Spline Wavelets in Numerical Resolution ofPartial Differential Equations

Jianzhong Wang1

Department of mathematics, Computer Sciences, and StatisticsSam Houston State University

Huntsville, Texas, USA

Abstract. We give a review of applications of spline wavelets in the resolution ofpartial differential equations. Two typical methods for numerical solutions of partial dif-ferential equations are Galerkin method and collocation method. Corresponding to thesetwo 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 numericalresolution 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, wefirst need to find a finite-dimensional approximation space for the solutions, then discretizethe PDE to a system of algebraic equations in this space so that the numerical solutionscan be obtained. In many cases, the wavelets provide better bases of the approximationspaces 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 ofthe discrete algebraic equations. Second, the wavelet representations are effective in theadaptive procedures so that the complexity of calculations can be reduced. Furthermore,when the solution has a certain singularity, its wavelet representation can automaticallycapture the singularity. Hence, the advantage of wavelet methods in numerical resolutionof PDE’s is quite significant. However, in general, the wavelets do not have a localizationas sharp as finite element method or finite difference method. Hence, the sparsity of thematrices associated to the discrete equations based on wavelet bases is no better than thesparsity 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 anindividual problem.

In this paper, we only introduce spline wavelets. We select spline wavelets becausethey 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 asspline 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-orthogonalspline wavelets. In Section 3, we introduce the semi-interpolation spline wavelets and theirproperties. The numerical solutions of Dirichlet boundary problems using wavelets will becontained in Section 4. Finally, in Section 5, we discuss the adaptive wavelet method fornumerical solutions of evolution equations.

2 Semi-Orthogonal Spline Wavelets

Orthonormal or semi-orthogonal wavelets are often utilized in the Galerkin method for theresolution of PDE’s. It is well-known that the useful orthogonal or semi-orthogonal waveletbases are obtained from multiresolution analyses. We first discuss one dimensional case.

2.1 Semi-Orthogonal Spline Wavelets on RThe 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(Vj)j∈Z of L2(R) such that

(1) ∩Vj = 0, and ∪Vj is dense in L2(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 theMRA. We now set

φjk(x) = 2j/2φ(2jx− k)

and φk = φ0k. Then φjkk∈Z forms a Riesz basis of Vj. The wavelet subspace Wj is definedas the orthogonal complement of Vj with respect to Vj+1,

Vj+1 = Vj ⊕Wj Wj ⊥ Vj. (1)

The space L2(R) therefore has a decomposition L2(R) = ⊕jWj. It has been proved thatthere exists a function ψ ∈ W0 such that ψjkk∈Z forms a basis of Wj, and thereforeψjk(j,k)∈Z2 is a basis of L2(R). The function ψ is called a semi-orthogonal wavelet since

2

< ψjk, ψj′k′ >= 0, j 6= j′.

We call ψjk(j,k)∈Z2 a wavelet basis of L2(R).Similarly, ψ is called an orthonormal wavelet if

< ψjk, ψj′k′ >= δjj′δkk′ ,

and the basis ψjk(j,k)∈Z2 created by an orthonormal wavelet ψ is called an orthonormalwavelet basis of L2(R).

Remark 2.1 Sometimes, for much flexibility, the wavelet subspace Wj defined by (1) isnot 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, ψ∗j′k′ >= δjj′δkk′ .

The pair of (ψ, ψ∗) 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 thanL2(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 φ satisfies a refinement equation

φ(x) =∑

pkφ(2x− k)

where the sequence (pk) is called the mask of φ, and the Laurent series p(z) = 12

∑pkz

k iscalled the symbol of φ. In most of applications, φ is required to be compactly supportedor to decay exponentially. Splines are good examples for scaling functions. Let χ be thecharacteristic function of the unit interval [0, 1). The function

bm =

m︷ ︸︸ ︷χ ∗ χ ∗ · · · ∗ χ

is called the cardinal B-spline of order m. In applications, we often use the splines with evenorder. Hence, in this paper, we always assume that the spline order is even. We sometimesalso omit the subscript m from the notations such as bm(x) if no vagueness arises. TheFourier transform of b(x) is

b(ω) =

(1− e−iω

ω

)m

.

3

It is easy to verify that b(x) satisfies the refinement equation

b(x) = 2−m+1

m∑

k=0

(mk

)b(2x− k). (2)

The B-spline b(x) generates an MRA of L2(R). The construction of the semi-orthogonalspline wavelet can be found in our paper [6]. We now quote the results in [6] below.

• Semi-orthogonal spline wavelet basis and its dual.

Let Πm(z) :=∑2m−2

k=0 b2m(k+1)zk, which is the well-known (normalized) Euler-Frobeniuspolynomial of order 2m−2. By the Poisson’s summation formula, we have ei(m−1)ωΠm(e−iω) =∑

k |b2m(ω + 2kπ)|2. Then the following proposition holds.

Proposition 1 Let b be the cardinal B-spline of order m, and the function w be defined by

w(ω) = (1− e−iω/2

2)mΠm(e−iω/2)b(ω/2) (3)

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 inthe bi-infinite form of

f(x) =∞∑

j=−∞

∞∑

k=−∞djkwjk, (4)

and another one is in the form of

f(x) =∞∑

k=−∞cJkbJk +

∞∑j=J

∞∑

k=−∞djkwjk. (5)

To decompose a function into its wavelet series by using semi-orthogonal wavelet basis, weneed to construct the dual scaling function of b and the dual wavelet of w respectively. Ascaling function b∗ ∈ V0 is called the dual of b with respect to V0 if

< bk, b∗k′ >= δkk′ ,

where < f, g > is the inner product in L2(R). Similarly, a function w∗ ∈ W0 is call thedual of w with respect to W0 if

< wk, w∗k′ >= δkk′ .

It is known that function b∗ is determined by

b∗(ω) =b(ω)

|Πm(e−iω)| .

Therefore, we have the following (see [6]).

4

Proposition 2 Let G(z) = (1+z2

)m Πm(z)zΠm(z2)

and H(z) = −(1−z2

)m 1zΠm(z2)

. Then the dual pair

(b∗, w∗) satisfies the following two-scale relations.

b∗(ω) = G(e−iω/2)b∗(ω/2),

w∗(ω) = H(e−iω/2)b∗(ω/2).

The coefficients djk and cJk in (4) and (5) can now be obtained by djk =< f,w∗jk > and

cJk =< f, b∗Jk > respectively. Thus, we have

f(x) =∞∑

j=−∞

∞∑

k=−∞< f, w∗

jk > wjk

and

f(x) =∞∑

k=−∞< f, b∗Jk > bJk +

∞∑j=J

∞∑

k=−∞< f, wjk > wjk.

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 functionη is determined by

η(ω) =b(ω)√|Πm(ω)| .

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

< ηk, ηk′ >= δkk′ .

The orthonormal wavelet ξ ∈ W0 is defined by

ξ(ω) = −e−iω/2(1− eiω/2

2)m

√|Πm(−e−iω/2)√

Πm(e−iω)η(ω/2),

which leads to

< ξjk, ξj′k′ >= δjj′δkk′ .

Both η and ξ exponentially decay. It is clear that f ∈ L2(R) can be decomposed into

f =∞∑

j=−∞

∞∑

k=−∞< f, ξjk > ξjk

and

f(x) =∞∑

k=−∞< f, ηJk > ηJk +

∞∑j=J

∞∑

k=−∞< f, ξjk > ξjk

respectively.

5

2.2 Semi-Orthogonal Periodic Spline Wavelet Bases

Meyer in [12] introduced the constructions of orthonormal periodic wavelet bases (alsosee [10]). Following his method, we can construct semi-orthogonal periodic spline waveletbases. we consider the wavelet bases in the space of periodic functions

L2 = f ; f(·+ 1) = f,

∫ 1

0

f 2dx < ∞.

Let b, b∗, w, w∗, η, and ξ be the functions defined in the preceding subsections.

Proposition 3 For j ≤ 0, let

wpjk =

∑

l∈Zwjk(x− l), w∗p

jk =∑

l∈Zw∗

jk(x− l).

Then the constant function and wpjk; k = 0, · · · , 2j − 1, j ≥ 0 form a semi-orthogonal

basis of L2, while the constant function and w∗pjk ; k = 0, · · · , 2j − 1, j ≥ 0 form the dual

basis with respect to wpjk; k = 0, · · · , 2j − 1, j ≥ 0. Similarly, let

ξpjk =

∑

l∈Zξjk(x− l).

Then the constant function and ξpjk; k = 0, · · · , 2j − 1, j ≥ 0 form an orthonormal basis

of L2.

A function f ∈ L2 now can be decomposed into

f =

∫ 1

0

f(x)dx +∞∑

j=0

2j−1∑

k=0

< f, w∗pjk > wp

jk

or

f =

∫ 1

0

f(x)dx +∞∑

j=0

2j−1∑

k=0

< f, ξpjk > ξp

jk

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 Vjj∈Z be the MRA in Subsection 2.1, which is generated by the cardinal B-splineb of order m. Let V I

j ⊂ L2(I) be the subspace which contains the restrictions to I of thefunctions in Vj. Then V I

j j≥0 is an MRA of L2(I) and dim Vj = m+2jN−1. To constructthe basis of Vj, we set bI

jk = bjkχI , and let Sj = 2−jk; suppbjk ∩ I 6= ∅. It is clear that|Sj| = dim Vj. We have the following.

6

Proposition 4 bIjk2−jk∈Sj

forms a basis of V Ij , and there are two positive constants C1

and C2 independent of j such that

C1

∑

2−jk∈Sj

|cjk|2 ≤ ||∑

2−jk∈Sj

cjkbIjk||2 ≤ C2

∑

2−jk∈Sj

|cjk|2.

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

Hs0(I) = f ∈ Cs−1

0 (I); f (s−1) ∈ L2(I), 0 ≤ s ≤ m− 1.

(We agree that H00 (I) = L2(I).) Since the truncated B-splines are not in the space Hs

0(I), s ≥1, we have to modify them in the construction of the wavelet bases of Hs

0(I). The B-splineswith 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 theknot sequence

m︷ ︸︸ ︷0, 0, · · · , 0, 1, 2, · · · ,

m︷ ︸︸ ︷N,N, · · · , N

by tim+N−1i=−m+1, and define

φIk(x) = [tk−m

2, tk+1, · · · , tk+m

2](· − x)m−1

+ , −m

2+ 1 ≤ k ≤ N − 1 +

m

2. (6)

The support of φIk is [tk−m

2, tk+m

2], and φI

k (x) = bk−m2(x), m

2≤ k ≤ N − m

2. For convenience,

the MRA of Hs0(I) is still denoted by V I

j ∞j=0. We have the following.

Proposition 5 Φ0 := φIk

N−1−s+m2

k=s+1−m2

is a basis of V I0 , 0 ≤ s ≤ min(m− 1, [m+N

2]− 1).

When m2≤ k ≤ N − m

2, φI

k is the shift of central B-spline, that is, φIk(x) = b(x−k + m

2).

To construct the bases of V Ij , we dilate the scaling functions in Φ0. We define φI

jk(x) =

2j/2φIk(2

jx) for s + 1 − m2≤ k ≤ m

2− 1 and 2jN − m

2+ 1 ≤ k ≤ 2jN − 1 − s + m

2, define

φIjk(x) = 2j/2b(2jx− k − m

2) for m

2≤ k ≤ 2jN − m

2. Then

Φj := φIjk

2jN−1−s+m2

k=s+1−m2

(7)

is a basis of V Ij . Write SL

j = 2−js + 1 − m2, · · · , m

2− 1, SC

j = 2−jm2, · · · , 2jN − m

2,

SRj = 2−j2jN−m

2+1, · · · , 2jN−1−s+ m

2. Then, ΦL

j := φIjk; 2

−jk ∈ SLj contains the left

boundary scaling functions in Φj, while ΦRj := φI

jk; 2−jk ∈ SR contains the right boundary

scaling functions in Φj. The central scaling functions are in ΦCj := φI

jk; 2−jk ∈ SC

j . Whens = m − 1, both ΦL

j and ΦRj are empty. Setting Sj = SL

j ∪ SCj ∪ SR

j ,we have a one-to-onemapping Sφ from Φj to Sj such that Sφ(φ

Ijk) = 2−jk ∈ Sj.

7

• Semi-orthogonal wavelet bases of space Hs0(I).

Let W Ij be the wavelet subspace of Hs

0(I) such that W Ij ⊕ V I

j = V Ij+1 and W I

j ⊥V Ij .

Setting Sj = (Sj+1 \ Sj) ∩ [0, N ], we have dim W Ij = |Sj|. The set Sj is symmetric with

respect to N2, 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.)It is clear that dim W I = min(N, (2N +m− 2s− 1)+). The bases of W I are not unique. Inapplications, two main properties are often required. First, the supports of the elements inthe wavelet bases of W I are required as small as possible. Second, the bases are expectedto have certain symmetry.

We now decompose the set S into the form of

S = SL ∪ SC ∪ SR, (8)

where SC = m − 12,m + 1

2, · · · , N −m + 1

2, SL = S ∩ (0,m − 1], and SR = N − SL =

S ∩ [N −m + 1, N). (If N < 2m− 1, then SC = ∅.) It can be verified that the matrix

M = (< φIl , φ

I1k >)l∈S, k

2∈S1

is a full-rank matrix.. The linear equation Md = 0 has N linear independent solutionsd1, · · ·dN . We now write S = k1, · · · , kN, where k1 < · · · < kN , and set ψI

kl= (dl)T Φ1.

Then ψIdd∈S is a basis of W I . By carefully selecting vectors d1, · · · ,dN , we can make

ψId(x) = ψI

N−d(N − x) for d ∈ SL and ψId(x) =

√2w(2(x− d) + m− 1

2) for d ∈ SC , where

w is defined by (3).

Example 2.1 Consider the wavelet bases of the space Hm−10 (I). In this case, The bases

W I can be constructed as follows.

1. If N < m− 1, then V I = 0 and V I1 = W I . A basis of V I

1 is also a basis of W I .

2. If m ≤ N ≤ 2m− 2, then, for any d ∈ S with d ≤ N2, we can find (qk

l )N−m+1+kl=k such

that the function

ψId =

N−m+k−1∑

l=k

qkl φ1l, k = 2d (9)

is orthogonal to the space V I1 and ||ψI

d||2 = 1. We define ψId(x) = ψI

N−d(N − x) ford ∈ S with d > N

2. Then ψI

dd∈S is a basis of W I .

3. If N ≥ 2m − 1, we have SL = m4, m

4+ 1

2, · · · , m

2− 1

2 ∪ m+1

2, m+3

2, · · · , 2m−3

2. Let

K = minN, 3m− 3. We can verify that the matrix

M = (<φk, φ1l >)K−m

2,K+m

2−1

k,l=m2

(10)

8

is a full-rank matrix. Therefore, the equation Mc = 0 has m−1 independent solutionscl, 1 ≤ l ≤ m− 1. We can choose cl so that its last (m− 1− l) components are zero.Let ψI

d = Φ1cl, where

d =

m4

+ l−12

, 1 ≤ l ≤ m2

l − 12, m

2+ 1 ≤ l ≤ m− 1

.

Then the support of ψId is

suppψId = [0,

K + l

2] ⊂ [0,

K − 1 + m

2].

Let ψId(x) = ψI

N−d(N − x) for d ∈ SR and let ψId(x) =

√2w(2(x − d) + m − 1

2), for

d ∈ SC . Thus, the set of ψIdd∈S forms a basis of W I .

Remark 2.3 If I is a dyadic interval, say I = [2−Jk, 2−J(k + l)] for a fixed J, the con-struction of the scaling function basis of V I

j and the wavelet basis of W Ij is quite similar.

We define the sets Sj and Sj in the same way as above. Then we still have dim W Ij = |Sj|.

An evident modification needs to make for the construction of the wavelet basis of W Ij .

We now consider the dual wavelet bases. Since the spaces V Ij and W I

j both are finite,the dual basis of Φj with respect to V I

j and the dual basis of Ψj with respect to W Ij can

be obtained using the Gram-Schmidt matrices generated by Φj = φIjk and Ψj = ψI

jkrespectively. For example, when j = 0, Let

Gψ = (< ψId, ψ

Id >)(d′,d)∈S,

Gφ = (< φId, φ

Id >)(d′,d)∈S.

Since Gψ and Gφ both are invertible, the vector Ψ∗ = (ψ∗d)d∈S defined by Ψ∗ = G−1ψ ΨI

forms a dual basis of ΨI with respect to W I , and the vector Φ∗ = (φ∗d)d∈S defined byΦ∗ = G−1

φ ΦI forms a dual basis of ΦI . Although the dual bases Ψ∗j and Φ∗

j are no longercompactly supported with respect to I, any element in Ψ∗

j or in Φ∗j exponentially decays

within I.Before we end this subsection, we briefly discuss how to construct semi-orthogonal

wavelet bases of the space Hs0(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 arbitrarydomain in R2.

Let Sj = 2−jk; suppφj,k ⊂ Γ and V Γj = spanφj,k; 2

−jk ∈ Sj. Then

V Γ0 ⊂ V Γ

1 ⊂ · · · ⊂ V Γj ⊂ · · · (11)

is an MRA of Hs0(Γ). If we set Γj =

⋃2−jk∈Sj supp(φj,k), The MRA

V Γ00 ⊂ V Γ1

1 ⊂ · · · ⊂ VΓj

j ⊂ · · ·

9

is the same as the MRA (11). Hence, W ΓJj = W Γ

j .For a fixed level j, there exist finite intervals Iλ such that Γj = ∪λIλ. It is obvious that

the end-point of each interval Iλ is in the dyadic form of 2−jk. Let Ψjλ be the wavelet basis

of W Iλj (see 2.3) and Φj

λ be the scaling function basis of V Iλj . Then Ψj = ∪λΨ

jλ forms the

wavelet basis of W ΓJj and Φj = ∪λΦ

jλ forms the scaling function basis of V ΓJ

j . Note that,when Γ is given, there is an one-to-one mapping form Sj 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 tensorproduct to construct the semi-orthogonal spline wavelet basis on Hs

0(Ω). (We now assumethat 1 ≤ s ≤ m− 1.) Since it is very straight forward, we will not discuss it in detail. Thereaders can refer to [8]. We now consider the spline wavelet basis of Hs

0(Ω) for the domainΩ other than the rectangle. For illustration, we only discuss the case of s = m − 1. Forother s, the discussion is similar except that more complicated indices will be introduced.Let Sj = (2−jk, 2−jl); supp(φ(j,k,l)) ⊂ Ω, where φ(j,k,l)(x, y) = φjk(x)φjl(y). Let V Ω

j =spanφ(j,k,l);j (2−jk, 2−jl) ∈ Sj. Then

V Ω0 ⊂ V Ω

1 ⊂ · · · ⊂ V Ωj ⊂ · · ·

is an MRA of Hm−10 (Ω) and ΦΩ

j := φ(j,k,l);j (2−jk, 2−jl) ∈ Sj is a basis of V Ωj , which

contains all scaling functions of level j. The wavelet subspace WΩj−1 of this MRA is defined

by WΩj−1⊥V Ω

j−1 and WΩj−1 ⊕ V Ω

j−1 = V Ωj . To construct a wavelet basis of WΩ

j−1, we set

Ωj =⋃

(2−jk,2−j l)∈Sj

supp(φ(j,k,l))

and define the y-cut and x-cut of Ωj at (d, d′) ∈ Sj by Γd′ = x; (x, d′) ∈ Ωj andΓd = y; (d, y) ∈ Ωj respectively. It is clear that either Γd′ or Γd is an union of finite

disjointed dyadic intervals. Let Φd′j−1 and Ψd′

j−1 be the bases of the subspaces VΓd′j−1 and

WΓd′j−1 respectively. Write Sd′

j = d; (d, d′) ∈ Sj and Sd′j−1 = Sd′

j \ Sd′j−1, then there is

an one-to-one mapping from Φd′j−1 to Sd′

j−1 and an one-to-one mapping from Ψd′j−1 to Sd′

j−1.

Similarly, let Φdj−1 and Ψd

j−1 be the bases of the subspaces V Γdj−1 and W Γd

j−1 respectively.

Write Sdj = d′; (d, d′) ∈ Sj and Sd

j−1 = Sdj \ Sd

j−1. Then there is an one-to-one mappingfrom Φd

j−1 to Sdj−1 and an one-to-one mapping from Ψd

j−1 to Sdj−1.

10

We now write

Shj−1 = (d, d′); (d, d′) ∈ Sj & d ∈ Sd′

j−1 & d′ ∈ Sdj−1,

Svj−1 = (d, d′); (d, d′) ∈ Sj & d ∈ Sd′

j−1 & d′ ∈ Sdj−1,

Sdiagj−1 = (d, d′); (d, d′) ∈ Sj & d ∈ Sd′

j−1 & d′ ∈ Sdj−1,

and let

Ψhj−1 = ψ(x)φ(y); ψ ∈ Ψd′

j−1, φ ∈ Φdj−1, (d, d′) ∈ Sh

j−1,

Ψvj−1 = φ(x)ψ(y); φ ∈ Φd′

j−1, ψ ∈ Ψdj−1, (d, d′) ∈ Sv

j−1,

Ψdiagj−1 = ψ1(x)ψ2(y); ψ1 ∈ Ψd′

j−1, ψ2 ∈ Ψdj−1, (d, d′) ∈ Sdiag

j−1 .

Then Ψhj−1 ∪ Ψv

j−1∪Ψdiagj−1 forms a basis of the subspace WΩ

j−1. Let W h,Ωj−1 = span(Ψh

j−1),

W v,Ωj−1 = (Ψv

j−1), and W diag,Ωj−1 = span(Ψdiag

j−1 ). Then the spaces W v,Ωj−1, W

v,Ωj−1,W

diag,Ωj−1 , and

V Ωj−1 are mutually orthogonal. Since all of these spaces are finite, the dual bases of

Ψhj−1, Ψ

vj−1, Ψ

diagj−1 and ΦΩ

j−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 notsuitable 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 continuouson R, equipped with the uniform norm ||||∞ . The MRA and the corresponding waveletstructure in Cu(R) can be found in [3], and [20]. The cardinal B-spline bm generates anMRA of Cu, that is, the nest of subspaces

Vj = ∑

k∈Zckbm(2jx− k); (ck) ∈ l∞, (12)

forms an MRA of Cu in the following sense: Cu = ∪Vj and ∩Vj = R. We define theinterpolation operator Ij from Cu to Vj : Ij(f) = sf by sf (2

−jk) = f(2−jk). It is knownthat the interpolation is unique if and only if the order m of the B-spline bm is even. Thefollowing theorem is well-known.

Theorem 3.1 For any f ∈ Cu, limj→∞ ||f − Ijf ||∞ = 0, and the approximation order ism.

11

The wavelet subspace Wj of Cu can be defined in the way that Vj ⊕ Wj = Vj+1

and Ij(g) = 0,∀g ∈ Wj. Let Ij be the interpolation operator from Cu to Wj such thatIj(f)(2−j−1(2k− 1)) = f(2−j−1(2k− 1)), k ∈ Z. Therefore, Ij+1 = Ij + Ij(I− Ij) = Ij +Hj,where Hj = Ij(I − Ij). Let

v(x) =∑

k∈Z

(−1)k−1b(k − 1)b(2x− k).

Then v(2jx− 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 aswe have done for semi-orthogonal ones. Let

C = f ; f(·+ 1) = f, f ∈ Cu.

Define vpjk(x) =

∑l∈Z vjk(x− l). Then the constant and vp

jk; k = 0, · · · , 2j−1, j ≥ 0 form

a basis of C. A function f ∈ C can be decomposed into

f = f(0) +∞∑

j=0

2j−1∑

k=0

djkvpjk.

We can find the dual basis of vpjk in the sense of Cu,which can be used to calculate the

coefficients djk. However, in practice, we get djk using interpolation scheme. Let Ipj

be the interpolation operator from C to V pj , and Ip

j be the interpolation operator from

C to W pj . Let Sj = 2−j−1(2k + 1); k = 0, · · · , 2j − 1, fj = f(Sj), and dj = (djk). Let

fj(x) = f(0) +∑j−1

i=0

∑2i−1k=0 dikv

pik Then we have

Gjdj = fj − fj(Sj),

where Gj =(vp

jk(2l+12j+1 )

)2j−1

k,l=0is invertible.

3.0.3 Semi-Interpolation Spline Wavelet Bases On Finite Intervals

The structure of semi-interpolation spline wavelet bases on finite intervals is simpler thanthat of semi-orthogonal ones. We still use V I

j to denote the MAR of the space Cs0(I) (I =

[0, N ]), which is generated by bm, and use W Ij to denote the wavelet subspace of Cs

0(I).

Then Φj in (7) (for Hs+10 (I)) is still a basis of V I

j . We assume that s ≤ m2−1, which is used

in most of applications. Under this assumption, Sj = (2−j−1(2Z+ 1)) ∩ [0, N ]. Withoutloss of generality, we only construct the bases for W I (i.e., j = 0.)

12

1. If N < m − 1, we set K = 1, · · · , 2N − 1, IN = 1, · · · , N − 1, and ΦK =φI

1k; k ∈ K. Then the matrix MK := ΦK(I) is a full-rank matrix. Let d1, · · · ,dN

be the N independent solutions of the equation MKd = 0. Let ψIk = ΦKdk. It is

obvious that ψIk(I

N) = 0. Hence, ψIkN

k=1 is the basis of W I .

2. If m − 1 ≤ N, we set Ll = l, · · · , 2l + m2− 2 and I l = 1, · · · , l + m

2− 2 for l =

1, · · · , m2− 1. Then the matrix Ml = (φI

1k(i))k∈Ll,i∈Il is a full-rank matrix so that theequation Mlq = 0 has an unique solution (up to a constant). Let ψI

d =∑

k∈Ll qkφI1k,

d = (2l − 1)/2. Then suppψId = [0, l + m

2− 1]. Define ψI

d(x) = ψIN−d(x), for d =

N− m−32

, · · · , N− 12, and define ψI

d(x) = v(x−d) for d = m−12

, m+12

, · · · , 2N−m+12

.Then ψI

dd∈S is a basis of W I .

We now develop the wavelet transform algorithms. We agree that, for any functionf ∈ Cs

0(I), the notation f(Sj) is the vector (f(d))d∈Sj, where f(d) = f

(−k)j (0) for d =

2−jk, k = −s, · · · ,−1, and f(d) = f(k−2jN)j (N) for k = 2jN +1, · · · , 2jN +s. We also write

Φj = φIjk2−jk∈Sj

, Ψj = ψIdd∈Sj

,Mj = Φj(Sj), Hj = Φj(Sj), and Gj = Ψj(Sj). We havethe following discrete wavelet transform algorithms for semi-interpolations.

Algorithm 1 (Two-level decomposition.) Let f = Φc0+Ψd0. 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

f(x) =∑

k∈S

c0kφk +

J−1∑j=0

2jN−1∑

k=1

djkψ

Ij,k(x) (13)

Algorithm 3 (Multi-level decomposition.)

c0 = M−1f(S)

d0 = G−1(f(S)−Hc0

)

for j = 1 : J − 1

cj = Lj−1cj−1 + Kj−1d

j−1

dj = G−1(f(Sj)−Hjc

j)

end (14)

13

where Lj−1 and Kj−1 are two-scale relation matrices generated by the relations Φjcj =

Φj−1cj−1 + Ψj−1d

j−1.

Algorithm 4 (Multi-level reconstruction.)

for j = 0 : J − 2

f(Sj) = Mjcj

f(Sj) = Gjdj + Vjc

j

cj+1 = Ljcj + Kjd

j

end

f(SJ−1) = GJ−1dJ−1 + VJ−1c

J−1 (15)

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

Theorem 3.2 Let Cα(I) be the Lip-α class on I (α ≥ 0) and Cαx (I) be the local Lip-α

class at c, that is, we say f ∈ Cαc (I) if there is a polynomial P ∈ πn, n = bαc, such that

f(x) = Pn(x− c) +©(|x− c|α).

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

|dj,k| ≤ C2−αj, 1 ≤ k ≤ nj, 0 ≤ j ≤ ∞.(2)If f ∈ Cα

c for c ∈ I and 0 ≤ α ≤ m− 1, then

|dj,k| ≤ C2−αj(1 + |2jc− k|α), 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

|f(x)− P (x)| ≤ C|x− c|α log2

|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. Thenwe can employee the method we used for semi-orthogonal ones to construct the bases ofHΩ

0 . 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 isknown that in the Galerkin methods the discrete system for the numerical resolution ofelliptic problems in a bounded domain is ill-conditioned if finite elements or finite differencemethods are used. Usually, the conditional number of the discrete system is ©(1/h2) fora second order elliptic problem in two dimensions. Using preconditioning methods, wecan reduce the conditional number to ©(1/h). However, if the wavelet bases are used, thepreconditioning yields a ©(1) conditional number. Therefore, wavelet methods lead tonumerical stabilities for the resolution. Besides, the iterative algorithms are very popularin numerical resolution of PDE’s. The wavelet method can accelerate the convergence ofthe iterative algorithms. As an example, we consider the following Dirichlet type boundaryvalue 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

A(u, v) =

∫

Ω

gv ∀v ∈ H10 (Ω). (17)

where A(u, v)=∫Ω(A(∇u) · ∇v + uv).

We now discretize the equation (17) using the spaces V Ωj and WΩ

j . Note that the dis-cretizations on V Ω

j for an arbitrary Ω or for a rectangle domain essentially are same, ex-cept that the former involves complicated indices. For simplicity, we now assume thatΩ = I2,where I is a finite interval. Let uj ∈ V Ω

j be the Galerkin approximation of thesolution u. We can expend uj in the terms of the wavelet basis

uj =∑

(d,d′)∈S2

udd′φId(x)φI

d′(y) +

j−1∑j=0

∑

(dj ,d′j)∈Sj×Sj

uhdjd′j

φIk(x)ψI

k′(y)

+∑

(dj ,d′j)∈Sj×Sj

uvdjd′j

ψIk(x)ψI

k′(y) +∑

(dj ,d′j)∈S2j

udiagdjd′j

ψIdj

(x)ψId′j

(y).

15

Let Pj be the orthogonal project from H10 (Ω) to V Ω

j . Then the projection of the functiong on V Ω

j has the expansion

Pjg =∑

(d,d′)∈S2

gdd′φId(x)φI

d′(y) +

j−1∑j=0

∑

(dj ,d′j)∈Sj×Sj

ghdjd′j

φIk(x)ψI

k′(y)

+∑

(dj ,d′j)∈Sj×Sj

gvdjd′j

ψIk(x)ψI

k′(y) +∑

(dj ,d′j)∈S2j

gdiagdjd′j

ψIdj

(x)ψId′j

(y)

Let uj and gj be the vectors of the coefficients of uj and Pjg respectively. Let Bj =(ΦΩ, ΨΩ, · · · , ΨΩ

j−1) and B∗j =

((ΦΩ

)∗,(ΨΩ

)∗, · · · ,

(ΨΩ

j−1

)∗). Let Aj be the matrix defined

by

Aj =

(∫

Ω

(A(∇b) · (∇b∗) + bb∗))

b∈Bj ,b∗∈B∗j

.

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 avery simple preconditioning to improve the condition number. Let Dj be the diagonal

matrix defined by D =(2jδdjd′j

), where dj and d′j are the indices of the bases Bj and B∗

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

)−1gj. (19)

We can solve Equation (19) by finding the inverse of Bj. In some cases, it costs a lotof 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, letα(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 slowconvergence. 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 discusshow to solve the equation (17) with α(x, y) À β(x, y) by multi-grid wavelet methods. Usingpenalty formulation introduced in [14], we can change (17) to

−εuxx − uyy + u = fnew, (x, y) ∈ Ω, 0 < ε ≤ 1 (20)

u = 0 (x, y) ∈ ∂Ω

where the unknown function u is a transformation of the original one.

16

• The Two-level Method

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

AjUj = Fj (21)

where Uj and Fj are the coefficient vectors of uj and Pjfnew on the basis ΦΩ

j respectively.Let Aj be the discretization of A(u, v) on ΦΩ

j . Its smoothing iteration is Lj :

U v+1j = Lj(U

vj , Fj) = SjU

vj + TjFj,

where Sj is the solver. The formation of Lj is dependent on both Aj and the type ofsmoothing iterations (Jacobi, Gauss-Seidel or Richardson. See [9].) After v times of it-erations, we obtain an approximate solution U v

j with initial guess U0j . 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) ε is close to 0, the errorEv

j = Uj −U vj 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 bemade for these two parts. Since the scaling subspace V Ω

j−1 contains the coarse version of Uj,and the wavelet space W h

j−1 contains the part which has high frequency in x-direction, wechoose V Ω

j−1∪W hj−1 for the coarse-grid correction. (See [14]). Let Ah

j−1 be the discretizationof A(u, v) on Ψh

j−1. The coarse grid correction is

U vj ←− U v

j +(Lj(Aj−1)

−1Rj−1 + Lhj (A

hj−1)

−1Rhj−1

)(Fj − AjU

vj ), (22)

where Rj−1 and Rhj−1 are come from two-level semi-orthogonal spline wavelet decomposi-

tion algorithms, in which Rj−1extracts the coefficients for ΦΩj−1, while Rh

j−1 extracts thecoefficients for Ψh

j−1, and Lj and Lhj are come from reconstruction algorithms, which recover

the coefficients of ΦΩj for the functions in V Ω

j−1 and in WΩ,hj−1 respectively.

According to Lemma 2.4.2 in [9], the iteration matrix of the two-grid iteration is

Mj(v) =(I − (

Lj(Aj−1)−1Rj−1 + Lh

j (Ahj−1)

−1Rhj−1

)Aj

)Sv

j , (23)

where Aj is a nonnegative matrix. We now define the Aj-norm for a matrix B with thesame size as Aj by

||B||Aj= ||A1/2

j BA−1/2j ||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 solverSj satisfies ||Sv

j ||Aj≤ 1 and

limv→∞

2−j||A1/2j Sv

j ||Aj= 0

17

holds uniformly for j. Then

||Mj(v)||Aj≤ Cη(v) +

√r(ε, j),

where η(v) → 0, as v →∞, 0 < r(ε, j) < σ < 1, and r(ε, j) → 0 for any fixed j as ε → 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 forwardmanner. We already know that, for Equation (21), the corresponding two-level methodcan be represented as Mj(v) in (23). There are two lower level matrices Aj−1 and Ah

j−1 in(23). The multilevel method repeats two-level method for these two matrices. For Aj−1,we can apply the two-level method Mj−1(v). However, for Ah

j−1, a slight modificationneeds to make. We now employee the wavelet packet structure for the modification. Inthe wavelet packet structure,(see [21] and [20]), the space W h

j−1 is decomposed into four

subspace W o,hj−2,W

h,hj−2,W

v,hj−2, and W diag,h

j−2 , where W o,hj−2 contains the functions in W h

j−1 with

lower frequency in both x and y-directions, W h,hj−2 contains the functions in W h

j−1 with

lower frequency in y-direction and higher frequency in x-direction, W v,hj−2 contains the func-

tions in W hj−1 with lower frequency in x-direction and higher frequency in y-direction, and

W diag,hj−2 .contains the functions in W h

j−1 with higher frequency in both x and y-directions.

Let Ao,hj−2 and Ah,h

j−2 be the discretizations of A(u, v) on W o,hj−2 and W h,h

j−2 respectively.

Then the two-level method for Ahj−1 is

Mhj−1(v) =

(I −

(Lo,h

j−1(Ao,hj−2)

−1Ro,hj−2 + Lh,h

j−1(Ah,hj−2)

−1Rh,hj−2

)Ah

j−1

) (Sh

j−1

)v.

The definitions of the operators Lo,hj−1, Ro,h

j−2, Lh,hj−1, and Rh,h

j−2 are similar to those for

Lj, Lhj , Rj−1, and Rh

j−1 in (22).

5 Adaptive Wavelet Method For Evolution Equations

Because wavelets have good localizations in both space and frequency domains, they canbe used for adaptive schemes in multiresolution approximation to obtain solutions whichdevelop singularity. Theorem 3.2 shows that, in the wavelet expansion, the magnitudesof 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 nonlinearwavelet approximation to dramatically reduce the number of wavelets in the representation.Furthermore, usually the singularity of the solutions in the problems of fluid dynamicsdevelop 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 thecurrent 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 sensitiveto dimensions. Hence, the method we describe here is also applied to high dimensions. Letus consider the MRA V I

j of the space Cs0(I), which is generated by the B-spline bm(x).

Let W Ij be the wavelet subspace, ΦI

j and ΨIj be the bases of V I

j and W Ij respectively.

Since the wavelet components add the details of the function to its “blur” version, thecomponents 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, thewavelet coefficient dj,k = ©(2−jα) tends to 0 as j tends to ∞. Hence we can reduce aquantity of the wavelet coefficients in order to save the operation time and the memoryspace. More precisely, we have the following. (See [19].)

Theorem 5.1 Let fJ =∑J−1

j=0

∑dj∈Sj

gdjψI

dj+

∑k∈S ckφk. For a given ε > 0, there exists

a constant cJ , which is only dependent on J , such that if gdjis selected by

gdj=

gdj

, gdj> cJε

0, gdj≤ cJε

j = 0, · · · , J − 1

and fJ =∑J−1

j=0

∑gdj

ψIdj

+∑

k∈S ckφk, then

||fJ − fJ ||C(I) < ε.

For a function fJ ∈ V IJ and a given tolerance ε, we say that the knot set Ξj ⊂ Sj is a

feasible set of level j if , for any dj ∈ Ξj, the coefficient gdj> cJε. The set Ξ(J) = ∪J−1

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 off. Furthermore, if we only use the wavelets corresponding all trees in Ξ(J) to approximatethe function f, we will obtain the same approximation order. The tree approximation willsave 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

ut + fx(u) = uxx + g(u), x ∈ [0, N ], t ≥ 0u(0, t) = g0(t)

u(N, t) = g1(t)u(x, 0) = f(x).

(24)

Here only Dirichlet boundary conditions are considered. However the methods can alsobe modified to treat Von Neuman type or Robin type boundary conditions. We use cubicB-spline b4 to create the MRA. The numerical solution uJ(x, t) will be represented by aunique decomposition in V0 ⊕W0 ⊕ · · · ⊕WJ−1, J − 1 ≥ 0,

uJ(x, t) =∑

k∈S

uk(t)φk(x) +J−1∑j=0

∑

k∈Sj

udj(t)ψdj

(x)

:= u−1(x) +

J−1∑j=0

uj(x), (25)

where the superscript I is omitted in all notations. We now identify the numerical solutionuJ(x, t) by its point values on all collocation points. We put all these values in vectoru = 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 pointsand obtain the following semi-discretized wavelet collocation method.Semi-Discretized Wavelet Collocation Methods

uJ t + fx(uJ) = R(uJxx) + g(uJ)|x=x

(j)k

,−1 ≤ j ≤ J − 1

uJ(0, t) = g0(t)uJ(L, t) = g1(t)

uJ(x(j)k , 0) = f(x

(j)k )

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 localuniform mesh.

Equation (5.2) involves a total of (2J − 1)N + 2 unknowns in u; two of them will bedetermined by the boundary conditions and the rest are the solutions of the ODE systemsubject to their initial conditions. In order to implement the time marching scheme for theODE’s system (for example Runge-Kutta type time integrator), we have to compute thederivative term in (5.2)

Assuming that the Euler forward difference scheme is used to discretize the time deriv-ative in (5.2), we obtain a fully discretized wavelet collocation method.

20

Fully discretized Wavelet Collocation Method

uJn+1 = uJ

n + ∆t[−fx(unJ) + R(un

Jxx) + g(unJ)]|

x=x(j)k

,−1 ≤ j ≤ J − 1

unJ(0) = g0(t

n)un

J(L) = g1(tn)

u0J(x

(j)k ) = f(x

(j)k ).

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 coefficients udj

for large j can be ignored within a given tolerance ε. So we can dynamically adjust thenumber and the locations of the collocation points used in the wavelet expansions, reducingsignificantly the cost of the scheme while providing enough resolution in the regions wherethe solution varies significantly. We can achieve this adaptability in the following way. Letε ≥ 0 be a prescribed tolerance

Step 1. First we locate the range for the index (j, k) such that

|u0j,k| ≥ cJε

to obtain the feasible set Ξ0 for the initial u (i.e.,at the time station 0).Step 2. We redefine u0

J(x) as

u0J(x) :=

∑

(j,k)∈Ξ0

u0j,kψj,k(x).

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

Ξ01 = ∪J−1j=0 Ξ01

j ,

where

Ξ01j = d; dist(d, Ξ0

j) ≤ 2−j.Step 4. We reduce u1

J(x) as

u01J (x) :=

∑

(j,k)∈Ξ01

u01j,kψj,k(x).

and solve the equation foru01

j,k

(j,k)∈Ξ01 .

Step 5. We use

|u01j,k| ≥ cJε

to obtain the (post) feasible set for the solution u.at the time t1.Then we repeat the steps to get un

j,k for the solution u at the time station tn, 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 thetwo-dimensional anisotropic diffusion equation

∂u

∂t= ∇ · (c(|∇u|)∇u), (x, y) ∈ Ω, (26)

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 inall notations.)

• Full-discretization of Type 1. In [13], the authors suggested the following dis-cretization of Equation (26).

un+1 = un +λ

4[cN∆N + cS∆S + cE∆E + cW ∆W )un

where the difference operators ∆N , ∆S, ∆E, ∆W are defined by

∆Nu(x, y) =1

∆x(u(x−∆x, y)− u(x, y)) ,

∆Su(x, y) =1

∆x(u(x + ∆x, y)− u(x, y)) ,

∆Eu(x, y) =1

∆y(u(x, y + ∆y)− u(x, y)) ,

∆W u(x, y) =1

∆y(u(x, y −∆y)− u(x, y)) ,

(note that, when un in SJ , the discrete differentials of x and y are ∆x = ∆y = 12J , )

and

cN(x, y) = c (|∆Nu(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 othertwo 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|)∆Nun) + ∆E(c(|~∆un|)∆Eun)],

where |~∆un| =√

(∆Nun)2 + (∆Eun)2.

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• 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 thesolution space. Assume that a function f ∈ VJ has the wavelet coefficients a0,d0, · · · ,dJ−1,where dj = dj,h,dj,v,dj,d. For a given tolerance ε, we reduce the wavelet coefficients

d0, · · · ,dJ−1 to d0, · · · , dJ−1. Let T be the mapping that maps the coefficient set dj

tothe index set of the wavelets.

T (dj,i) = (j, k, l); dj,ik,l 6= 0 i ∈ h, v, d.

Write S2Dj = Sj × Sj, Sh,2D

j = Sj × Sj, Sv,2Dj = Sj × Sj, Sd,2D

j = Sj × Sj,and S2Dj =

Sh,2Dj ∪ Sv,2D

j ∪ Sd,2Dj . We now define the feasible set for the j-level wavelets by

Ξ2Dj = S2D

j

⋂ ⋃

i∈h,v,d

⋃

(j,k,l)∈T (dj,i)

supp(ψj,ik.l)

Then Ξ2D = S2D0 ∪ Ξ2D

0 ∪ · · · ∪ Ξ2DJ−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 alllater time stations, that will save the time for searching the feasible sets for different timestations.

References

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[7] A. Cohen, I. Daubechies, and J. C. Feauveau, Biorthogonal bases of compactly sup-ported wavelets, Comm. Pure and Appl. Math. 45(1992).

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[11] S. Mallat, Multiresolution approximation and wavelet orthonormal bases of L2(R),Trans. Amer. Math. Soc. 315 (1989), 69–87.

[12] Y. Meyer, Ondelettes sur I’intervalle, Revista Mathematica Iberoamericana 7 (1991),115–133.

[13] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,IEEE PAML, 12 no.7 (1990).

[14] A. Rieder, R. Wells, X. Zhou, A Wavelet Approach to Robust Multilevel Solvers forAnisotropic Elliptic Equations, preprint, 1993.

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[16] Jianzhong Wang, Stability and linear independence associated with scaling vectors,SIAM Mathematical Analysis 29 (1998), 1140–1156.

[17] Jianzhong Wang, On solutions of two-scale difference equations, Chinese Ann. Math.B, 15(1994) 23–34.

[18] Jianzhong Wang, Study of linear independence and accuracy of scaling vectors viascaling divisors, Advances in Wavelets ed. by K.S. Lau et al., Springer (1998) 229–260.

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[21] V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, 1995.

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