13 research outputs found
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 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 -dimensional manifolds and investigate some analytical properties
(such as, e.g., embeddings and best -term approximation rates) of the
resulting quasi-Banach spaces. In particular, we prove that different prominent
constructions of biorthogonal wavelet systems on domains or manifolds
which admit a decomposition into smooth patches actually generate the
same Besov-type function spaces , provided that
their univariate ingredients possess a sufficiently large order of cancellation
and regularity (compared to the smoothness parameter of the space).
For this purpose, a theory of almost diagonal matrices on related sequence
spaces 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 such as, e.g., boundaries of
polyhedral domains . Using suitable biorthogonal
wavelet bases , we introduce a new class of Besov-type spaces
of functions
. Special attention is paid on the
rate of convergence for best -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
into , ,
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 .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
Besov regularity of solutions to the p-Poisson equation
In this paper, we study the regularity of solutions to the -Poisson
equation for all . In particular, we are interested in smoothness
estimates in the adaptivity scale , , of Besov spaces. The regularity in this scale determines the
order of approximation that can be achieved by adaptive and other nonlinear
approximation methods. It turns out that, especially for solutions to
-Poisson equations with homogeneous Dirichlet boundary conditions on bounded
polygonal domains, the Besov regularity is significantly higher than the
Sobolev regularity which justifies the use of adaptive algorithms. This type of
results is obtained by combining local H\"older with global Sobolev estimates.
In particular, we prove that intersections of locally weighted H\"older spaces
and Sobolev spaces can be continuously embedded into the specific scale of
Besov spaces we are interested in. The proof of this embedding result is based
on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure
An analysis of electrical impedance tomography with applications to Tikhonov regularization
This paper analyzes the continuum model/complete electrode model in the electrical
impedance tomography inverse problem of determining the conductivity parameter from
boundary measurements. The continuity and differentiability of the forward operator with
respect to the conductivity parameter in
Lp-norms are proved. These analytical results
are applied to several popular regularization formulations, which incorporate a
priori information of smoothness/sparsity on the inhomogeneity through Tikhonov
regularization, for both linearized and nonlinear models. Some important properties,
e.g., existence, stability, consistency and
convergence rates, are established. This provides some theoretical justifications of their
practical usage
Piecewise Tensor Product Wavelet Bases by Extensions and Approximation Rates
Following [Studia Math., 76(2) (1983), pp. 1-58 and 95-136] by Z. Ciesielski and T. Figiel and [SIAM J. Math. Anal., 31 (1999), pp. 184-230] by W. Dahmen and R. Schneider, by the application of extension operators we construct a basis for a range of Sobolev spaces on a domain from corresponding bases on subdomains that form a non-overlapping decomposition. As subdomains, we take hypercubes, or smooth parametric images of those, and equip them with tensor product wavelet bases. We prove approximation rates from the resulting piecewise tensor product basis that are independent of the spatial dimension of . For two- and three-dimensional polytopes we show that the solution of Poisson type problems satisfies the required regularity condition. The dimension independent rates will be realized numerically in linear complexity by the application of the adaptive wavelet-Galerkin scheme