Approximation by Tensor Product Wavelets

3.4 Approximation by Tensor Product Wavelets

For the isotropically supported higher-dimensional wavelets{Ψλ: λ∈ ∇^(d)}, one obtains estimates for linear approximation that are completely analogous to (3.9). Let u∈ H^s(R^d) for s > 0, and let the underlying scaling function ϕ∈ H^t(R) satisfy the polynomial reproduction property of order m− 1. Then for 0 < t < s ≤ m, we have the direct estimate

inf

uj∈span{Ψλ: λ∈∇^(d), |λ|≤j}kuj− ukH^t(R^d). 2^−(s−t)j|u|H^s(R^d).

To obtain an approximation to u, we additionally need to take spatial decay properties of u into account. For this discussion, let us assume for the sake of simplicity that u is compactly supported, then with the notation Λ_u :={ν ∈ ∇^(d):hu, Ψνi 6= 0} we obtain

# {λ ∈ ∇^(d):|λ| ≤ j} ∩ Λu

∼ # {λ ∈ ∇^(d):|λ| ≤ j} ∩ Λu

∼ 2^dj,

where the constants depend on the wavelet bases and on|supp u|. In terms of the number N of nonzero coefficients of uj, we thus obtain the estimate

kuj − ukH^t(R^d). N^−s−td ,

or conversely, the number of coefficients required for an approximation error ε is proportional to ε^−s−td .

In this setting, a reduction of the approximation error by a prescribed factor requires a growth of the number of coefficients by a factor increasing exponentially with d. We shall briefly review a construction that addresses this problem based on dimension-dependent regularity information, or more specificially, on integrability of high-order mixed derivatives.

To this end, we introduce certain standard Sobolev spaces of dominating mixed derivatives. For s, k > 0 and d, D ∈ N, we define the Sobolev space H^s,k_mix(R^d; D) to comprise those f ∈ L2(R^dD) we have norm equivalences for fully and partially anisotropic tensor product wavelet bases,

kuk²s,k ∼ X

3 Higher-Dimensional Approximation and Adaptive Wavelet Methods

Proofs are given in [154], see also [68, 44] for the case of bounded domains.

The norm equivalences (3.25) give an indication of the relevance of such Sobolev spaces of mixed smoothness for higher-dimensional approximation. The coefficients with respect to anisotropic tensor product bases of functions with this type of regularity have a certain decay that can be exploited for the construction of so-called sparse grid approximations. This approach has its origins in higher-dimensional quadrature [133] and was later applied in the discretization of partial differential equations [158]; see also [21] for a review. In the context of wavelet approximation, one also finds the term hyperbolic wavelets [43].

In order to illustrate the idea by a simple example, we consider approximation by tensor product wavelets in H¹(R^D). Defining for j ∈ Z the sparse grid or hyperbolic wavelet subspace

Λˆ_j :=n In combination with suitable decay properties of u, this enables the construction of approximations u_j for which the scaling of the number of unknowns with respect to j for a given accuracy is very close to the one-dimensional case. To give a specific example, let in addition u be compactly supported, and let Λ_u :={ν ∈ ∇^D:hu, Ψνi 6= 0}, then one obtains

#(ˆΛj∩ Λu)∼ j^D−12^j.

Thus one obtains an almost – up to a logarithmic factor – dimension-independent convergence rate. For functions that are not compactly supported but decay exponentially, such as electronic Schr¨odinger eigenfunctions, approximations of this type will be considered in Section 4.1.1. For certain combinations of mixed regularity and approximation norm, such sparse grid constructions can be modified so as to remove the dimension-dependent logarithmic factor j^D−1 and to yield a dimension-independent convergence rate, cf. [21].

For Besov spaces in higher dimensions, there exist wavelet characterizations similar to Theorem 3.9. For the further discussion, we need two dimensional parameters d, D ∈ N, where the total space dimension is dD. For the standard Besov spaces B^s_p,p(R^dD), under assumptions similar to those of Theorem 3.9, as a further special case of [29, Theorem 3.7.7] one obtains

kfkB^s_p,p(R^dD) ∼ the case of adaptive approximation by such isotropic wavelet bases, we obtain convergence of order N^−dDs . The deterioration of the convergence rate with increasing dimension is therefore the same as in the case of Sobolev regularity and uniform refinement. In other words, adaptive approxima-tion with basis funcapproxima-tions of isotropic support reduces the regularity requirements for achieving a certain rate in comparison to linear approximation, but does not improve the dependence on space dimension.

3.4 Approximation by Tensor Product Wavelets

The situation is different, however, in the case of nonlinear approximation by anisotropic tensor product bases{Ψν}_ν∈(∇^(d)₎^D. The rates of best N -term approximation by such bases are governed by tensor product Besov spaces, which have been studied in [116, 131, 77].

Here we shall only consider those tensor product Besov spaces relevant for nonlinear approxima-tion in H¹(R^dD), which we denote by ˜B^s_p(R^d; D). These spaces can be characterized as intersections of tensor products of lower-dimensional standard Besov spaces,

B˜^s_p(R^d; D) = B^s+1_p,p (R^d)⊗ B^sp,p(R^d)⊗ · · · ⊗ B^sp,p(R^d) ∩ . . .

∩ B^sp,p(R^d)⊗ · · · ⊗ B^sp,p(R^d)⊗ B^s+1p,p (R^d) , p⁻¹ = s d+1

2. (3.28) These spaces thus provide an analogue to the Sobolev spaces H^s,1_mix(R^d; D). In the case 0 < p < 1, (3.28) requires an appropriate notion of tensor product, cf. [116].

It can be shown that for elements of ˜B^s_p(R^d; D) one obtains rates N^−ds for best N -term ap-proximation in H¹(R^dD) by the tensor product basis {Ψν}ν∈(∇^(d))^D, that is, a convergence rate independent of D.

We shall later use the families of spaces ˜B^s_p(R; 2) with p = (s + ¹₂)⁻¹ and ˜B^s_p(R³; 2) with p = (^s₃ +¹₂)⁻¹, for which – again assuming the tensor product basis is constructed from a sufficiently regular univariate wavelet – one obtains the following norm equivalences analogous to (3.11), (3.27).

Proposition 3.18. Letϕ be a compactly supported, continuous, and L2–orthonormal scaling func-tion of a multiresolufunc-tion analysis, let{ψν}ν∈∇ be a corresponding wavelet basis, and lets > 0. Let ϕ∈ B^tp,p for at > s + 1, and let ψ have at least bs + 1c + 1 vanishing moments.

(i) Let p = (s + ¹₂)⁻¹, then for u∈ ˜B^s_p(R; 2), kukB˜^sp(R;2)∼

(2^max|ν|hu, Ψνi)ν∈∇²

`p, (3.29)

and thus ˜B^s_p(R; 2)⊂ A^s∞(H¹(R²)) with respect to the tensor product basis{Ψν}ν∈∇². (ii) Let p = (^s₃ +¹₂)⁻¹, then foru∈ ˜B^s_p(R³; 2),

kukB˜^s_p(R³;2)∼

(2^max|ν|hu, Ψνi)_ν∈(∇⁽³⁾₎²

`p, (3.30)

and hence ˜B^s_p(R³; 2)⊂ A^s/3∞ (H¹(R⁶)) with respect to the tensor product basis{Ψν}ν∈(∇⁽³⁾)². For details, we refer to [147, 116, 131]; note that the particular norm equivalence (3.30) also plays a central role in [51].

Whereas the standard Besov spaces B^s_p,p are invariant under coordinate rotations, this is not the case for the tensor product spaces ˜B^s_p. In other words, the measure of regularity that governs the approximation rates achievable by anisotropic tensor product wavelets depends on the choice of coordinates.

Note that even if a dimension-independent convergence rate can be achieved in certain cases, by the constructions outlined above one generally does not obtain dimension-independent complexity:

even if the convergence rate remains unchanged, the constants in the estimates can blow up with increasing dimension. This problem actually does arise in basic examples of the adaptive solution of constant-coefficient elliptic partial differential equations on higher-dimensional product domains, cf. [44]. It is therefore to be expected that in order to keep the complexity of approximations in higher dimensions in check, it will in general be necessary to exploit further structural properties of the concrete problem at hand. In Section 4.3 we study examples where such problem-specific structure – here, approximability by separable functions – can be used to reduce the approximation complexity.

3 Higher-Dimensional Approximation and Adaptive Wavelet Methods