General Construction Principles

1.3. WAVELET BASES ON BOUNDED DOMAINS 21

S = (Sj)j≥j0 of linear, closed and nested subspaces that is asymptotically dense in V,

closV

[∞ j=j0

Sj =V. (1.3.1)

The concept of a multiresolution analysis was originally introduced in a shift–

invariant setting [116, 118]. There, the spaces Sj consist of functions on Ω = R^d,Sj

is closed under integer translationf 7→f(· −k),k ∈Z^d, and the consecutive spaces Sj and Sj+1 are related by dilation, i.e., f ∈ Sj if and only if f(2·) ∈Sj+1. Due to Ω = R^d, the powerful tool of the Fourier transform may then be exploited for the construction of wavelet bases, see [37, 65, 66, 117, 118] for classical examples.

However, the setting of bounded domains Ω we are interested in clearly inhibits shift–invariance and the use of Fourier transform techniques. Nevertheless, some basic principles from classical wavelet theory still carry over to wavelet constructions for more general domain geometries. In the following, we will outline the basic steps for the design of a biorthogonal wavelet basis over a bounded domain Ω, as propagated, e.g., in [32, 59, 61]. Moreover, we assume from now on thatV =L2(Ω), which is sufficient for all cases of practical interest.

In a first step, one identifies stable generator bases Φj ={φj,k : k ∈ ∆j} of the spacesSj, so that

Sj =S(Φj) = closL2(Ω)Φj, j ≥j0 (1.3.2) and likewise ˜S_j =S( ˜Φ_j), with a system of dual generators ˜Φ_j ={φ˜_j,k :k ∈∆_j}. In view of the later steps of the construction, we will have to require that the stability of the primal generators Φj holds uniformly in j

kckℓ2(∆j)hkc^⊤Φ_jkL2(Ω), c∈ℓ₂(∆_j), j ≥j₀. (1.3.3) In a shift–invariant setting, ∆j =Z^dis an infinite set and one would typically assume that the space S0 is spanned by the integer translates φ0,k = φ(· −k) of a single functionφ,

S0 = closL2(Ω)span{φ(· −k) :k∈Z^d}. (1.3.4) Then the additional requirement (1.3.3) is automatically fulfilled for allj ∈Z when-ever one has stability inV0, since in the shift–invariant setting,Sj is spanned by the integer translates

φj,k(x) = 2^−jd/2φ(2^jx−k), k ∈Z^d, x∈R^d (1.3.5) and an application of the transformation rule fory= 2^jxtraces back theℓ2–stability of Φj inVj to that of Φ0 inV0.

On a bounded domain Ω, however, the index sets ∆j will be finite and of increas-ing cardinality, as j tends to infinity. Moreover, Φj will not consist of the integer translates of a single function alone, so that the uniform stability (1.3.3) of Φj indeed has to be proved separately, see [54, 59]. A sufficient criterion for (1.3.3) to hold is the uniform boundedness ofkφj,kkL2(Ω) and kφ˜j,kkL2(Ω) in combination with uniform locality of the primal and dual generators, i.e., for a fixed constant C <∞ we have

#{supp_j,k ∩supp_j,k^′ 6=∅} ≤C, diam_j,k .2^−j, (1.3.6)

1.3. WAVELET BASES ON BOUNDED DOMAINS 23 where_j,k is the smallest cube containing the supports ofφj,k and ˜φj,k, respectively.

The locality assumption is useful anyway in view of the numerical realization, and it can in turn be assured by some structural properties of the abstract nestedness condition, which shall be discussed next.

The inclusionS_j ⊂S_j+1 connects the generators of two consecutive approximat-ing spaces by the so–called two–scale orrefinement relation

Φ_j =M^⊤_j,0Φ_j+1, j ≥j₀. (1.3.7) HereM_j,0 ∈R|∆j+1|×|∆j| is a matrix holding in itsk–th column the expansion coef-ficients ofφj,k with respect to the generators on the finer scalej+ 1. Since also the spaces ˜Sj are nested, the corresponding dual refinement matrices shall be denoted analogously by ˜M_j,0.

In a shift–invariant setting, the generatorφ fulfills a special refinement equation of the form

φ(x) =X

k∈Z

a_kφ(2x−k), x∈R^d (1.3.8) with refinement coefficients ak ∈ R. For most of the practically relevant cases, one may assume that φ is compactly supported and that only finitely many ak are nonzero. Hence M_j,0 is a quasi–banded biinfinite matrix with entries (M_j,0)_k,l = 2^−1/2ak−2l. Since the size and the entries of M_j,0 are independent of the current refinement level in the shift–invariant case, M_j,0 is then calledstationary.

For wavelet constructions on a bounded domain Ω, the structure of the refinement matricesM_j,0 will be more complicated. There, we can still hope thatM_j,0 isquasi–

stationary, meaning that only the matrix dimensions change with j but, away from some level–independent corner blocks, M_j,0 is still quasi–banded. This has the consequence that the interior generators will again be given as the dyadic dilates and translates of a single function, analogous to the shift–invariant case. From the quasi–bandedness of M_j,0, we can hence easily infer the uniform stability (1.3.3) of the generators Φj. Moreover, quasi–stationary matrices M_j,0 are uniformly sparse.

By this we mean that the number of nonzero entries per row and column of these matrices remains uniformly bounded inj, which is useful in numerical computations.

In Figure 1.1, the nonzero pattern of M_j,0 is visualized, for the special case of a wavelet basis on the unit interval.

Moreover, the systems Φj and ˜Φj shall be connected by thebiorthogonality con-dition

hΦj,Φ˜ji=I, j ≥j0. (1.3.9) This relation has several important consequences. By (1.3.3) and (1.3.9), also the stability of the dual system ˜Φj will hold uniformly in j. Moreover, concerning the refinement matrices M_j,0 and ˜M_j,0, the biorthogonality condition implies that M^⊤_j,0M˜_j,0 =Iand we can introduce projectorsQ_j and ˜Q_j onto the spaces S_j and ˜S_j via

Qjf :=hf,Φ˜jiΦj, Q^′_jf :=hf,ΦjiΦ˜j, f ∈L2(Ω). (1.3.10) Note that Q^′_j is the L2–adjoint of Qj and these projectors will of course coincide with the projectorsQj,Q^′_j from (1.1.16) and (1.1.17). The uniform stability (1.3.3)

0 2 4 6 8 10 0

2

4

6

8

10

12

14

16

18

nz = 34

Figure 1.1: Nonzero pattern of the primal refinement matrix M_j,0 for the spline wavelet basis on Ω = (0,1) from [126] withm = 3, ˜m= 3, j = 3 and free boundary conditions.

of Φ_j corresponds to the uniform boundedness of the projectors Q_j, as needed in Theorem 1.1.

For the proof of the Riesz stability as well as for the characterization of function spaces, we need in particular specific approximation properties of the spaces S_j, S˜j. To guarantee these, one will try to construct the generator bases Φj and ˜Φj

in such a way that they admit the reproduction of polynomials of order m and ˜m, respectively,

P_m

−1 ⊂Sj, P_m˜

−1 ⊂S˜j, j ≥j0. (1.3.11) Here P_k denotes the set of all polynomials with total degree less or equal than k. In a shift–invariant setting, the inclusions (1.3.11) are meant in the sense of pointwise convergence. It can be shown that under the biorthogonality, stability and locality assumptions stated so far, the reproduction of polynomials (1.3.11) implies the validity of a Jackson estimate (1.2.14) for the values m_S =m, mS˜= ˜m, respectively, see [55]. In the case that Dirichlet boundary conditions of some order have to be incorporated into the primal multiresolution spaces Sj, the inclusion P_m

−1 ⊂Sj is usually relaxed to merely hold in the interior of Ω, up to a boundary layer of thicknessc2^−j.

Concerning the validity of the Bernstein estimate (1.2.15), we only note that due to the quasi–stationary refinement matrices, it is sufficient to verify that the generators Φj and ˜Φj have a corresponding L2–Sobolev regularity of order γ_S and γS˜, respectively, and we refer to [53] for the relevant proofs.

In the next step of constructing a wavelet basis over Ω, we have to pick some algebraic complement spaces Wj, ˜Wj such that

Sj+1 =Sj⊕Wj, S˜j+1 = ˜Sj⊕W˜j, j ≥j0. (1.3.12) Moreover, the complement spaces should bebiorthogonal in the sense that

Wj ⊥S˜j, W˜j ⊥Sj, j ≥j0, (1.3.13)

1.3. WAVELET BASES ON BOUNDED DOMAINS 25 which determines Wj and ˜Wj uniquely. Then, due to the nestedness of the spaces Sj and ˜Sj, the operatorsQj −Qj−1 and Q^′_j−Q^′_j−1 are projectors onto Wj and ˜Wj, respectively.

Analogous to the generator sets Φj that span the spaces Sj, one then tries to find stable wavelet bases ˇΨj = {ψˇj,k : k ∈ ∇j} for the spaces Wj = S( ˇΨj). Again we assume here that the systems ˇΨ_j are uniformly stable

kck^ℓ2(∇j) hkc^⊤Ψˇjk, c∈ℓ2(∇^j), j ≥j0. (1.3.14) In view of the nestednessWj ⊂Sj+1, we can then express the wavelets on the level j with respect to the generator basis of the next higher scale

Ψˇj = ˇM^⊤_j,1Φj+1, j ≥j0, (1.3.15) where ˇM_j,1 ∈R|∆j+1|×|∇^j|. Moreover, it turns out that the construction of uniformly stable complement bases ˇΨj is equivalent tocompleting the refinement matricesM_j,0 to invertible mappings

Mˇ_j = (Mj,0,Mˇ_j,1) :ℓ2(∆j)⊕ℓ2(∇^j)→ℓ2(∆j+1), (1.3.16) where the operator norms of ˇM_j and of the inverse ˇG_j := ˇM⁻_j¹ stay uniformly bounded inj. This construction principle is known as the method of stable comple-tion [28]. It is useful to block also ˇG_j with

Gˇ_j =

M˜^⊤_j,0 Gˇ^⊤_j,1

, (1.3.17)

where ˇG_j,1 contains in its columns the expansion coefficients of some dual wavelet basis ˜ˇΨj for ˜Wj with respect to ˜Φj+1. The invertibility of ˇM_j then implies the matrix equation

I= ˇM_jGˇ_j =M_j,0M˜^⊤_j,0+ ˇM_j,1Gˇ^⊤_j,1. (1.3.18) It should be noted that (1.3.18) is not uniquely solvable. In particular, it is not a priorily clear whether there exist biorthogonal wavelet bases, i.e., whether one may find special wavelet bases Ψj =M^⊤_j,1Φj+1, ˜Ψj = ˜M^⊤_j,1Φ˜j+1 that satisfy

hΨj,Ψ˜ji=I (1.3.19)

and that are moreover also uniformly locally supported. The delicate problem here is that one has to determine a sparse matrix M_j of the above form such that its inverse G_j is also sparse, which is nontrivial. It could be shown in [28] how the solution space of (1.3.18) may be parametrized by a family of linear transformations as soon as one initial stable completion ˇM_j,1 is known. In the latter case, equation (1.3.18) has exactly one biorthogonal solution

M_j,1 = (I−M_j,0M˜^⊤_j,0) ˇM_j,1, (1.3.20) and the corresponding dual block in G_j can be computed as

M˜^⊤_j,1 :=G^⊤_j,1 = (I+ ˜M^⊤_j,0Mˇ_j,1) ˇG^⊤_j,1. (1.3.21)

Moreover, if the initial stable completion is uniformly locally supported, then this also holds for the biorthogonal completion (1.3.20). By (1.3.19), we can infer the matrix equationM^⊤_j,1M˜_j,1 =I.

The overall wavelet Riesz basis Ψ is obtained by aggregating the single comple-ment bases Ψj and the generators Φj0 from the coarsest level. In order to write this down in the most convenient way, we will use multiindicesλ= (j, e, k) with an additionaltype parameter e∈ {0,1}, and we set

ψ(j,0,k) :=φj,k, k∈∆j, ψ(j,1,k) :=ψj,k, k ∈ ∇j. (1.3.22) The overall wavelet index set J is then given by

J :={j0} × {0} ×∆j0 ∪ [

j≥j0

{j} × {1} × ∇^j, (1.3.23) so that we can define Ψ :={ψλ}^λ∈J. Analogous abbreviations shall be used for the dual wavelets ˜ψλ. For eachλ= (j, e, k)∈ J, we denote by|λ|=j the corresponding scale and bye(λ) = e the type.

Im Dokument Adaptive Wavelet and Frame Schemes for Elliptic and Parabolic Equations (Seite 35-40)