Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms

Nonlinear Approximation and Adaptive Techniques for Solving Elliptic Operator Equations

This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions the best possible approximation rate, which a function can have for a given number of degrees of freedom, is characterized in te...

Piecewise tensor product wavelet bases by extensions and approximation rates

In this chapter, we present some of the major results that have been achieved in the context of the DFG-SPP project “Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Problems”. This project has been concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with related inve...

Nonlinear and Adaptive Frame Approximation Schemes for Elliptic PDEs: Theory and Numerical Experiments

This paper is concerned with adaptive numerical frame methods for elliptic operator equations. We show how specific non-canonical frame expansions on domains can be constructed. Moreover, we study the approximation order of best n-term frame approximation which serves as the benchmark for the performance of adaptive schemes. We also discuss numerical experiments for second order elliptic bounda...

Adaptive frame methods for elliptic operator equations: the steepest descent approach

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent itera...

An optimal adaptive wavelet method - for strongly elliptic operator equations