1.3 Wavelet Bases on Bounded Domains
1.3.3 Wavelet Constructions on the Interval
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.
1.3. WAVELET BASES ON BOUNDED DOMAINS 27 interval Ω = (0,1). In view of the application of these bases in the numerical treatment of operator equations, some obstructions arise which shall be listed first.
On the one hand, we are particularly interested in those biorthogonal wavelet bases that admit the incorporation of Dirichlet boundary conditions into the pri-mal multiresolution spacesSj, while the dual spaces ˜Sj should fulfillcomplementary boundary conditions, if possible. The bases should be able to characterize the appro-priate scales of Sobolev and Besov spaces that are needed in the numerical examples.
Moreover, at least the primal waveletsψλ and their derivatives should be accessible analytically, making arbitrary point evaluations and quadrature cheap.
These requirements already exclude various classical wavelet constructions on the interval. Just to state some examples,periodic wavelet bases are useless for our purpose since they do not fulfill the appropriate boundary conditions nor do they reproduce the correct set of polynomials. In orthogonal wavelet constructions like [38], the primal wavelets are only given implicitly which makes them difficult to han-dle numerically. Moreover it is neither possible to increase the number of vanishing moments of the primal wavelets independent from the order of accuracy, nor can any orthogonal wavelet construction realize complementary boundary conditions.
For the same reasons, the orthogonal splinemultiwavelet bases from [71, 72] cannot be used. Though the latter constructions look interesting for their relatively high order of accuracy, their numerical realization seems not easy. Other biorthogonal multiwavelet bases on the interval were constructed in [56], where the primal gen-erators are the cubic Hermite interpolatory splines. Here it is not fully clear at the moment whether also complementary boundary conditions can be realized with the primal and dual generators. Further constructions one may look at use so–called prewavelets. There, the complement spaces Wj are mutually orthogonal with re-spect to theL2 or another energy inner product, drastically simplifying the stability analysis. The construction of prewavelets is relatively easy, see, e.g., [30, 100], but the dual wavelets are often not explicitly known and they are globally supported.
In the case that the prewavelet spacesWj are chosen orthogonal with respect to an-other inner product than that ofL2, the dual generators and wavelets may moreover have only a low Sobolev regularity and they mey even fail to belong toL2, as is the case for the dual prewavelet basis from [100]. Especially from the viewpoint of tight Riesz bounds, also the construction of spline wavelets on the interval in [15] looks quite interesting, though the corresponding dual generators are unknown so far.
In principle, the interval bases we shall consider in this thesis are based on the constructions in [59, 61, 126]. There, the primal generatorsφj,k live in a spline space of order m with respect to some knot sequence {tjk}k ⊂[0,1]. In the interior of the domain Ω = (0,1), the primal generators in the mentioned constructions coincide with dilates and translates of cardinal B–splines of order m, up to a factor. As a consequence of the embedding into spline spaces, the criticalL2–Sobolev regularity of the primal multiresolution spaces is given by m− 12.
For the verification of the polynomial exactness of order m, some modifications of the generators at the boundaries are necessary. Either one uses specific linear combinations of restricted cardinal B–splines [59, 61] or one resorts to B–splines related to the Schoenberg knot sequence [126] with m–fold boundary knots,
tjk= min
1,max{0,2−jk} , k=−m+ 1, . . . ,2j+m−1, (1.3.26)
where the correspondingB–splines of orderm are defined as
Bk,mj (x) := (tjk+m−tjk)[tjk, . . . , tjk+m]t(t−x)m+−1. (1.3.27) Here (x)+ := max{0, x} and [t0, . . . , tk]t denotes the divided difference operator corresponding to the knots ti acting in the t variable. In the construction of [126], the primal spline generators of orderm are then simply defined as φj,k := 2j/2Bk,mj . Examples of these generators in the casem = 3 can be found in Figure 1.2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2 2.5
Figure 1.2: Primal spline generatorsφj,k from [126] withm= 3, j = 3 and homoge-neous Dirichlet boundary conditions.
In any of the mentioned spline wavelet constructions, the primal refinement matrices Mj,0 can be computed by simply solving some triangular systems of linear equations, see [59, 126] for details.
The recent approach from [126] seems favorable for several reasons. Firstly, the primal multiresolution spaces Sj coincide with the full spline spaces related to the knot sequences {tjk}2k=j+m−m+1−1 , which is not always the case for the construction in [59]. Moreover, the incorporation of homogeneous Dirichlet boundary conditions of some order into the primal multiresolution spaces is easily done by omitting the corresponding number of boundary knots. Point evaluation of the primal generators and their derivatives is painless sinceB–splines fulfill numerous recurrence relations that allow for fast and stable evaluation algorithms [67]. Add to this, using onlyB– splines as primal generators drastically simplifies the computation of supports and singular supports in [126] compared with the constructions in [59]. Finally, choosing the shortest possible supports of the boundary generators seems to have a favorable impact on the L2–Riesz constants of the wavelet system.
In order to realize complementary boundary conditions for the multiresolution spacesSj and ˜Sj, the dual generators ˜φj,k are chosen to be exact of the full order ˜m and to be biorthogonal to the primal generators. Since the concrete construction is quite subtle, we refer the reader to [59, 61, 126] for the technical details. Let us only note that the dual refinement matrices ˜Mj,0 can again be set up by solving several triangular systems, see [59, 126] for examples. Generally speaking, in all known biorthogonal spline wavelet constructions on the interval, the dual generators are given as linear combinations of some dual scaling functions from the shift–invariant
1.3. WAVELET BASES ON BOUNDED DOMAINS 29 construction [37]. As a consequence, also the critical L2–Sobolev regularity of the dual generators is known and can be computed with well–known algorithms1 [99], see Table 1.1 for various example values.
m me ν2(φ)e 2 2 0.44076544507035 2 4 1.17513151026734 2 6 1.79313390050964 3 3 0.17513151026735 3 5 0.79313390050989 3 7 1.34408387241967 3 9 1.86201980785003 4 6 0.34408387241950 4 8 0.86201980784985 4 10 1.36282957312823
Table 1.1: Critical L2–Sobolev regularity of some dual scaling functions from [37].
Some specific dual generators from [126] are given in Figure 1.3. However, it should be noted that at least in the numerical discretization of elliptic operator equations with wavelets, the dual generators and wavelets are never needed explic-itly. There they merely serve to ensure the corresponding number ˜m of vanishing moments for the primal waveletsψλ and to establish an appropriate negative lower bound −˜γ in (1.2.16).
Concerning the construction of the primal wavelets, the approaches [59, 62, 63, 126] utilize the method of stable completion, as introduced in the previous subsec-tion. In [59] it has been shown how an initial stable completion can immediately be derived from a specific factorization of Mj,0. After performing the biorthogo-nalization step (1.3.20), the interior wavelets coincide with those presented in [37], whereas the boundary wavelets look more complicated. However, the factorization algorithm used in [59] does not automatically lead to themaximal number of interior wavelets. This is of particular importance since the Riesz constants of the overall wavelet basis crucially depend on the shape of the boundary functions. Moreover, due to the small supports of the interior wavelets, the sparsity of stiffness matrices is potentially higher. In [126], a refined algorithm was developed that produces the optimal number of interior wavelets. Figure 1.4 shows some primal wavelets from the latter construction.
From the viewpoint of numerical stability, in particular concerning tight L2– and Hs–Riesz bounds, the wavelet bases from [126] clearly outperform those of the original construction in [59, 63]. It should be mentioned that in [9], the latter wavelet bases have been substantially stabilized by orthogonalizing the boundary wavelets with respect to some energy inner product. However, a comparison with the benchmark values in [58] reveals that the Riesz constants as well as the spectral
1A software package for the estimation of critical Sobolev exponents of (multi)wavelets can be found on the website http://www.mathematik.uni-marburg.de/∼dahlke/ag-numerik/
research/software.
properties of stiffness matrices from a wavelet–Galerkin discretization of elliptic dif-ferential equations are still favorable in [126], as long as the spline order is restricted to low values m ≤ 3. As indicated by the recent construction [15], future research in the direction of high order biorthogonal spline wavelets on the interval remains potentially fruitful.
1.3. WAVELET BASES ON BOUNDED DOMAINS 31
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−5 0 5 10 15 20 25 30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−30
−25
−20
−15
−10
−5 0 5 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−5 0 5 10 15 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−10
−8
−6
−4
−2 0 2 4 6 8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−4
−2 0 2 4 6 8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−4
−2 0 2 4 6 8
Figure 1.3: Some dual spline generators ˜φj,k from [126] with m = 3, ˜m = 5, j = 4 and complementary boundary conditions.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Figure 1.4: Some primal spline wavelets ψj,k from [126] with m = 3, ˜m = 3, j = 3 and complementary boundary conditions.
1.3. WAVELET BASES ON BOUNDED DOMAINS 33