Spline Wavelets in Numerical Resolution of Partial Differential Equations Jianzhong Wang1 Department of mathematics, Computer Sciences, and Statistics Sam Houston State University Huntsville, Texas, USA Abstract. We give a review of applications of spline wavelets in the resolution of partial differential equations. Two typical methods for numerical solutions of partial dif- ferential equations are Galerkin method and collocation method. Corresponding to these two methods, we present the constructions of semi-orthogonal spline wavelets and semi- interpolation spline wavelets respectively. We also show how to use them in the numerical resolution of various partial differential equations. 1 Introduction In this paper, we show how wavelets, particularly the spline wavelets, can be used in the nu- merical resolution of partial differential equations (PDE’s). To solve a PDE numerically, we first need to find a finite-dimensional approximation space for the solutions, then discretize the PDE to a system of algebraic equations in this space so that the numerical solutions can be obtained. In many cases, the wavelets provide better bases of the approximation spaces than other bases in the following sense. First, the representations of the differen- tial operators are almost diagonal on the wavelet bases, that improves the conditioning of the discrete algebraic equations. Second, the wavelet representations are effective in the adaptive procedures so that the complexity of calculations can be reduced. Furthermore, when the solution has a certain singularity, its wavelet representation can automatically capture the singularity. Hence, the advantage of wavelet methods in numerical resolution of PDE’s is quite significant. However, in general, the wavelets do not have a localization as sharp as finite element method or finite difference method. Hence, the sparsity of the matrices associated to the discrete equations based on wavelet bases is no better than the sparsity of the matrices obtained from finite element method or finite difference method. The readers have to consider with more care to select the methods in the resolution of an individual problem. In this paper, we only introduce spline wavelets. We select spline wavelets because they have very simple structure and useful properties. Other wavelets, such as Daubeshies’ 1This paper is supported by SHSU FRC 1998. 1 wavelets, coiflets, etc., can play the similar role in the numerical resolution of PDE’s as spline wavelets. This paper is written as a brief survey. All results in this paper are given concisely, and their proofs are either sketched or referred to the references. The outline of the pa- per as follows. In Section 2, we give the constructions of 1-D and 2-D semi-orthogonal spline wavelets. In Section 3, we introduce the semi-interpolation spline wavelets and their properties. The numerical solutions of Dirichlet boundary problems using wavelets will be contained in Section 4. Finally, in Section 5, we discuss the adaptive wavelet method for numerical solutions of evolution equations. 2 Semi-Orthogonal Spline Wavelets Orthonormal or semi-orthogonal wavelets are often utilized in the Galerkin method for the resolution of PDE’s. It is well-known that the useful orthogonal or semi-orthogonal wavelet bases are obtained from multiresolution analyses. We first discuss one dimensional case. 2.1 Semi-Orthogonal Spline Wavelets on R The materials in this subsection are well-known, we present them here for the readers’ convenience. Definition 2.1 A multiresolution analysis (MRA) of L2(R) is a nest of closed subspaces 2 (Vj)j2Z of L (R) such that 2 (1) \Vj = f0g; and [Vj is dense in L (R): (2) f(x) 2 Vj () f(2x) 2 Vj+1: (3) There is a function Á 2 V0; such that fÁ(¢ ¡ k)gk2Z forms a Riesz basis of V0: The function Á in Definition 2.1 is called the generator (or the scaling function) of the MRA. We now set j=2 j Ájk(x) = 2 Á(2 x ¡ k) and Ák = Á0k: Then fÁjkgk2Z forms a Riesz basis of Vj: The wavelet subspace Wj is defined as the orthogonal complement of Vj with respect to Vj+1; Vj+1 = Vj © Wj Wj ? Vj: (1) 2 2 The space L (R) therefore has a decomposition L (R) = ©jWj: It has been proved that there exists a function à 2 W0 such that fÃjkgk2Z forms a basis of Wj, and therefore 2 fÃjkg(j;k)2Z2 is a basis of L (R). The function à is called a semi-orthogonal wavelet since 2 0 < Ãjk;Ãj0k0 >= 0; j 6= j : 2 We call fÃjkg(j;k)2Z2 a wavelet basis of L (R): Similarly, à is called an orthonormal wavelet if < Ãjk;Ãj0k0 >= ±jj0 ±kk0 ; and the basis fÃjkg(j;k)2Z2 created by an orthonormal wavelet à is called an orthonormal wavelet basis of L2(R): Remark 2.1 Sometimes, for much flexibility, the wavelet subspace Wj defined by (1) is not required to satisfy Wj ? Vj: Then the corresponding wavelet à is not semi-orthogonal. In this case, a function ä 2 L2(R) is call the dual wavelet of à with respect to L2(R) if ¤ < Ãjk;Ãj0k0 >= ±jj0 ±kk0 : The pair of (Ã; ä) is said to be biorthogonal. In this paper we do not discuss the biorthog- onal wavelets. Readers can refer to [7]. Remark 2.2 The multiresolution analyses and wavelet bases of the spaces other than L2(R) can be defined in a similar way as above. Since this kind of generalization is al- most trivial, we will not give those definitions in the paper. It is clear that the scaling function Á satisfies a refinement equation X Á(x) = pkÁ(2x ¡ k) 1 P k where the sequence (pk) is called the mask of Á; and the Laurent series p(z) = 2 pkz is called the symbol of Á: In most of applications, Á is required to be compactly supported or to decay exponentially. Splines are good examples for scaling functions. Let  be the characteristic function of the unit interval [0; 1): The function z }|m { bm =  ¤  ¤ ¢ ¢ ¢ ¤  is called the cardinal B-spline of order m: In applications, we often use the splines with even order. Hence, in this paper, we always assume that the spline order is even. We sometimes also omit the subscript m from the notations such as bm(x) if no vagueness arises. The Fourier transform of b(x) is µ ¶ 1 ¡ e¡i! m ˆb(!) = : ! 3 It is easy to verify that b(x) satisfies the refinement equation m µ ¶ X m b(x) = 2¡m+1 b(2x ¡ k): (2) k k=0 The B-spline b(x) generates an MRA of L2(R). The construction of the semi-orthogonal spline wavelet can be found in our paper [6]. We now quote the results in [6] below. ² Semi-orthogonal spline wavelet basis and its dual. P2m¡2 k Let Πm(z) := k=0 b2m(k+1)z , which is the well-known (normalized) Euler-Frobenius i(m¡1)! ¡i! Ppolynomial of order 2m¡2: By the Poisson’s summation formula, we have e Πm(e ) = ˆ 2 k jb2m(! + 2k¼)j . Then the following proposition holds. Proposition 1 Let b be the cardinal B-spline of order m, and the function w be defined by 1 ¡ e¡i!=2 wˆ(!) = ( )mΠ (e¡i!=2)ˆb(!=2) (3) 2 m Then w is a semi-orthogonal wavelet with respect to b: A function f 2 L2(R) can be decomposed into a wavelet series in two ways. One is in the bi-infinite form of X1 X1 f(x) = djkwjk; (4) j=¡1 k=¡1 and another one is in the form of X1 X1 X1 f(x) = cJkbJk + djkwjk: (5) k=¡1 j=J k=¡1 To decompose a function into its wavelet series by using semi-orthogonal wavelet basis, we need to construct the dual scaling function of b and the dual wavelet of w respectively. A ¤ scaling function b 2 V0 is called the dual of b with respect to V0 if ¤ < bk; bk0 >= ±kk0 ; 2 ¤ where < f; g > is the inner product in L (R). Similarly, a function w 2 W0 is call the dual of w with respect to W0 if ¤ < wk; wk0 >= ±kk0 : It is known that function b¤ is determined by ˆb(!) b¤ b (!) = ¡i! : jΠm(e )j Therefore, we have the following (see [6]). 4 1+z m Πm(z) 1¡z m 1 Proposition 2 Let G(z) = ( ) 2 and H(z) = ¡( ) 2 : Then the dual pair 2 zΠm(z ) 2 zΠm(z ) (b¤; w¤) satisfies the following two-scale relations. bb¤(!) = G(e¡i!=2)bb¤(!=2); wc¤(!) = H(e¡i!=2)bb¤(!=2): ¤ The coefficients djk and cJk in (4) and (5) can now be obtained by djk =< f; wjk > and ¤ cJk =< f; bJk > respectively. Thus, we have X1 X1 ¤ f(x) = < f; wjk > wjk j=¡1 k=¡1 and X1 X1 X1 ¤ f(x) = < f; bJk > bJk + < f; wjk > wjk: k=¡1 j=J k=¡1 The fast wavelet transform algorithms can be found in [12], [11], and [8]. ² Orthonormal spline wavelet basis. The construction of orthonormal spline wavelets can be found in [11]. Let the function ´ is determined by ˆb(!) ´ˆ(!) = p : jΠm(!)j Then ´ 2 V0 is an orthonormal scaling function in the sense that < ´k; ´k0 >= ±kk0 : The orthonormal wavelet » 2 W0 is defined by p 1 ¡ ei!=2 jΠ (¡e¡i!=2) »ˆ(!) = ¡e¡i!=2( )m p m ´ˆ(!=2); ¡i! 2 Πm(e ) which leads to < »jk;»j0k0 >= ±jj0 ±kk0 : Both ´ and » exponentially decay. It is clear that f 2 L2(R) can be decomposed into X1 X1 f = < f; »jk > »jk j=¡1 k=¡1 and X1 X1 X1 f(x) = < f; ´Jk > ´Jk + < f; »jk > »jk k=¡1 j=J k=¡1 respectively.