Stephan Dahlke

Besov regularity for elliptic boundary value problems

This paper studies the regularity of solutions to boundary value problems for the Laplace operator on Lipschitz domains {Omega} in R{sup d} and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinear approximation in L{sub p}({Omega}) are the Besov spaces B{sub {tau}}{sup {alpha}}(L{sub {tau}}({Omega})), {tau} := ({alpha}/d + 1/p){sup -1}. Thus, the regularity of the solution in T this scale of Besov spaces is investigated with the aim of determining the largest a for which the solution is in B{sub {tau}}{sup {alpha}}(L{sub {tau}}({Omega})). The regularity theorems given in this paper build upon the recent results of Jerison and Kenig. The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions. 14 refs., 1 fig.

Stable multiscale bases and local error estimation for elliptic problems

Abstract This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient a-posteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms of wavelet coefficients leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order.

Optimal approximation of elliptic problems by linear and nonlinear mappings I

We study the optimal approximation of the solution of an operator equation Au=f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs on an arbitrary bounded Lipschitz domain. Here we prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. Our results are concerned with approximation, not with computation. Our goal is to understand better the possibilities of nonlinear approximation.

THE UNCERTAINTY PRINCIPLE ASSOCIATED WITH THE CONTINUOUS SHEARLET TRANSFORM

Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.

Adaptive Wavelet Schemes for Elliptic Problems---Implementation and Numerical Experiments

Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution behaves asymptotically like the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets. On one hand, the results are purely asymptotic. On the other hand, the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet. It is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results. We briefly review first the main theoretical facts, describe the main ingredients of the algorithm, highlight the essential data structures, and illustrate the results by one- and two-dimensional numerical examples including comparisons with an adaptive finite element scheme.

Adaptive frame methods for elliptic operator equations

Abstract This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.

Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates

In this paper an adaptive wavelet scheme for saddle point problems is developed and analyzed. Under the assumption that the underlying continuous problem satisfies the inf-sup condition, it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. This means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the error of the best wavelet N-term approximation of the solution with respect to the relevant norms. Moreover, the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom. It is remarkable that compatibility constraints on the trial spaces such as the Ladyzhenskaya--Babuska--Brezzi (LBB) condition do not arise. In the second part the general results are applied to the Stokes problem. Aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon, the regularity of the solutions in certain Besov scales is analyzed. These results reveal under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements. This in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the Stokes problem.

Adaptive wavelet methods and sparsity reconstruction for inverse heat conduction problems

This paper is concerned with the numerical treatment of inverse heat conduction problems. In particular, we combine recent results on the regularization of ill-posed problems by iterated soft shrinkage with adaptive wavelet algorithms for the forward problem. The analysis is applied to an inverse parabolic problem that stems from the industrial process of melting iron ore in a steel furnace. Some numerical experiments that confirm the applicability of our approach are presented.

Multiresolution analysis and wavelets on S 2 and S 3

In this paper, we construct a multiresolution analysis and a wavelet basis on two specific compact manifolds. Using special charts, the problem is reduced to finding appropriate nested spaces on rectangular domains. The claim of C 1-continuity gives rise to certain boundary conditions on the rectangles. To satisfy these conditions, we use a tensor product approach in which one factor is an exponential spline.

Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere

This paper is concerned with the construction of generalized Banach frames on homogeneous spaces. The major tool is a unitary group representation which is square integrable modulo a certain subgroup. By means of this representation, generalized coorbit spaces can be defined. Moreover, we can construct a specific reproducing kernel which, after a judicious discretization, gives rise to atomic decompositions for these coorbit spaces. Furthermore, we show that under certain additional conditions our discretization method generates Banach frames. We also discuss nonlinear approximation schemes based on the atomic decomposition. As a classical example, we apply our construction to the problem of analyzing and approximating functions on the spheres.