Spline Wavelets in Numerical Resolution of Partial Differential

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 differential 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 numerical 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 differential 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 paper 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)j∈Z of L (R) such that 2 (1) ∩Vj = {0}, and ∪Vj is dense in L (R). (2) f(x) ∈ Vj ⇐⇒ f(2x) ∈ Vj+1. (3) There is a function φ ∈ V0, such that {φ(· − k)}k∈Z forms a Riesz basis of V0.

The function φ in Deﬁnition 2.1 is called the generator (or the scaling function) of the MRA. We now set

j/2 j φjk(x) = 2 φ(2 x − k) and φk = φ0k. Then {φjk}k∈Z 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 ψ ∈ W0 such that {ψjk}k∈Z forms a basis of Wj, and therefore 2 {ψjk}(j,k)∈Z2 is a basis of L (R). The function ψ is called a semi-orthogonal wavelet since

2 0 < ψjk, ψj0k0 >= 0, j 6= j .

2 We call {ψjk}(j,k)∈Z2 a wavelet basis of L (R). Similarly, ψ is called an orthonormal wavelet if

< ψjk, ψj0k0 >= δjj0 δkk0 , and the basis {ψjk}(j,k)∈Z2 created by an orthonormal wavelet ψ 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 ψ is not semi-orthogonal. In this case, a function ψ∗ ∈ L2(R) is call the dual wavelet of ψ with respect to L2(R) if

∗ < ψjk, ψj0k0 >= δjj0 δkk0 . The pair of (ψ, ψ∗) is said to be biorthogonal. In this paper we do not discuss the biorthogonal 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 almost trivial, we will not give those deﬁnitions in the paper.

It is clear that the scaling function φ satisﬁes a reﬁnement equation X φ(x) = pkφ(2x − k)

1 P k where the sequence (pk) is called the mask of φ, and the Laurent series p(z) = 2 pkz is called the symbol of φ. In most of applications, φ is required to be compactly supported or to decay exponentially. Splines are good examples for scaling functions. Let χ be the characteristic function of the unit interval [0, 1). The function

z }|m { bm = χ ∗ χ ∗ · · · ∗ χ 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−iω m ˆb(ω) = . ω

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. • 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)ω −iω polynomialP of order 2m−2. By the Poisson’s summation formula, we have e Πm(e ) = ˆ 2 k |b2m(ω + 2kπ)| . 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−iω/2 wˆ(ω) = ( )mΠ (e−iω/2)ˆb(ω/2) (3) 2 m Then w is a semi-orthogonal wavelet with respect to b. A function f ∈ L2(R) can be decomposed into a wavelet series in two ways. One is in the bi-inﬁnite form of X∞ X∞ f(x) = djkwjk, (4) j=−∞ k=−∞ and another one is in the form of X∞ X∞ X∞ f(x) = cJkbJk + djkwjk. (5) k=−∞ j=J k=−∞ 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 ∈ V0 is called the dual of b with respect to V0 if ∗ < bk, bk0 >= δkk0 , 2 ∗ where < f, g > is the inner product in L (R). Similarly, a function w ∈ W0 is call the dual of w with respect to W0 if ∗ < wk, wk0 >= δkk0 . It is known that function b∗ is determined by

ˆb(ω) b∗ b (ω) = −iω . |Πm(e )| 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∗(ω) = G(e−iω/2)bb∗(ω/2), wc∗(ω) = H(e−iω/2)bb∗(ω/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

X∞ X∞ ∗ f(x) = < f, wjk > wjk j=−∞ k=−∞ and X∞ X∞ X∞ ∗ f(x) = < f, bJk > bJk + < f, wjk > wjk. k=−∞ j=J k=−∞ The fast wavelet transform algorithms can be found in [12], [11], and [8]. • Orthonormal spline wavelet basis. The construction of orthonormal spline wavelets can be found in [11]. Let the function η is determined by ˆb(ω) ηˆ(ω) = p . |Πm(ω)|

Then η ∈ V0 is an orthonormal scaling function in the sense that

< ηk, ηk0 >= δkk0 .

The orthonormal wavelet ξ ∈ W0 is deﬁned by p 1 − eiω/2 |Π (−e−iω/2) ξˆ(ω) = −e−iω/2( )m p m ηˆ(ω/2), −iω 2 Πm(e ) which leads to

< ξjk, ξj0k0 >= δjj0 δkk0 . Both η and ξ exponentially decay. It is clear that f ∈ L2(R) can be decomposed into X∞ X∞ f = < f, ξjk > ξjk j=−∞ k=−∞ and X∞ X∞ X∞ f(x) = < f, ηJk > ηJk + < f, ξjk > ξjk k=−∞ j=J k=−∞ respectively.

5 2.2 Semi-Orthogonal Periodic Spline Wavelet Bases Meyer in [12] introduced the constructions of orthonormal periodic wavelet bases (also see [10]). Following his method, we can construct semi-orthogonal periodic spline wavelet bases. we consider the wavelet bases in the space of periodic functions

Z 1 L˜2 = {f; f(· + 1) = f, f 2dx < ∞}. 0 Let b, b∗, w, w∗, η, and ξ be the functions deﬁned in the preceding subsections. Proposition 3 For j ≤ 0, let X X p ∗p ∗ wjk = wjk(x − l), wjk = wjk(x − l). l∈Z l∈Z

p j Then the constant function and {wjk; k = 0, ··· , 2 − 1, j ≥ 0} form a semi-orthogonal ˜2 ∗p j basis of L , while the constant function and {wjk; k = 0, ··· , 2 − 1, j ≥ 0} form the dual p j basis with respect to {wjk; k = 0, ··· , 2 − 1, j ≥ 0}. Similarly, let X p ξjk = ξjk(x − l). l∈Z

p j Then the constant function and {ξjk; k = 0, ··· , 2 − 1, j ≥ 0} form an orthonormal basis of L˜2. A function f ∈ L˜2 now can be decomposed into

j Z 1 X∞ 2X−1 ∗p p f = f(x)dx + < f, wjk > wjk 0 j=0 k=0 or

j Z 1 X∞ 2X−1 p p f = f(x)dx + < f, ξjk > ξjk 0 j=0 k=0

2.3 Semi-Orthogonal Spline Wavelet bases on ﬁnite intervals We now construct semi-orthogonal bases of the space L2(I), where I is a ﬁnite interval. Without loss of generality, we assume I = [0,N],N ∈ N. Let {Vj}j∈Z be the MRA in Subsection 2.1, which is generated by the cardinal B-spline I 2 b of order m. Let Vj ⊂ L (I) be the subspace which contains the restrictions to I of the I 2 j functions in Vj. Then {Vj }j≥0 is an MRA of L (I) and dim Vj = m+2 N −1. To construct I −j the basis of Vj, we set bjk = bjkχI , and let Sj = {2 k; suppbjk ∩ I 6= ∅}. It is clear that |Sj| = dim Vj. We have the following.

6 I I −j Proposition 4 {bjk}2 k∈Sj forms a basis of Vj , and there are two positive constants C1 and C2 independent of j such that X X X 2 I 2 2 C1 |cjk| ≤ || cjkbjk|| ≤ C2 |cjk| . −j −j −j 2 k∈Sj 2 k∈Sj 2 k∈Sj

In the resolution of PDE’s, the solutions are sometimes required to satisfy certain homogeneous boundary conditions. Thus, we need the wavelet bases in the space

s s−1 (s−1) 2 H0 (I) = {f ∈ C0 (I); f ∈ L (I), 0 ≤ s ≤ m − 1.

0 2 s (We agree that H0 (I) = L (I).) Since the truncated B-splines are not in the space H0 (I), s ≥ s 1, we have to modify them in the construction of the wavelet bases of H0 (I). The B-splines with multiple knots are the appropriate tools for the modiﬁcation. Let [x1, ··· , xn]f be the divided diﬀerence of function f at x1, ··· , xn. We denote the knot sequence

m z }|m { z }| { 0, 0, ··· , 0, 1, 2, ··· , N,N, ··· ,N

m+N−1 by {ti}i=−m+1, and deﬁne

I m−1 m m φ (x) = [tk− m , tk+1, ··· , tk+ m ](· − x) , − + 1 ≤ k ≤ N − 1 + . (6) k 2 2 + 2 2

I I m m The support of φ is [t m , t m ], and φ (x) = b m (x), ≤ k ≤ N − . For convenience, k k− 2 k+ 2 k k− 2 2 2 s I ∞ the MRA of H0 (I) is still denoted by {Vj }j=0. We have the following.

m I N−1−s+ 2 I m+N Proposition 5 Φ0 := {φk} m is a basis of V0 , 0 ≤ s ≤ min(m − 1, [ ] − 1). k=s+1− 2 2

m m I I m When 2 ≤ k ≤ N − 2 , φk is the shift of central B-spline, that is, φk(x) = b(x − k + 2 ). I I To construct the bases of Vj , we dilate the scaling functions in Φ0. We deﬁne φjk(x) = j/2 I j m m j m j m 2 φk(2 x) for s + 1 − 2 ≤ k ≤ 2 − 1 and 2 N − 2 + 1 ≤ k ≤ 2 N − 1 − s + 2 , deﬁne I j/2 j m m j m φjk(x) = 2 b(2 x − k − 2 ) for 2 ≤ k ≤ 2 N − 2 . Then

j m I 2 N−1−s+ 2 Φj := {φjk} m (7) k=s+1− 2

I L −j m m C −j m j m is a basis of Vj . Write Sj = 2 {s + 1 − 2 , ··· , 2 − 1},Sj = 2 { 2 , ··· , 2 N − 2 }, R −j j m j m L I −j L Sj = 2 {2 N − 2 +1, ··· , 2 N −1−s+ 2 }. Then, Φj := {φjk; 2 k ∈ Sj } contains the left R I −j R boundary scaling functions in Φj, while Φj := {φjk; 2 k ∈ S } contains the right boundary C I −j C scaling functions in Φj. The central scaling functions are in Φj := {φjk; 2 k ∈ Sj }. When L R L C R s = m − 1, both Φj and Φj are empty. Setting Sj = Sj ∪ Sj ∪ Sj ,we have a one-to-one I −j mapping Sφ from Φj to Sj such that Sφ(φjk) = 2 k ∈ Sj.

7 s • Semi-orthogonal wavelet bases of space H0 (I).

I s I I I I I Let Wj be the wavelet subspace of H0 (I) such that Wj ⊕ Vj = Vj+1 and Wj ⊥Vj . ¯ I ¯ ¯ Setting Sj = (Sj+1 \ Sj) ∩ [0,N], we have dim Wj = |Sj|. The set Sj is symmetric with N ¯ ¯ respect to 2 , i.e., if d ∈ Sj, then N −d ∈ Sj. Without loss of generality, we only discuss the construction of the wavelet basis of W I . (For convenience, any subscript 0 will be omitted.) I I It is clear that dim W = min(N, (2N + m − 2s − 1)+). The bases of W are not unique. In applications, two main properties are often required. First, the supports of the elements in the wavelet bases of W I are required as small as possible. Second, the bases are expected to have certain symmetry. We now decompose the set S¯ into the form of

S¯ = S¯L ∪ S¯C ∪ S¯R, (8)

¯C 1 1 1 ¯L ¯ ¯R ¯L where S = {m − 2 , m + 2 , ··· ,N − m + 2 }, S = S ∩ (0, m − 1], and S = N − S = S¯ ∩ [N − m + 1,N). (If N < 2m − 1, then S¯C = ∅.) It can be veriﬁed that the matrix

I I M = (< φl , φ1k >) k l∈S, 2 ∈S1 is a full-rank matrix.. The linear equation Md = 0 has N linear independent solutions d1, ··· dN . We now write S¯ = {k , ··· , k }, where k < ··· < k , and set ψI = (dl)T Φ . 1 N 1 N kl 1 I I 1 N Then {ψd}d∈S¯ is a basis of W . By carefully selecting√ vectors d , ··· , d , we can make I I ¯L I 1 ¯C ψd(x) = ψN−d(N − x) for d ∈ S and ψd(x) = 2w(2(x − d) + m − 2 ) for d ∈ S , where w is deﬁned by (3).

m−1 Example 2.1 Consider the wavelet bases of the space H0 (I). In this case, The bases W I can be constructed as follows.

I I I I I 1. If N < m − 1, then V = {0} and V1 = W . A basis of V1 is also a basis of W . ¯ N k N−m+1+k 2. If m ≤ N ≤ 2m − 2, then, for any d ∈ S with d ≤ 2 , we can ﬁnd (ql )l=k such that the function N−Xm+k−1 I k ψd = ql φ1l, k = 2d (9) l=k

I I I I is orthogonal to the space V1 and ||ψd||2 = 1. We deﬁne ψd(x) = ψN−d(N − x) for ¯ N I I d ∈ S with d > 2 . Then {ψd}d∈S¯ is a basis of W . ¯L m m 1 m 1 m+1 m+3 2m−3 3. If N ≥ 2m − 1, we have S = { 4 , 4 + 2 , ··· , 2 − 2 } ∪ { 2 , 2 , ··· , 2 }. Let K = min{N, 3m − 3}. We can verify that the matrix

m m K− 2 ,K+ 2 −1 M = (<φk, φ1l >) m (10) k,l= 2

8 is a full-rank matrix. Therefore, the equation Mc = 0 has m−1 independent solutions cl, 1 ≤ l ≤ m − 1. We can choose cl so that its last (m − 1 − l) components are zero. I l Let ψd = Φ1c , where ½ m l−1 m 4 + 2 , 1 ≤ l ≤ 2 d = 1 m . l − 2 , 2 + 1 ≤ l ≤ m − 1

I Then the support of ψd is K + l K − 1 + m suppψI = [0, ] ⊂ [0, ]. d 2 2 √ I I ¯R I 1 Let ψd(x) = ψN−d(N − x) for d ∈ S and let ψd(x) = 2w(2(x − d) + m − 2 ), for ¯C I I d ∈ S . Thus, the set of {ψd}d∈S¯ forms a basis of W . Remark 2.3 If I is a dyadic interval, say I = [2−J k, 2−J (k + l)] for a fixed J, the con- I I struction of the scaling function basis of Vj and the wavelet basis of Wj is quite similar. ¯ I ¯ We define the sets Sj and Sj in the same way as above. Then we still have dim Wj = |Sj|. I An evident modification needs to make for the construction of the wavelet basis of Wj .

I I We now consider the dual wavelet bases. Since the spaces Vj and Wj both are ﬁnite, I I the dual basis of Φj with respect to Vj and the dual basis of Ψj with respect to Wj can I I be obtained using the Gram-Schmidt matrices generated by Φj = {φjk} and Ψj = {ψjk} respectively. For example, when j = 0, Let

I I Gψ = (< ψd, ψd >)(d0,d)∈S¯, I I Gφ = (< φd, φd >)(d0,d)∈S.

∗ ∗ ∗ −1 I Since Gψ and Gφ both are invertible, the vector Ψ = (ψd)d∈S¯ defined by Ψ = Gψ Ψ I I ∗ ∗ forms a dual basis of Ψ with respect to W , and the vector Φ = (φd)d∈S defined by ∗ −1 I I ∗ ∗ Φ = Gφ Φ forms a dual basis of Φ . Although the dual bases Ψj and Φj are no longer ∗ ∗ compactly supported with respect to I, any element in Ψj or in Φj exponentially decays within I. Before we end this subsection, we briefly discuss how to construct semi-orthogonal s wavelet bases of the space H0 (Γ), where Γ is the union of finite disjointed intervals. We shall use them in the construction of the semi-orthogonal wavelet bases on an arbitrary domain in R2. j −j ¯ Γ −j j Let S = {2 k; suppφj,k ⊂ Γ} and Vj = span{φj,k; 2 k ∈ S }. Then

Γ Γ Γ V0 ⊂ V1 ⊂ · · · ⊂ Vj ⊂ · · · (11)

s j S is an MRA of H0 (Γ). If we set Γ = 2−j k∈Sj supp(φj,k), The MRA

Γ0 Γ1 Γj V0 ⊂ V1 ⊂ · · · ⊂ Vj ⊂ · · ·

9 ΓJ Γ is the same as the MRA (11). Hence, Wj = Wj . For a ﬁxed level j, there exist ﬁnite intervals Iλ such that Γj = ∪λIλ. It is obvious that −j j the end-point of each interval Iλ is in the dyadic form of 2 k. Let Ψλ be the wavelet basis Iλ j Iλ j j of Wj (see 2.3) and Φλ be the scaling function basis of Vj . Then Ψ = ∪λΨλ forms the ΓJ j j ΓJ wavelet basis of Wj and Φ = ∪λΦλ forms the scaling function basis of Vj . Note that, when Γ is given, there is an one-to-one mapping form S¯j to Ψj (from Sj to Φj).

2.4 Semi-Orthogonal Spline Wavelet on an arbitrary domain We now construct the semi-orthogonal spline wavelet bases on an arbitrary domain in R2 by tensor product. For high dimensions, the discussion is similar. Let Ω be an arbitrary boundary domain in R2. We always assume that the boundary of Ω, say ∂Ω, is a Lipschitz curve. If Ω is a rectangle, then we can directly use tensor s product to construct the semi-orthogonal spline wavelet basis on H0 (Ω). (We now assume that 1 ≤ s ≤ m − 1.) Since it is very straight forward, we will not discuss it in detail. The s readers can refer to [8]. We now consider the spline wavelet basis of H0 (Ω) for the domain Ω other than the rectangle. For illustration, we only discuss the case of s = m − 1. For other s, the discussion is similar except that more complicated indices will be introduced. −j −j ¯ Ω Let Sj = {(2 k, 2 l); supp(φ(j,k,l)) ⊂ Ω}, where φ(j,k,l)(x, y) = φjk(x)φjl(y). Let Vj = −j −j span{φ(j,k,l);j (2 k, 2 l) ∈ Sj}. Then

Ω Ω Ω V0 ⊂ V1 ⊂ · · · ⊂ Vj ⊂ · · ·

m−1 Ω −j −j Ω is an MRA of H0 (Ω) and Φj := {φ(j,k,l);j (2 k, 2 l) ∈ Sj} is a basis of Vj , which Ω contains all scaling functions of level j. The wavelet subspace Wj−1 of this MRA is deﬁned Ω Ω Ω Ω Ω Ω by Wj−1⊥Vj−1 and Wj−1 ⊕ Vj−1 = Vj . To construct a wavelet basis of Wj−1, we set [ Ωj = supp(φ(j,k,l)) −j −j (2 k,2 l)∈Sj

0 0 and deﬁne the y-cut and x-cut of Ωj at (d, d ) ∈ Sj by Γd0 = {x;(x, d ) ∈ Ωj} and Γd = {y;(d, y) ∈ Ωj} respectively. It is clear that either Γd0 or Γd is an union of ﬁnite 0 0 d d Γd0 disjointed dyadic intervals. Let Φj−1 and Ψj−1 be the bases of the subspaces Vj−1 and 0 0 0 0 Γd0 d 0 ¯d d d Wj−1 respectively. Write Sj = {d;(d, d ) ∈ Sj} and Sj−1 = Sj \ Sj−1, then there is d0 d0 d0 ¯d0 an one-to-one mapping from Φj−1 to Sj−1 and an one-to-one mapping from Ψj−1 to Sj−1. d d Γd Γd Similarly, let Φj−1 and Ψj−1 be the bases of the subspaces Vj−1 and Wj−1 respectively. d 0 0 ¯d d d Write Sj = {d ;(d, d ) ∈ Sj} and Sj−1 = Sj \ Sj−1. Then there is an one-to-one mapping d d d ¯d from Φj−1 to Sj−1 and an one-to-one mapping from Ψj−1 to Sj−1.

10 We now write ¯h 0 0 ¯d0 0 d Sj−1 = {(d, d ); (d, d ) ∈ Sj & d ∈ Sj−1 & d ∈ Sj−1}, ¯v 0 0 d0 0 ¯d Sj−1 = {(d, d ); (d, d ) ∈ Sj & d ∈ Sj−1 & d ∈ Sj−1}, ¯diag 0 0 ¯d0 0 ¯d Sj−1 = {(d, d ); (d, d ) ∈ Sj & d ∈ Sj−1 & d ∈ Sj−1}, and let h d0 d 0 ¯h Ψj−1 = {ψ(x)φ(y); ψ ∈ Ψj−1, φ ∈ Φj−1, (d, d ) ∈ Sj−1, v d0 d 0 ¯v Ψj−1 = {φ(x)ψ(y); φ ∈ Φj−1, ψ ∈ Ψj−1, (d, d ) ∈ Sj−1, diag d0 d 0 ¯diag Ψj−1 = {ψ1(x)ψ2(y); ψ1 ∈ Ψj−1, ψ2 ∈ Ψj−1, (d, d ) ∈ Sj−1 .

h v diag Ω h,Ω h Then Ψj−1 ∪ Ψj−1∪Ψj−1 forms a basis of the subspace Wj−1. Let Wj−1 = span(Ψj−1), v,Ω v diag,Ω diag v,Ω v,Ω diag,Ω Wj−1 = (Ψj−1), and Wj−1 = span(Ψj−1 ). Then the spaces Wj−1,Wj−1,Wj−1 , and Ω Vj−1 are mutually orthogonal. Since all of these spaces are ﬁnite, the dual bases of h v diag Ω Ψj−1, Ψj−1, Ψj−1 and Φj−1 can be obtained by the Gram-Schmidt orthogonal procedure.

3 Semi-interpolation Spline Wavelets

In collocation method, the solutions of PDE’s are represented by their sample data. Hence, we often assume that the solution is in L∞. Therefore, the orthogonal structure is not suitable for this model. Instead, the interpolation scheme is a powerful tool for the method. It leads to the semi-interpolation spline wavelets.

3.0.1 Semi-Interpolation Spline Wavelets on R Let Cu(R) be the space of the functions, which are bounded and uniformly continuous on R, equipped with the uniform norm ||||∞ . The MRA and the corresponding wavelet u structure in C (R) can be found in [3], and [20]. The cardinal B-spline bm generates an MRA of Cu, that is, the nest of subspaces X j ∞ Vj = { ckbm(2 x − k); (ck) ∈ l }, (12) k∈Z

u u forms an MRA of C in the following sense: C = ∪Vj and ∩Vj = R. We deﬁne the u −j −j interpolation operator Ij from C to Vj : Ij(f) = sf by sf (2 k) = f(2 k). It is known that the interpolation is unique if and only if the order m of the B-spline bm is even. The following theorem is well-known.

u Theorem 3.1 For any f ∈ C , limj→∞ ||f − Ijf||∞ = 0, and the approximation order is m.

11 The wavelet subspace Wj of Cu can be deﬁned in the way that Vj ⊕ Wj = Vj+1 ¯ u and Ij(g) = 0, ∀g ∈ Wj. Let Ij be the interpolation operator from C to Wj such that ¯ −j−1 −j−1 ¯ Ij(f)(2 (2k − 1)) = f(2 (2k − 1)), k ∈ Z. Therefore, Ij+1 = Ij + Ij(I − Ij) = Ij + Hj, ¯ where Hj = Ij(I − Ij). Let X v(x) = (−1)k−1b(k − 1)b(2x − k). k∈Z

j Then {v(2 x − k)}k∈Z is a basis of Wj. We have suppv = [0, m − 1].

3.0.2 Semi-Interpolation Periodic Spline Wavelet Bases The semi-interpolation periodic spline wavelets can be constructed in the similar way as we have done for semi-orthogonal ones. Let

C˜ = {f; f(· + 1) = f, f ∈ Cu}.

p P p j Deﬁne vjk(x) = l∈Z vjk(x−l). Then the constant and {vjk; k = 0, ··· , 2 −1, j ≥ 0} form a basis of C.˜ A function f ∈ C˜ can be decomposed into

X∞ 2Xj −1 p f = f(0) + djkvjk. j=0 k=0

p u We can ﬁnd the dual basis of {vjk} in the sense of C ,which can be used to calculate the p coeﬃcients {djk}. However, in practice, we get {djk} using interpolation scheme. Let Ij ˜ p ¯p be the interpolation operator from C to Vj , and Ij be the interpolation operator from ˜ p ¯ −j−1 j ¯ ¯ C to Wj . Let Sj = {2 (2k + 1); k = 0, ··· , 2 − 1}, fj = f(Sj), and dj = (djk). Let Pj−1 P2i−1 p fj(x) = f(0) + i=0 k=0 dikvik Then we have ¯ ¯ Gjdj = fj − fj(Sj),

¡ ¢ j where G = vp ( 2l+1 ) 2 −1 is invertible. j jk 2j+1 k,l=0

3.0.3 Semi-Interpolation Spline Wavelet Bases On Finite Intervals The structure of semi-interpolation spline wavelet bases on ﬁnite intervals is simpler than I s that of semi-orthogonal ones. We still use {Vj } to denote the MAR of the space C0(I)(I = I s [0,N]), which is generated by bm, and use Wj to denote the wavelet subspace of C0(I). s+1 I m Then Φj in (7) (for H0 (I)) is still a basis of Vj . We assume that s ≤ 2 −1, which is used ¯ −j−1 in most of applications. Under this assumption, Sj = (2 (2Z + 1)) ∩ [0,N]. Without loss of generality, we only construct the bases for W I (i.e., j = 0.)

12 N 1. If N < m − 1, we set K = {1, ··· , 2N − 1}, I = {1, ··· ,N − 1}, and ΦK = I 1 N {φ1k; k ∈ K}. Then the matrix MK := ΦK (I) is a full-rank matrix. Let d , ··· , d I k be the N independent solutions of the equation MK d = 0. Let ψk = ΦK d . It is I N I N I obvious that ψk(I ) = 0. Hence, {ψk}k=1 is the basis of W . l m l m 2. If m − 1 ≤ N, we set L = {l, ··· , 2l + 2 − 2} and I = {1, ··· , l + 2 − 2} for l = m I 1, ··· , 2 − 1. Then the matrix Ml = (φ1k(i))k∈Ll,i∈Il is a full-rank matrix so that the I P I equation Mlq = 0 has an unique solution (up to a constant). Let ψd = k∈Ll qkφ1k, I m I I d = (2l − 1)/2. Then suppψd = [0, l + 2 − 1]. Deﬁne ψd(x) = ψN−d(x), for d = m−3 1 I m−1 m+1 2N−m+1 {N − 2 , ··· ,N − 2 }, and deﬁne ψd(x) = v(x−d) for d = { 2 , 2 , ··· , 2 }. I I Then {ψd}d∈S¯ is a basis of W . We now develop the wavelet transform algorithms. We agree that, for any function s (−k) f ∈ C0(I), the notation f(Sj) is the vector (f(d))d∈Sj , where f(d) = fj (0) for d = j −j (k−2 N) j j 2 k, k = −s, ··· , −1, and f(d) = fj (N) for k = 2 N +1, ··· , 2 N +s. We also write I I −j ¯ ¯ Φj = {φjk}2 k∈Sj ,Ψj = {ψd}d∈S¯j ,Mj = Φj(Sj),Hj = Φj(Sj), and Gj = Ψj(Sj). We have the following discrete wavelet transform algorithms for semi-interpolations.

0 0 Algorithm 1 (Two-level decomposition.) Let f = Φc +Ψd . Assume that f(S1) is known. Then we have c0 = M −1f(S) ¡ ¢ d0 = G−1 f(S¯) − Hc0 . Algorithm 2 (Two-level Reconstruction.) Assume that the vector c0 and d0 are known. Then f(S) = Mc0 f(S¯) = Gd0 + Hc0 We now assume X XJ−1 2jXN−1 0 j I f(x) = ckφk + dkψj,k(x) (13) k∈S j=0 k=1 Algorithm 3 (Multi-level decomposition.) c0 = M −1f(S) ¡ ¢ d0 = G−1 f(S¯) − Hc0 for j = 1 : J − 1 cj = L cj−1 + K dj−1 j−1¡ j−1 ¢ j −1 ¯ j d = G f(Sj) − Hjc end (14)

13 j where Lj−1 and Kj−1 are two-scale relation matrices generated by the relations Φjc = j−1 j−1 Φj−1c + Ψj−1d .

Algorithm 4 (Multi-level reconstruction.)

for j = 0 : J − 2 j f(Sj) = Mjc ¯ j j f(Sj) = Gjd + Vjc j+1 j j c = Ljc + Kjd end ¯ J−1 J−1 f(SJ−1) = GJ−1d + VJ−1c (15)

Although the vanishing moment property is no longer valid for the semi-interpolation spline wavelets, The coeﬃcients of the semi-interpolation spline wavelet series still indicate the singularity of the functions. The following theorem can be proved in the way we used in [19].

α α Theorem 3.2 Let C (I) be the Lip-α class on I (α ≥ 0) and Cx (I) be the local Lip-α α class at c, that is, we say f ∈ Cc (I) if there is a polynomial P ∈ πn, n = bαc, such that

α f(x) = Pn(x − c) + °(|x − c| ).

Then we have the followings. (1) For any α, 0 ≤ α ≤ m − 1, the function f in (13) is in Cα(I) if and only if −αj |dj,k| ≤ C2 , 1 ≤ k ≤ nj, 0 ≤ j ≤ ∞. α (2)If f ∈ Cc for c ∈ I and 0 ≤ α ≤ m − 1, then

−αj j α |dj,k| ≤ C2 (1 + |2 c − k| ), 1 ≤ k ≤ nj, 0 ≤ j ≤ ∞. (16)

Conversely, if (16) holds and f ∈ Cβ(I), β > 0, then there exists a polynomial P ∈ πn, n = bαc, such that 2 |f(x) − P (x)| ≤ C|x − c|α log |x − c|

These theorems form a foundation of adaptive collocation method for solving PDE’s.

3.0.4 Semi-Interpolation Spline Wavelet Bases On An Arbitrary Domain Let Ω be an arbitrary boundary domain in R2 with a Lipschitz curve boundary ∂Ω. Then we can employee the method we used for semi-orthogonal ones to construct the bases of Ω H0 . Since the method is similar, we will not repeat it here.

14 4 Numerical Resolutions of Dirichlet Boundary Prob- lems

In this section, we will show the application of wavelets in the Galerkin methods. It is known that in the Galerkin methods the discrete system for the numerical resolution of elliptic problems in a bounded domain is ill-conditioned if finite elements or finite difference methods are used. Usually, the conditional number of the discrete system is °(1/h2) for a second order elliptic problem in two dimensions. Using preconditioning methods, we can reduce the conditional number to °(1/h). However, if the wavelet bases are used, the preconditioning yields a °(1) conditional number. Therefore, wavelet methods lead to numerical stabilities for the resolution. Besides, the iterative algorithms are very popular in numerical resolution of PDE’s. The wavelet method can accelerate the convergence of the iterative algorithms. As an example, we consider the following Dirichlet type boundary value problem −∇ · (A∇v) + v = f, f ∈ H1(Ω) v = h, (x, y) ∈ ∂Ω, h2 ∈ H(∂Ω), where Ω is a bounded domain with a Lipschitz boundary, and A(x, y) is a positive matrix. We can homogenize the equation by smoothly extending h to Ω. Setting u = v − h,we have −∇ · (A∇u) + u = g u = 0, (x, y) ∈ ∂Ω, where g = f − ∇ · (A∇h) + h. The variational form of this problem is Z 1 A(u, v) = gv ∀v ∈ H0 (Ω). (17) Ω R where A(u, v)= Ω(A(∇u) · ∇v + uv). Ω Ω We now discretize the equation (17) using the spaces Vj and Wj . Note that the dis- Ω cretizations on Vj for an arbitrary Ω or for a rectangle domain essentially are same, except that the former involves complicated indices. For simplicity, we now assume that 2 Ω Ω = I ,where I is a finite interval. Let uj ∈ Vj be the Galerkin approximation of the solution u. We can expend uj in the terms of the wavelet basis X Xj−1 X I I h I I uj = udd0 φ (x)φ 0 (y) + u 0 φ (x)ψ 0 (y) d d dj dj k k 0 2 0 (d,d )∈S j=0 (dj ,d )∈Sj ×S¯j X j X v I I diag I I + ud d0 ψk(x)ψk0 (y) + u 0 ψd (x)ψd0 (y). j j dj dj j j 0 ¯ 0 ¯2 (dj ,dj )∈Sj ×Sj (dj ,dj )∈Sj

15 1 Ω Let Pj be the orthogonal project from H0 (Ω) to Vj . Then the projection of the function Ω g on Vj has the expansion

X Xj−1 X I I h I I Pjg = gdd0 φ (x)φ 0 (y) + g 0 φ (x)ψ 0 (y) d d dj dj k k 0 2 0 (d,d )∈S j=0 (dj ,d )∈Sj ×S¯j X j X v I I diag I I + gd d0 ψk(x)ψk0 (y) + g 0 ψd (x)ψd0 (y) j j dj dj j j 0 ¯ 0 ¯2 (dj ,dj )∈Sj ×Sj (dj ,dj )∈Sj

j j Let u and g be the vectors¡¡ of the¢ coefficients¡ ¢ of¡ uj ¢and¢ Pjg respectively. Let Bj = Ω Ω Ω ∗ Ω ∗ Ω ∗ Ω ∗ j (Φ , Ψ , ··· , Ψj−1) and Bj = Φ , Ψ , ··· , Ψj−1 . Let A be the matrix defined by µZ ¶ Aj = (A(∇b) · (∇b∗) + bb∗) . Ω ∗ ∗ b∈Bj ,b ∈Bj We have Ajuj = gj. (18) Solving the linear equation (18), we obtain the Galerkin approximation of the solution. Note that the matrix Aj has the conditional number κ = °(22j). But we can add a j very simple preconditioning³ to´ improve the condition number. Let D be the diagonal j 0 ∗ matrix defined by D = 2 δ 0 , where d and d are the indices of the bases B and B dj dj j j j j respectively. Let Aj = (Dj) Bj (Dj) . Then the conditional number of Bj is °(1). (See [10]). The equation (18) can be changed into ¡ ¢ ¡ ¢ Bj Djuj = Dj −1 gj. (19) We can solve Equation (19) by finding the inverse of Bj. In some cases, it costs a lot of time. Hence, sometimes iterative methods are used for solving Equation (17). However, we will face to very slow convergence of the iteration. in some cases. For example, let α(x, y) and β(x, y) be two eigenvalues of A(x, y) in (17). Since A(x, y) is a positive matrix, α and β both are positive functions. If α(x, y) À β(x, y) or β(x, y) À α(x, y), that is, the differential equation (17) is anisotropic, then the iterative methods lead to a very slow convergence. In this case, multi-grid methods are effective because of their fast convergence. The wavelet bases provide a good structure for using multi-grid methods. We now discuss how to solve the equation (17) with α(x, y) À β(x, y) by multi-grid wavelet methods. Using penalty formulation introduced in [14], we can change (17) to new −²uxx − uyy + u = f , (x, y) ∈ Ω, 0 < ² ≤ 1 (20) u = 0 (x, y) ∈ ∂Ω where the unknown function u is a transformation of the original one.

16 • The Two-level Method

Ω We ﬁrst discretize Equation (20) on the space Vj so that (20) becomes a linear system

AjUj = Fj (21)

new Ω where Uj and Fj are the coeﬃcient vectors of uj and Pjf on the basis Φj respectively. Ω Let Aj be the discretization of A(u, v) on Φj . Its smoothing iteration is Lj :

v+1 v v Uj = Lj(Uj ,Fj) = SjUj + TjFj, where Sj is the solver. The formation of Lj is dependent on both Aj and the type of smoothing iterations (Jacobi, Gauss-Seidel or Richardson. See [9].) After v times of it- v 0 erations, we obtain an approximate solution Uj with initial guess Uj . It is known that the smoothing iterations only reduce error components well in the direction of eigenvec- tors, which are corresponding to large eigenvalues. Since in (20) ² is close to 0, the error v v Ej = Uj − Uj may contain not only the low frequency part in both the x and y-directions, but also the high frequency part in x-direction. Hence, the coarse correction should be Ω made for these two parts. Since the scaling subspace Vj−1 contains the coarse version of Uj, h and the wavelet space Wj−1 contains the part which has high frequency in x-direction, we Ω h h choose Vj−1 ∪Wj−1 for the coarse-grid correction. (See [14]). Let Aj−1 be the discretization h of A(u, v) on Ψj−1. The coarse grid correction is

v v ¡ −1 h h −1 h ¢ v Uj ←− Uj + Lj(Aj−1) Rj−1 + Lj (Aj−1) Rj−1 (Fj − AjUj ), (22)

h where Rj−1 and Rj−1 are come from two-level semi-orthogonal spline wavelet decomposi- Ω h tion algorithms, in which Rj−1extracts the coefficients for Φj−1, while Rj−1 extracts the h h coefficients for Ψj−1, and Lj and Lj are come from reconstruction algorithms, which recover Ω Ω Ω,h the coefficients of Φj for the functions in Vj−1 and in Wj−1 respectively. According to Lemma 2.4.2 in [9], the iteration matrix of the two-grid iteration is

¡ ¡ −1 h h −1 h ¢ ¢ v Mj(v) = I − Lj(Aj−1) Rj−1 + Lj (Aj−1) Rj−1 Aj Sj , (23) where Aj is a nonnegative matrix. We now deﬁne the Aj-norm for a matrix B with the same size as Aj by

1/2 −1/2 ||B||Aj = ||Aj BAj ||2. Similar to [14], we can prove the following.

Theorem 4.1 Let Mj(v) be the iteration matrix deﬁned in (23). Assume that the solver v Sj satisﬁes ||Sj ||Aj ≤ 1 and

−j 1/2 v lim 2 ||Aj Sj ||Aj = 0 v→∞

17 holds uniformly for j. Then p

||Mj(v)||Aj ≤ Cη(v) + r(², j), where η(v) → 0, as v → ∞, 0 < r(², j) < σ < 1, and r(², j) → 0 for any ﬁxed j as ² → 0.

The theorem implies that the wavelet iteration method is robust.

• Multilevel Method.

The multilevel method can be derived from two-level method in a straight forward manner. We already know that, for Equation (21), the corresponding two-level method h can be represented as Mj(v) in (23). There are two lower level matrices Aj−1 and Aj−1 in (23). The multilevel method repeats two-level method for these two matrices. For Aj−1, h we can apply the two-level method Mj−1(v). However, for Aj−1, a slight modiﬁcation needs to make. We now employee the wavelet packet structure for the modiﬁcation. In h the wavelet packet structure,(see [21] and [20]), the space Wj−1 is decomposed into four o,h h,h v,h diag,h o,h h subspace Wj−2,Wj−2,Wj−2, and Wj−2 , where Wj−2 contains the functions in Wj−1 with h,h h lower frequency in both x and y-directions, Wj−2 contains the functions in Wj−1 with v,h lower frequency in y-direction and higher frequency in x-direction, Wj−2 contains the func- h tions in Wj−1 with lower frequency in x-direction and higher frequency in y-direction, and diag,h h Wj−2 .contains the functions in Wj−1 with higher frequency in both x and y-directions. o,h h,h o,h h,h Let Aj−2 and Aj−2 be the discretizations of A(u, v) on Wj−2 and Wj−2 respectively. h Then the two-level method for Aj−1 is ³ ³ ´ ´ h o,h o,h −1 o,h h,h h,h −1 h,h h ¡ h ¢v Mj−1(v) = I − Lj−1(Aj−2) Rj−2 + Lj−1(Aj−2) Rj−2 Aj−1 Sj−1 .

o,h o,h h,h h,h The deﬁnitions of the operators Lj−1,Rj−2,Lj−1, and Rj−2 are similar to those for h h Lj,Lj ,Rj−1, and Rj−1 in (22).

5 Adaptive Wavelet Method For Evolution Equations

Because wavelets have good localizations in both space and frequency domains, they can be used for adaptive schemes in multiresolution approximation to obtain solutions which develop singularity. Theorem 3.2 shows that, in the wavelet expansion, the magnitudes of most coefficients are very small, only a small quantity of them will be relatively large. The large coefficients indicate the singularity of the functions. Hence, we can use nonlinear wavelet approximation to dramatically reduce the number of wavelets in the representation. Furthermore, usually the singularity of the solutions in the problems of fluid dynamics develop continuously along with the time. Thus, the wavelet coefficients of the solution at

18 a previous time can be used to predict the positions of the large wavelet coeﬃcients at the current time so that the computational costs will be reduced. Before we apply the adaptive wavelet method in the resolution of evolution problems, we ﬁrst introduce the wavelet adaptive approximation.

5.1 Adaptive Wavelet Approximation. For simplicity, we only consider 1-D approximation. The adaptive schemes are not sensitive to dimensions. Hence, the method we describe here is also applied to high dimensions. Let I s us consider the MRA {Vj } of the space C0(I), which is generated by the B-spline bm(x). I I I I I Let Wj be the wavelet subspace, Φj and Ψj be the bases of Vj and Wj respectively. Since the wavelet components add the details of the function to its “blur” version, the components with small coefficients can be deleted without causing a big error. Recall that, if function f has certain smoothness, for instance f ∈ Lipα, then, by Theorem 3.2, the −jα wavelet coefficient dj,k = °(2 ) tends to 0 as j tends to ∞. Hence we can reduce a quantity of the wavelet coefficients in order to save the operation time and the memory space. More precisely, we have the following. (See [19].) P P P Theorem 5.1 Let f = J−1 g ψI + c φ . For a given ² > 0, there exists J j=0 dj ∈S¯j dj dj k∈S k k a constant cJ , which is only dependent on J, such that if g¯dj is selected by ½ gdj , gdj > cJ ² g¯dj = j = 0, ··· ,J − 1 0, gdj ≤ cJ ² P P P and f¯ = J−1 g¯ ψI + c φ , then J j=0 dj dj k∈S k k ¯ ||fJ − fJ ||C(I) < ².

I ¯ For a function fJ ∈ VJ and a given tolerance ², we say that the knot set Ξj ⊂ Sj is a J−1 feasible set of level j if , for any dj ∈ Ξj, the coeﬃcient gdj > cJ ². The set Ξ(J) = ∪j=0 Ξj is called the feasible set for fJ .

Remark 5.1 According to Theorem 3.2, the feasible set Ξ(J) has the tree structure. Roughly, ¯ if the function f has a certain singularity at x0 ∈ I, then at each level j, the points in Sj, which near x0, are in the feasible set Ξ(J). Thus, a tree in Ξ(J) indicates a singular point of f. Furthermore, if we only use the wavelets corresponding all trees in Ξ(J) to approximate the function f, we will obtain the same approximation order. The tree approximation will save a lot of search time for ﬁnding all points in the feasible set.

19 5.2 Adaptive Wavelet Collocation Methods for 1-D PDE’s We use the wavelet collocation methods to solve time dependent PDE’s. Let u = u(x, t) be the solution of the following initial boundary value problem   u + f (u) = u + g(u), x ∈ [0,N], t ≥ 0  t x xx u(0, t) = g (t) 0 (24)  u(N, t) = g (t)  1 u(x, 0) = f(x). Here only Dirichlet boundary conditions are considered. However the methods can also be modiﬁed to treat Von Neuman type or Robin type boundary conditions. We use cubic B-spline b4 to create the MRA. The numerical solution uJ (x, t) will be represented by a unique decomposition in V0 ⊕ W0 ⊕ · · · ⊕ WJ−1,J − 1 ≥ 0,   X XJ−1 X XJ−1   uJ (x, t) = uˆk(t)φk(x) + uˆdj (t)ψdj (x) := u−1(x) + uj(x), (25)

k∈S j=0 k∈S¯j j=0 where the superscript I is omitted in all notations. We now identify the numerical solution uJ (x, t) by its point values on all collocation points. We put all these values in vector u = u(t), i.e.

u = u(t) = (u(−1), u(0), ··· , u(J−1))>.

To solve the unknown vector u(t), we collocate the PDE (24) on all collocation points and obtain the following semi-discretized wavelet collocation method. Semi-Discretized Wavelet Collocation Methods   (j)  uJ t + fx(uJ ) = R(uJ xx) + g(uJ )|x=x , −1 ≤ j ≤ J − 1  k uJ (0, t) = g0(t)  uJ (L, t) = g1(t)  (j) (j) uJ (xk , 0) = f(xk ) The average operator R on the second derivative is used to take advantage of the super- convergence of the splines at the knot points. However, R should only be used at a local uniform mesh. Equation (5.2) involves a total of (2J − 1)N + 2 unknowns in u; two of them will be determined by the boundary conditions and the rest are the solutions of the ODE system subject to their initial conditions. In order to implement the time marching scheme for the ODE’s system (for example Runge-Kutta type time integrator), we have to compute the derivative term in (5.2) Assuming that the Euler forward diﬀerence scheme is used to discretize the time derivative in (5.2), we obtain a fully discretized wavelet collocation method.

20 Fully discretized Wavelet Collocation Method  n+1 n n n n  uJ = uJ + ∆t[−fx(uJ ) + R(uJ xx) + g(uJ )]| (j) , −1 ≤ j ≤ J − 1  x=xk  n n uJ (0) = g0(t )  un(L) = g (tn)  J 1  0 (j) (j) uJ (xk ) = f(xk ). where tn = n∆t is the time station and ∆t is the time step. Adaptive Choice of Collocation Points

As discussed in the previous subsection, most of the wavelet expansion coefficientsu ˆdj for large j can be ignored within a given tolerance ². So we can dynamically adjust the number and the locations of the collocation points used in the wavelet expansions, reducing significantly the cost of the scheme while providing enough resolution in the regions where the solution varies significantly. We can achieve this adaptability in the following way. Let ² ≥ 0 be a prescribed tolerance Step 1. First we locate the range for the index (j, k) such that

0 |uˆj,k| ≥ cJ ² to obtain the feasible set Ξ0 for the initial u (i.e.,at the time station 0). 0 Step 2. We redeﬁne uJ (x) as X 0 0 uJ (x) := uˆj,kψj,k(x). (j,k)∈Ξ0 where Ξ0 denotes the feasible set for the solution u at the time station 0. Step 3. We deﬁne the pre-feasible set for the solution u at the time station t1 by setting

01 J−1 01 Ξ = ∪j=0 Ξj , where 01 0 −j Ξj = {d; dist(d, Ξj ) ≤ 2 }. 1 Step 4. We reduce uJ (x) as X 01 01 uJ (x) := uˆj,kψj,k(x). (j,k)∈Ξ01 © ª and solve the equation for uˆ01 . j,k (j,k)∈Ξ01 Step 5. We use

01 |uˆj,k| ≥ cJ ² to obtain the (post) feasible set for the solution u.at the time t1. n n Then we repeat the steps to get {uˆj,k} for the solution u at the time station t , n = 1, 2, ··· .

21 5.3 Adaptive Wavelet Collocation Methods for 2-D PDE’s The adaptive wavelet collocation method for 2-D PDE’s is similar to that for 1-D PDE’s. We only give one example to show the method. Consider the numerical solution of the two-dimensional anisotropic diﬀusion equation ∂u = ∇ · (c(|∇u|)∇u), (x, y) ∈ Ω, (26) ∂t where Ω is the square [0,N]2. The equation (26) is used to enhance the edge of an image[13]. We assume that the initial time t0 = 0, the time step is ∆t = λ, and the discrete solution un at time station tn is in the space VJ . (Here, again we omit the superscript Ω in all notations.)

• Full-discretization of Type 1. In [13], the authors suggested the following discretization of Equation (26). λ u = u + [c ∆ + c ∆ + c ∆ + c ∆ )u n+1 n 4 N N S S E E W W n

where the diﬀerence operators ∆N , ∆S, ∆E, ∆W are deﬁned by 1 ∆ u(x, y) = (u(x − ∆x, y) − u(x, y)) , N ∆x 1 ∆ u(x, y) = (u(x + ∆x, y) − u(x, y)) , S ∆x 1 ∆ u(x, y) = (u(x, y + ∆y) − u(x, y)) , E ∆y 1 ∆ u(x, y) = (u(x, y − ∆y) − u(x, y)) , W ∆y

1 (note that, when un in SJ , the discrete diﬀerentials of x and y are ∆x = ∆y = 2J , ) and

cN (x, y) = c (|∆N u(x, y)|) , cS(x, y) = c (|∆Su(x, y)|) ,

cE(x, y) = c (|∆Eu(x, y)|) , cW (x, y) = c (|∆W u(x, y)|) . Taking the advantage of spline representations of the solutions, we also adopt other two discretizations for the equation (26).

• Full-discretization of Type 2. The equation (26) is discretized in a natural way. ~ ~ un+1 = un + λ[∆N (c(∆|un|)∆N un) + ∆E(c(|∆un|)∆Eun)], q ~ 2 2 where |∆un| = (∆N un) + (∆Eun) .

22 • Semi-discretization. In this case, only the scale t is discretized.

un+1 = un + λIJ (c(|∇un|)∆un + ∇c(|∇un|) · ∇un) , (27)

where IJ is the interpolation operator on VJ . We use the same adaptive collocation method in the previous subsection to reduce the 0 0 J−1 solution space. Assume that a function f ∈ VJ has the wavelet coefficients {a , d , ··· , d }, where dj = {dj,h, dj,v, dj,d}. For a given tolerance ², we reduce the wavelet coefficients j d0, ··· , dJ−1 to {d˜0, ··· , d˜J−1}. Let T be the mapping that maps the coefficient set d˜ to the index set of the wavelets. ˜j,i ˜j,i T (d ) = {(j, k, l); dk,l 6= 0} i ∈ {h, v, d}.

2D h,2D ¯ v,2D ¯ d,2D ¯ ¯ ¯2D Write Sj = Sj × Sj,Sj = Sj × Sj,Sj = Sj × Sj,Sj = Sj × Sj,and Sj = h,2D v,2D d,2D Sj ∪ Sj ∪ Sj . We now deﬁne the feasible set for the j-level wavelets by   \ [ [ 2D ¯2D  j,i  Ξj = Sj supp(ψk.l) i∈{h,v,d}(j,k,l)∈T (d˜j,i)

2D 2D 2D 2D Then Ξ = S0 ∪ Ξ0 ∪ · · · ∪ ΞJ−1 is a feasible set for f. The steps for solving the discrete equation are similar to those in the previous subsection. One thing here is different from Equation (24). The singularity of the solution of Equation (26) is independent of the time t. Hence, after a few steps, we can fix the feasible set for all later time stations, that will save the time for searching the feasible sets for different time stations.

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