Almost diagonal matrices and Besov-type spaces based on wavelet expansions

This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of $d$-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best $n$-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds $\Gamma$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter $\alpha$ of the space). For this purpose, a theory of almost diagonal matrices on related sequence spaces $b^\alpha_{p,q}(\nabla)$ of Besov type is developed. Keywords: Besov spaces, wavelets, localization, sequence spaces, adaptive methods, non-linear approximation, manifolds, domain decomposition.Comment: 38 pages, 2 figure

Besov regularity for operator equations on patchwise smooth manifolds

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in $\Omega$.Comment: 42 pages, 3 figures, updated after peer review. Preprint: Bericht Mathematik Nr. 2013-03 des Fachbereichs Mathematik und Informatik, Universit\"at Marburg. To appear in J. Found. Comput. Mat

Adaptive Wavelet BEM for boundary integral equations. Theory and numerical experiments

In this paper, we are concerned with the numerical treatment of boundary integral equations by means of the adaptive wavelet boundary element method (BEM). In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in $\mathbb{R}^3$. The corresponding operator equations are treated by means of adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions that can be made on the basis of a systematic investigation of the Besov regularity of the exact solution

Adaptive Wavelet BEM for boundary integral equations. Theory and numerical experiments

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ℝ 3 . The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution

Adaptive Wavelet BEM for boundary integral equations. Theory and numerical experiments

In this paper, we are concerned with the numerical treatment of boundary integral equations by means of the adaptive wavelet boundary element method (BEM). In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in $\mathbb{R}^3$. The corresponding operator equations are treated by means of adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions that can be made on the basis of a systematic investigation of the Besov regularity of the exact solution

A fast direct solver for nonlocal operators in wavelet coordinates

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix. To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields

Compression of boundary integral operators discretized by anisotropic wavelet bases

The present article is devoted to wavelet matrix compression for boundary integral equations when using anisotropic wavelet bases for the discretization. We provide a compression scheme which amounts to only $O(N)$ relevant matrix coefficients in the system matrix without deteriorating the accuracy offered by the underlying Galerkin scheme. Here, $N$ denotes the degrees of freedom in the related trial spaces. By numerical results we validate our theoretical findings

Wavelet solution of variable order pseudodifferential equations

Sobolev spaces H m(x)(I) of variable order 0<m(x)<1 on an interval I⊂ℝ arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X t∈ℝ with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary boundary conditions, we establish multilevel norm equivalences in H m(x)(I) and prove preconditioning and wavelet matrix compression results for the variable order pseudodifferential generators A of X. Sufficient conditions on A to satisfy a Gårding inequality in H m(x)(I) and time-analyticity of the semigroup T t associated with the Feller process X t are established. As application, wavelet-based algorithms of log-linear complexity are obtained for the valuation of contingent claims on pure jump Feller-Lévy processes X t with state-dependent jump intensity by numerical solution of the corresponding Kolmogoroff equation