Adaptive Wavelet Methods for Elliptic Operator Equations - Convergence Rates
Abstract
This paper is concerned with the construction and analysis of wavelet-basedadaptive algorithms for the numerical solution of elliptic equations. These algorithmsapproximate the solution u of the equation by a linear combination of Nwavelets. Therefore, a benchmark for their performance is provided by the rate ofbest approximation to u by an arbitrary linear combination of N wavelets (so calledN-term approximation), which would be obtained by keeping the N largest waveletcoefficients of the real solution (which of course is unknown). The main result of thepaper is the construction of an adaptive scheme which produces an approximationto u with error O(Ns) in the energy norm, whenever such a rate is possible byN-term approximation. The range of s > 0 for which this holds is only limited bythe approximation properties of the wavelets together with their ability to compressthe elliptic operator. Moreover, it is shown that the number of arithmetic operationsneeded to compute the approximate solution stays proportional to N. Theadaptive algorithm applies to a wide class of elliptic problems and wavelet bases.The analysis in this paper puts forward new techniques for treating elliptic problemsas well as the linear systems of equations that arise from the wavelet discretization.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 19, 1998
- Accession Number
- AD1000113
Entities
People
- Albert Cohen
- Ronald DeVore
- Wolfgang Dahmen
Organizations
- University of South Carolina