On the Compressibility of Integral Operators in Anisotropic Wavelet Coordinates
Helmut Harbrecht, Remo von Rickenbach
TL;DR
The article demonstrates that boundary integral operators discretised with anisotropic tensor-product wavelets are $s^{\star}$-compressible, enabling adaptive $N$-term approximations with near-optimal complexity. By constructing and analyzing a multi-stage compression scheme (diagonal cutoff, far-field, mixed-field, and near-field), it provides explicit decay and sparsity bounds that yield $s^{\star}=\min\{\sigma,\alpha(d-q),\tilde{d}/2+q-\max\{0, q\alpha(d-q)/(\gamma-q)\}\}$, and shows that for large enough regularity and vanishing moments, $s^{\star}>d-q$, allowing the adaptive method to achieve the rate $N^{-s}$ for any $s<s^{\star}$ with $\mathcal{O}(N)$ work. The framework extends to Lipschitz manifolds, where patchwise anisotropic wavelet representations retain the same compression properties. Overall, the work establishes a rigorous foundation for quasi-optimal adaptive boundary element methods using anisotropic wavelets and provides practical guidance on wavelet design and manifold extensions.
Abstract
The present article is concerned with the s*-compressibility of classical boundary integral operators in anisotropic wavelet coordinates. Having the s*-compressibility at hand, one can design adaptive wavelet algorithms which are asymptotically optimal, meaning that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom (within the setting determined by an underlying wavelet basis) that would ideally be necessary for realising that target accuracy if full knowledge about the unknown solution were given. As we consider here anisotropic wavelet coordinates, we can achieve higher convergence rates compared to the standard, isotropic setting. Especially, edge singularities of anisotropic nature can be resolved.