bases is their relatively large condition number. Modifications of these constructions that lead to better conditioned bases can be found in Ref. 4, 5, 6, 22, 42. Biorthogonal cubic Hermite spline multiwavelet bases on the interval with local duals were designed in Ref. 18, 45. Several cubic spline wavelet and multiwavelet bases with nonlocal duals have been constructed and adapted to various types of boundary conditions in Ref. 7, 8, 9, 10, 15, 24, 30, 34, 35, 36, 38, 41, 43. The main advantages of these types of bases in comparison with bases with local duals are usually the shorter support of wavelets, the lower condition number of the basis and the corresponding stiffness matrices, and the simplicity of the construction. We recall the concept of a wavelet basis and introduce the notation. Let (a, b) be a bounded interval. Let L2 (a, b) be a Hilbert space of all Lebesgue measurable real-valued functions defined on (a, b) such that their L2-norm Let H1 (a, b) be the Sobolev space, i.e. the space of all functions from I? (a, b) for which their first-order weak derivatives also belong to L2 (a, b). (1.1) is finite. This space is equipped with the inner product b (1.2) a Cubic spline wavelets with four vanishing moments 3 We consider four spaces, the space VN (a, b) = I? (a, b) and the subspaces of I? (a, b) incorporating homogeneous boundary conditions in one or both endpoints, namely the spaces VD (a, b) = {v e H1 (a, b) : v (a) = v (b) = 0} , (1.3) VL (a, b) = {v e H1 (a, b) : v (a) = 0} , (1.4) VR (a, b) = {v G H1 (a, b) : v (b) = 0} , (1.5) equipped with inner product (1.2). We use the shorthand notation Vr = Vr (0,1), r = N, D, L, N. We construct wavelet bases for these spaces and prove their L2-stability. Let J be a finite or countably infinite index set and let IIVH = JE^' for v = HeJ,»Aet, (1.6) and I2 (J) = {v : v = {vx}XeJ ,vxeR, ||v|| < oo} . (1.7) For the operator M : I2 (J) —> l2 (J) we define its spectral norm as llMvll ||M|| = sup " 11. (1.8) 0/v6i2(J) llvll Schur's Theorem implies that if M is symmetric and its 1-norm defined as \\MW-y = sup V IMj j\ (1.9) is finite, then M is a bounded operator on I2 (J) and ||M|| < HM^. Let Aj, i G J, be eigenvalues of M and let us denote Xmax (M) = sup |Ad , Xmin (M) = inf |Ad . (1.10) iej ^ Let H C L2 (a, 6) be a real Hilbert space equipped with the inner product (•,•)# and the norm Our aim is to construct a wavelet basis for H in the sense of the following definition. Definition 1.1. Let J be at most countable index set where each index A G J takes the form A = (j, k) and let denote |A| = j G Z. A family ^/ = {ip\, A G J) is called a wavelet basis of H, if i) is a Riesz basis for H, i.e. the span o/^F is dense in H and there exist constants c,C & (0, oo) such that for allh = {bx}X€jEl2 (J). ii) The functions are local in the sense that diam supp ipx < C2_|A|, A G J, (1.12) where the constant C does not depend on X, and at a given level j the supports of only finitely many wavelets overlap at any point x. 4 Dana Černá Hi) The family ^ has the hierarchical structure K 3=3 o for some K E N U {oo}. iv) There exists L > 1 suc/i i/iai all functions tfj\ E ^>j, jo < j < K, have L vanishing moments, i.e. b J xktfjx(x) = 0, fc = 0,...,L-l. (1.14) a The definition of a wavelet basis is not unified in the mathematical literature and the conditions i) — iv) from Definition 1.1 can be generalized. The functions from the set $J0 are called scaling functions and the functions from the set fyj, j > jo, are called wavelets on the level j. Wavelets in the inner part of the interval are typically translations and dilations of one function tfj or several functions ... ,tfjp also called wavelets and the functions near the boundary are derived from functions called boundary wavelets. The Riesz basis property (1.11) is crucial for stability and accuracy of the computation and for many types of operator equations it guaranties that the diagonally preconditioned system matrix is well conditioned.17'19 For the two countable sets of functions 1, 8 C L2 (£}) the symbol (r, 0) denotes the matrix (r,e) = {(7,(9)}7er;eee. (i.i5) Remark 1.1. The constants cy = sup{c : c satisfies (1.11)} and C$ = inf{C : C satisfies (1.11)} (1-16) are called (optimal) Riesz bounds and the number cond 4* = /c^ is called the condition number of 4'. In some papers the squares of norms are used in (1.11) and Riesz bounds are defined as c|, and C|,. The Gram matrix (\I>, \I>) can be finite or biinfinite and it is known that it represents a linear operator which is continuous, positive definite, and self-adjoint, and that the constants c^ and satisfy C* = V^min C* = V^max «*,*))• (1-17) Remark 1.2. The set of functions is called a Riesz sequence in H if there exist positive constants c and C that satisfy (1.11) but the closure of this set is not necessarily H. 2. Construction of a cubic spline wavelet basis on the unit interval We define a scaling basis as a basis of cubic B-splines in the same way as in Ref. 5, 15, 42. Let 4> be a cubic B-spline defined on knots [0,1,2,3,4]. It can be written explicitly as 2d. 6 ' 4>(x) + 2x2 xE [0,1], 2x + §, x E [1,2], \ -4x2 + 10x- f, x E [2,3], íá#, *e [3,4], 0, otherwise. (2.1) Cubic spline wavelets with four vanishing moments 5 Then satisfies a scaling equation 4>(x) = (2x) + 0 (2x - 1) + 30 (2x - 2) + 0 (2x - 3) + 0 (2x - 4) We define three boundary scaling functions. Let 0?,o be a cubic B-spline defined on knots [0,0,0,0,1], i.e. 4>bo{x) (1 — x) , x G [0,1], 0, otherwise. (2.3) Furthermore, let 0?,i be a cubic B-spline defined on knots [0, 0, 0,1, 2] and 0?,2 be a cubic B-spline defined on knots [0, 0,1, 2, 3]. The explicit forms of and <^2 are 4>bi{x) 7x-> 4 2f- + 3x, x e [0,1], and 4>b2{x) 7x 12 i£[l,2], 0, otherwise, Ux3 i 3x2 12 ' 2 ' x G [0,1], 3x ~\~ ~2~ 2" j 3/ £e [lj 2|. xG [2,3], otherwise. (3-^)J 6 o, (2.4) (2-5) Then the functions 0?,o, 0?,i, and 0?,2 satisfy scaling equations i / \ , /0 x . bi (2x) b2 {x) 2 %2 {2x) 16 110 (2x) 0(2x - + (2x - 2) 4 16 2 The graphs of the functions 0&o, 0&i, 0&2, and 0 are displayed in Figure 1. (2.6) (2.7) (2.8) Fig. 1. The boundary scaling functions (f>bo, (f>bi, and 0f)2 (left) and the scaling function f> (right). 6 Dana Černá For j > 3 and x E [0,1] we set , . . 2ßt24>(2ßx - k + 3) IHI II 4>w\\ 2^2«; hi{2jx) II 4>bi\\ llj,k ,k = 0,. .,2J+3} <$>D 3 = {4>j,k ,k = i,. .,2^+2} $L 3 = {4>j,k ,k = i,. .,2^+3} 3 = {4>j,k ,k = 0,. .,2^+2} for j > 3. We define a wavelet ip as ip{x) = 4>{2x - 1) - 44>{2x - 2) + 6(p(2x - 3) - 4^(2x - 4) + 4>{2x - 5). Then supp-0 = [0.5,4.5] and ip has four vanishing moments, i.e. 4.5 xkip{x)dx = 0, A; = 0,1,2,3. (2.14) (2.15) 0.5 It can be verified easily using substitution of (2.1) and (2.14) into (2.15). In the following lemma we show that the wavelet ip has the shortest possible support among all wavelets with four vanishing moments that are generated from cubic B-splines. Lemma 2.1. Let the function

{2x - 1), (2.21) and 57 919 116 4,b2(x) = Q^bl(2x) - —M2x) + — (2x) - —{2x - 1) + 0(2x - 2). (2.22) 5 100 25 Then supp-i/^i = [0,2.5], supp-0^2 = [0,3], and both wavelets have four vanishing moments, i.e. 2.5 3 fc„/. --n „„J / ™fc„ x Tpbl(x)dx = 0 and J x Tpb2(x)dx = 0, (2.23) o o for k = 0,1,2,3. Using the similar argument as in the proof of Lemma 2.1 it is easy to see that also the boundary wavelets tfjbi and tfjb2 have the shortest possible supports among all boundary wavelets with four vanishing moments corresponding to scaling functions defined by (2.9) and that other such boundary wavelets with the supports in [0, 3] are linear combinations of ipbl and ipb2. Then the wavelet basis on the level j > 3 is defined as = {i-iM,k = l,...,2j}, (2.24) i,1Ax) = 2"2*(2':-k + 2). * = 3,...,*-2, (2.25) where 2il^bx(Vx) Tl^bx(T(\-x)) ^1(X)= ||^|| ' ^{X) =-JhMJ-' ^ y/^b2(2ix) 2?/^h2{2?{l-x)) ^2(x)= Ub2\\ ' *i>»-M =--• ^ Hence, the basis functions tfjj^ are also normalized with respect to the L2-norm. To adapt the set W^ to homogeneous Dirichlet boundary conditions we have to replace the function tfjbi which is not equal to zero in the point 0 with another function. We denote this function as tfjb3 and we define it as 7 319 101 25 = o^(2x) - —0(2x) + —0(2x - 1) - —0(2x - 2) + 0(2x - 3). (2.28) 6 60 15 6 8 Dana Černá Then the wavelet tfjbs has also four vanishing moments and its support is [0,3.5]. We define the boundary functions on the level j > 3 that are adapted to homogeneous Dirichlet boundary conditions as D 2^(^ 2^(2^(1-x)) ' ^[X)- ||^21| ' [ ^ We set "63 Wb3 *f = fc = 1, 2, 2>' - 1, 2''} U fc, fc = 3,..., 2' - 2} , (2.30) ^ = fc = 1, 2} U fc, fc = 3,..., 2'} , (2.31) *f = {^)Jfc, fc = 1,..., 2' - 2} U fc = 2' - 1, 2'} . (2.32) The graphs of wavelets tfjbi, tfjb2, ipb3, and tfj are displayed in Figure 2. Fig. 2. The boundary wavelets "i/^i, ipb2? "063) and the wavelet tp. Our aim is to show that for r = N, D, L, R, the set oo *r = $5 U |J (2.33) J=3 is a Riesz basis of the space Vr. We denote the finite-dimensional subset of tyr with s levels of wavelets as 2+s ^ = % U U (2-34) 3=3 First we define auxiliary bases ,Fr and r = N, D, L, R, that contain the same inner wavelets and generate the same spaces, i.e. span'J/j = span^, but boundary wavelets are different. The reason is that some constants characterizing the bases that are used in the proof of the Riesz basis property are too large for the bases ^/r. Hence, let us define ipbl(x) = ipbl(x) - Q.3ifjb2(x), ipb2(x) = 0.3ipbl(x) + 0Aipb2(x), (2.35) Cubic spline wavelets with four vanishing moments 9 and fc(i) = -rpr n-, ^j,2j{x) =-,-y—1|-, (2.36) || H^&lll 7 y/^b2{yx) - 2^2(2^(i-x)) ^j,2lx) =-rp? n-, ^j,2i-i{x) =-jpp-fi-• (2.37) ||-0&2|| ||^62 || and the wavelet basis on the level j > 3 is defined as tff = {$j>k, k = 1, 2, 2J' - 1, 2j} U fc = 3,..., 2j - 2} . (2.38) We define ■iksix) = 0.9^62(a;) + 1.63^63(a:), Mx) = O.Q9^b2(x) - 0A^b3(x), (2.39) and 4>°i(x) = ~ nTm"^ ^Ax) = " "T.T V.T (2-4°) 2^/2^3(2^) -d . . 2^/2^3(2^(1 -x)) hA &3 ^) = ^frP- ^-i(*) = ~ "7-7 V (2-4i) 2?IH}A{2?x) -£) ,,_ 2^/2^(2^(1-3;)) 64 and we set 3 2j -2}, (2.42) 2j (2.43) 2j -2}, (2.44) tff = (^,1,^,2,^-1,^ j U {^,jfc,fc = 3, and for r = N,D, L, R, we set 00 2+s fr = $5 U [J 4frs = $r3 U J s e N. (2.45) j=3 j=3 Now we formulate sufficient conditions under which the sum of Riesz sequences is a Riesz sequence. We employ frame theory from Ref. 13, 14, 27. Let us recall that {A}fceX is a frame sequence, if there exist constants Cf,Cf > 0 such that M/ii2<£k/>/*>i2<g = (1 - cos (F, G)) ■ min {cf,cg). (2.48) The consequence of this theorem is the following theorem about the Riesz sequences. Theorem 2.2. Let I and J be countable index sets, {fk}kex be a Riesz sequence with a Riesz lower bound Cf and {gi}l€j be a Riesz sequence with a Riesz lower bound cg, F = span ({fk}k€x) and G = span ({gi}l€j). If cos (F, G) < 1, then {fk}k€X U {gi}l€j is a Riesz sequence and its Riesz lower bound c satisfies c> y/l — cos (F, G) • min [cf,cg) . (2.49) Proof. It is known that if {fk}k€x 1S a Riesz sequence, then {fk}k€x 1S a frame sequence.14 From Theorem 2.1 and the fact that cos (F, G) < cos(F, G) < 1 it follows that {fk}kex U {9i}iej 1s a frame sequence. Furthermore, cos (F,G) < 1 implies that F Hi G = {0} and thus the set {fk}k€x U 1S "^-independent, i.e. £ c^/a = 0 (2-5°) implies that c\ = 0 for all A G J. Indeed, if / G {fk}kex ^ {d^iej 1S nonzero function, then cos (F, G) = sup -—rr—rr—j-p > = 1, (2.51; 0ytU£F,0ytv£G \\u\\ \\v\\ which is the contradiction. Since a frame sequence of ^-independent functions with a frame lower bound b is a Riesz sequence with a Riesz lower bound \/b,14: the Theorem 2.2 is proved. □ In the following theorem we derive the upper bound for cos (F, G). Theorem 2.3. Let I and J be countable index sets, {fk}kex be a Riesz sequence with a Riesz lower bound Cf and {gi}l€j be a Riesz sequence with a Riesz lower bound cg. Then IIGII cos (F,G) < -—-, (2.52) C.fC9 where the entries of the matrix G are defined by Gkti = (fk,gi), k El, I G J'. Proof. For u = Y^k&i (c«)fc fr and v = 2~2i^j (cv)i 9i we nave \(u,v)\ ^ \(Gcu,cv}\ \\Gcu\\ \\G\\\\cu\\ \\G\\\\u\\ sup —j-—j-— < sup--—-— = - < - < -. (2.53) \\v\\ 0/c„6!2(J) c9 IMI C9 C9 C.fC9 Hence, we have cos(F,G) = sup ^1^ 3} is a Riesz sequence with a Riesz lower bound ct > 0.783. Proof. It was shown in Ref. 11 and Ref. 28 that |2j'/2^ (pix -k) ,j,ke z} (2.55) is a Riesz basis of the space L2 (M). Since \I>j is its subset, it is a Riesz sequence. We estimate its Riesz lower bound. Let cf be a Riesz lower bound of the Riesz sequence tyf = {ipj,k, k = 3,..., 2°' — 2, 3 < j < K} which is clearly a subset of \I>j. We estimate the Riesz lower bounds cf using Remark 1.1. Due to the structure of the set tyf, the Gram matrix Gf = (^f, ^f) has the block structure Gf / ^3,3 ^4,3 ^3,4 G4;4 G4yk (2.56) where Gij \Gr,3 Gk,4 ■ ■ ■ Gk,k/ i iTr-f \ r^r. lTr-f — |-0J>fc, k = 3,..., 2J' — 2}. The matrix Gf is symmetric and due to the normalization of basis functions it has ones on the diagonal. Let Ff be a matrix that contains the diagonal blocks and the blocks next to the diagonal blocks of the matrix Gf, but without the diagonal entries, i.e. k /G3<3 G3)4 0 ^4,3 Gr4;4 G4;5 0 G5)4 G5;5 V 0 0 G k,k-1 0 Gk-\,k Gk,k (2-57) where I is an identity matrix and 0 are zero matrices of appropriate sizes. Let Ff -Ff — I. Let x be a normalized eigenvector corresponding to Xmin {Gf) - Then Gf Ar (Gf) xTGfx xTIx + xTFf x + xTFf x (2.5? > 1 - |xTFf x| - |x For k > 3 direct computation yields TFfx| 1, i,3 __7_ 2652 ' _J_ 78' 7 2652 ' I = J, 1, 2, 3, (2.59) 0, otherwise. The nonzero entries of the matrix Gk+\,k are given by rfc+i,fc, hi. 2j + l, (2.60) 12 Dana Černá for i,j en such that l 1 - 0.375 - 0.011 = 0.614 and cf > ^0.614 for all K > 3 and thus cj > v/0.614 > 0.783. □ Now we estimate Riesz lower bounds for the sets of boundary wavelets. Theorem 2.5. a) The set V% = {^,1,^,2, j > 3} is a Riesz sequence with a Riesz lower bound cf > 0.490. b) The set 'I'f = {"0fi, V^2>3 — ^} ^s a Riesz sequence with a Riesz lower bound cf > 0.380. Proof, a) According to Remark 1.1 the set 'I'f is a Riesz sequence if and only if the extremal eigenvalues of the (bimfmite) matrix Gf = (wf,'I'f) satisfy 0 < \mm (Gf) < Xmax (Gf) < 00. (2.67) We denote f,n,k {4iA2,3 5. (2.69) Cubic spline wavelets with four vanishing moments 13 Hence, the matrix G^'K is banded. Moreover it is symmetric, it has repeated structure and it is known in an explicit form. Therefore, it is easy to compute an estimate of its 1-norm. We have ,n,k X rN,K , < max \ v:r_£/ ' < 1.763. (2.70) Let HK = G — I, where I is the identity matrix of the appropriate size. We have 1/m Xr, ,N,K > 1 |H*||>1 H > 0.241 2.71 for m = 4. Since the estimates in (2.70) and (2.71) do not depend on the maximal level K, the condition (2.67) is satisfied and therefore the set W^ is a Riesz sequence and its Riesz lower bound satisfies > \/0.241 > 0.490. b) The proof follows the lines of the proof of the part a) with m = 8. □ Corollary 2.1. Since {i>j,2i-l,i>j,2J,J > 3} (2.72) has similar structure as W^ , the set ^„ is a Riesz sequence with a Riesz lower bound CR Cj?. Due to the non-overlapping supports ofipj^, k = 1,2, and ipj,i> I = 23 — 1,23, the set W^ = W^ U W^ is also a Riesz sequence with a Riesz lower bound c^ = = Cn. Similarly, we define and i>3 (2.73) D N ,R (2.74) We denote their Riesz lower bounds by c®, c^, and cf, respectively. We have cV = c L > cf = min (cf ,cf) For r = N, D, L, R we denote the set of all wavelets as ^rm theorem we prove that ^rm is a Riesz sequence. WruW/. In the following Theorem 2.6. The sets ^>rm, r N, D, L, R, are Riesz sequences in the space Vr. Proof. From Theorem 2.4 and Corollary 2.1 we already know that \I>j and ^>rb for r = N,D, L, R, are Riesz sequences. Thus, due to Theorem 2.2 and Theorem 2.3 to prove that ^rm is a Riesz sequence it remains to show that the matrix Hr = (W^,W/) satisfies |Hr chci < 1. Using the relation |Hr < l/2m (2-75) (2.76) for m = 8 we obtain H < 0.293 and H < 0.278 and we may conclude that (2.75) is satisfied and that W^ and W^ are Riesz sequences. Hence, also W^ U \I>j, \I>j U W^, and i U Wj;, are Riesz sequences and due to the non-overlapping supports of the functions from tyrL and tyrR, the sets W^ and W^j are also Riesz sequences. □ 14 Dana Černá Theorem 2.7. For r = N,D, L, R, the set \řr is a Riesz basis of the space Vr. Proof. First we show that the set ^ = %U^m (2.77) is a Riesz basis of the space Vr. We already know that ^rm is a Riesz sequence. Since $3 is a finite set of functions, it is a Riesz sequence too. Therefore, the spaces F = spanW^ and G = span$g (2.78) are closed and since G is finite-dimensional, the set F + G is also closed. The closedness of F + G and the fact that F n G = {0} implies cos (F, G) < l.21 Since it is known that the spline spaces Vf span $ J spanW^_3 are nested and their union is dense in Vr, see e.g. Ref. 5, 42, the set W is dense in Vr. Due to this fact and Theorem 2.2, the set Wr is a Riesz basis of Vr. Since 4>r = MW, where M is a biinfinite block diagonal matrix such that the first diagonal block of M is an identity matrix and other diagonal blocks are of the form /flia2 0...0 0 0\ 03 04 0 0 0 1 (2.79) where aj and ftj, i 0 10 0 0 0 61 62 \ 0 0 0 ... 0 63 64/ ,4, are determined by the relations (2.35) and (2.39). Since ||M|| and M || are bounded and (Wr,Wr) = M"1 (Wr, Wr) (M-the Remark 1.1 implies that W is a Riesz basis of Vr. (2.80) □ Remark 2.1. A wavelet basis on a general domain can be constructed in the following way: First, the wavelet basis for the space Vr (a, b) is derived from W using a simple linear transformation y = {x — a) / {b — a). Then a wavelet basis on the hyperrectangle can be constructed using an isotropic, anisotropic, or sparse tensor product. Finally, by splitting the domain into subdomains which are images of the hyperrectangle under appropriate parametric mappings one can obtain a wavelet basis on a fairly general domain. 3. Numerical results In this section we use the Galerkin method with the constructed wavelets for valuation of options under a double exponential jump-diffusion model proposed by Kou in Ref. 39 and we compare the results with other approaches and other cubic spline wavelet bases. Let S be the price of the underlying asset, t represent time to maturity, r be a risk-free rate and U (S, t) be the market price of the option. Then the general jump-diffusion models are represented by the equation ^r-V{U)-l{U)=0,S>0,te{0,T), (3.1) Cubic spline wavelets with four vanishing moments 15 where the operators T> and X are given by n2 c2 a2jj ajj vW = —dsi~ + (r-Ak) s ds {r + A) ^ (3-2) and oo 1(U)=X J U (bex,t)g(x)dx. (3.3) — oo The parameter A is the intensity of the price jumps, i.e. the average number of jumps per unit time. The function g represents the probability density function which in the model of Kou is given by g(x)=p rjie~rilXH (x) + q r^e^H {-x) , x G R, (3.4) where H denotes the Heaviside function, p E (0,1) represents the probability of the upward jump, q = 1 — p represents the probability of the downward jump, 771 > 1, and f]2 > 0. The parameter n in this model is given by - PV1 +-^--1. (3.5) 7/i - 1 T]2 + 1 The initial and boundary conditions depend on the type of the option. We present the approach for a European put option. The value of a European call option can be computed using the put-call parity.1 The initial condition for a European vanilla put option is given by U{S,0) = max (K-5,0), (3.6) where K is the strike price, and the boundary conditions have the form BU U (0, t) = Ke~rt, — (S, t) « 0 for S -> 00. (3.7) ob We choose the maximal value bmax large enough and approximate the unbounded domain (0, 00) by a domain Vt = (0, bmax). We replace the boundary conditions with BU U(0,t) = Ke-rt, —(bmax,t)=Q. (3.8) Furthermore we have 00 00 ( y \ 1{U)=\ j U [bex,t) g{x)dx = \ j U {y, t) 9 ^ °g ^ dy. (3.9) -00 0 Since U {y, t) 9 (*°g ^ ^0 for y -> 00, (3.10) we define 1(U) = X I U(y,t)9(l°g^dy (3.11) 0 and we approximate 1{U) ~ Z(C7). 16 Dana Černá Let U = U — W, where U is the solution of the equation (3.1) satisfying the initial and boundary conditions defined above and W is defined by W (S, t) = Ke~rt for S G [0, Smax] and t G [0, T]. Then U G VL (0, Smax) and U is the solution of the equation ^-v(u)-l(u)=f{W), (3.12) with dW f{W) = -—+V{W)+X{W) (3.13) satisfying the initial condition U(S,G) = U(S,G)-K, Se[0,Smax], (3.14) and boundary conditions dU U(0,t)=0, —(Smax,t) = 0, te[0,T\. (3.15) oS We use the Crank-Nicolson scheme for temporal discretization. Let Men, t = U =W, l = 0,...,M, (3.16) and let us denote Ul(S) = U(S,tl), fi(S)=f(W(S,tl)). (3.17) The Crank-Nicolson scheme has the form r 2 2 2 2 2 1 ' J for Z = 0, ...,M-1. Let 'I'1, be a smooth enough wavelet basis for the space VL (0,Smax), ^/f be its finite-dimensional subset with s levels of wavelets, and VSL = span^. We define a{u,v) = (V{u),v) + (l{u),v}, (3.19) for all u,v G V/', s > 1. The Galerkin method consists in finding U,s+1 G VSL such that fi + fi+i + v 0 7 + r ' >) (3-20) r 2 r 2 \ 2 for all -y G V^. If we set v = ip^ G ^f' and we expand ř7/*+1 in a basis 'J'f, i.e. t/?+1 = (3.21) then the vector of coefficients us = {usx} is the solution of the system of linear algebraic equations Asus = fs, where and f-%^ + %^ + (^,*,.). (3,3, Cubic spline wavelets with four vanishing moments 17 It is clear that fs and us depend on the time level ti, but for simplicity we omit the index I. For preconditioning we use the Jacobi diagonal preconditioner Ds, where the diagonal elements of Ds satisfy DA>A = yjA*x>x. (3.24) We obtain the preconditioned system Asus = fs (3.25) with As = (Ds)_1 As (Ds)_1 , fs = (Ds)_1fs, us = Dsus. (3.26) It is well-known that due to the compact support of the wavelets and a hierarchical structure of the wavelet basis the matrices arising from discretization of the differential operator T> have so-called finger-band pattern.44 Hence, we focus on the properties of the matrix Cs with entries C*;A = (z(^a),^), ^A,^G*f. (3.27) For the standard Galerkin method with the standard spline basis such matrix is full. However, it is known that for integral equations with some types of kernels and for wavelet bases with vanishing moments many entries of discretization matrices are small and can be thresholded and the matrices arising from discretization of the integral term can be approximated with a matrix that is sparse or quasi-sparse.3'12 The following theorem provides the decay estimates for the entries of the matrix Cs corresponding to a general wavelets with L vanishing moments. Theorem 3.1. Let and ifjjj be wavelets with L vanishing moments, i.e. the conditions i) — iv) from Definition 1.1 are satisfied, and let us denote Qiyk = supp-0i)fc, = supply, fti,j,k,i = £li,k x tijj. (3.28) Let e be some small parameter such that 0 < e < Smax and let us denote ne = {(S,y):S^y,S,ye [0, Smax]}\ [0, e]2 . (3.29) // interior(Qitjtk,i) C (3.30) then (x(^a),^)| itk {S)i/>jtl (y)dSdy = 0 ,j,k ,1 and Q (S, V) i>i,k (S) tpjti (y) dS dy = 0. (3.35) (3.36) ,j,k,l Due to the property ii) from Definition 1.1 there exists a constant C\ independent on i, j, k, I such that \S - Slyk\L < CX2~L\ \y - yjtl\L < CX2~LK (3.37) From (2.25) and (2.29) there exists a constant C2 independent on i,j, k,l such that \A,k (S)\dS < C22~1/2, / \i/>jtl (y)\dy < C22~^' (3.3S Hence, we have K (S, y) tpitk (S) tpjti (y) dS dy 3,k,l (3.39)