Relationships among Interpolation Bases of Wavelet Spaces and Approximation Spaces

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis

Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations

We study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses "time derivatives of order one half" on the bi-infinite time axis. We show that for linear, parabolic initial-boundary value problems on $(0,\infty)$, the corresponding bilinear form admits an inf-sup condition with sparse tensor product trial and test function spaces. We deduce optimality of compressive, space-time Galerkin discretizations, where stability of Galerkin approximations is implied by the well-posedness of the parabolic operator equation. The variational setting adopted here admits more general Riesz bases than previous work; in particular, no stability in negative order Sobolev spaces on the spatial or temporal domains is required of the Riesz bases accommodated by the present formulation. The trial and test spaces are based on Sobolev spaces of equal order $1/2$ with respect to the temporal variable. Sparse tensor products of multi-level decompositions of the spatial and temporal spaces in Galerkin discretizations lead to large, non-symmetric linear systems of equations. We prove that their condition numbers are uniformly bounded with respect to the discretization level. In terms of the total number of degrees of freedom, the convergence orders equal, up to logarithmic terms, those of best $N$-term approximations of solutions of the corresponding elliptic problems.Comment: 26 page

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