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   -> Volume 7, Issue 7


Preprint: Nonlinear Functionals of Wavelet Expansions
 
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"Angela Kunoth" (kunoth@igpm.rwth-aachen.de)
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PostPosted: Wed Jul 15, 1998 6:30 pm    
Subject: Preprint: Nonlinear Functionals of Wavelet Expansions
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#4 Preprint: Nonlinear Functionals of Wavelet Expansions

Nonlinear Functionals of Wavelet Expansions
- Adaptive Reconstruction and Fast Evaluation

Wolfgang Dahmen, Reinhold Schneider, Yuesheng Xu

Abstract:

This paper is concerned with the efficient evaluation of nonlinear
expressions of wavelet expansions obtained through an adaptive
process. In particular, evaluation covers here the computation of
inner products of such expressions with wavelets which arise, for
instance, in the context of Galerkin or Petrov Galerkin schemes for
the solution of differential equations. The central objective is to
develop schemes that facilitate such expansions at a computational
expense exceeding the complexity of the given expansion, i.e., the
number of nonzero wavelet coefficients, as little as possible. The
following issues are addressed. First, motivated by previous
treatments of the subject, we discuss the type of regularity
assumptions that are appropriate in this context and explain the
relevance of Besov norms. The principal strategy is to relate the
computation of inner products of wavelets with compositions to
approximations of compositions in terms of possibly few dual
wavelets. The analysis of these approximations finally leads to a
concrete evaluation scheme which is shown to be in a certain sense
asymptotically optimal. We conclude with a simple numerical example.

IGPM-Report 160, May 1998, RWTH Aachen
http://www.igpm.rwth-aachen.de/~dahmen/rep_98.html
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