Adaptive wavelet methods for PDEs

 

Host Institution:

Australian National University

Title of Seminar:

Adaptive Wavelet Methods for PDEs

Speaker's Name:

Prof. Dr Wolfgang Dahmen

Speaker's Institution:

RWTH Aachen, Germany

Time and Date:

Wednesday 6 October 2010 at 3:00 pm

Seminar Abstract:

We begin by giving a brief introduction to wavelet decompositions. Using the Haar basis as a simple example, those key features of wavelets are highlighted that are most relevant for the solution of partial differential or singular integral equations, namely cancellation properties and norm equivalences. Some principles are outlined which allow one to construct bases with these properties on nontrivial domains.

These key properties of wavelets can be exploited best for a wide class of well-posed variational problems. This means that the induced operator is a norm isomorphism from what one may call the "energy space" associated with the problem onto its dual. This covers coercive as well as indefinite problems such as saddle point problems. The main objective of this lecture is to explain an adaptive solution paradigm for this scope of problems which is based on an (idealized) iteration of an equivalent formulation of the full infinite dimensional problem in wavelet coordinates. In fact, whenever a Riesz basis for the corresponding energy space is available, one can show that in this formulation the problem can be shown to be well conditioned so that the iterative schemes exhibit a fixed error reduction per step. It is briefly indicated how this relates to residual based error estimators for finite element discretizations. The numerical realization can then be understood as a perturbation of such ideal iterations where the involved operators are applied adaptively with suitably updated accuracy tolerances. These concepts apply to certain nonlinear problems as well. The main algorithmic ingredients are explained. These developments are closely intertwined with some concepts from nonlinear approximation which are briefly indicated. A central role is played by tree approximation and coarsening schemes. They also lead to quantitative estimates for the action of nonlinear operators on multiscale decompositions. The central message is that such methods can be shown to generate an approximate solution of the underlying variational problem for any target accuracy at an expense that stays uniformly proportional to the smallest number of degrees of freedom (with respect to the underlying Riesz basis) needed to realize that target accuracy.

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