Advances in Computational Mathematics

, Volume 1, Issue 3, pp 259–335

Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution

  • W. Dahmen
  • S. Prössdorf
  • R. Schneider
Articles

DOI: 10.1007/BF02072014

Cite this article as:
Dahmen, W., Prössdorf, S. & Schneider, R. Adv Comput Math (1993) 1: 259. doi:10.1007/BF02072014

Abstract

This is the second part of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in ℝn. This setting covers classical Galerkin methods, collocation, and quasi-interpolation. The numerical methods are based on a general framework of multiresolution analysis, i.e. of sequences of nested spaces which are generated by refinable functions. In this part, we analyse compression techniques for the resulting stiffness matrices relative to wavelet-type bases. We will show that, although these stiffness matrices are generally not sparse, the order of the overall computational work which is needed to realize a certain accuracy is of the formO(N(logN)b), whereN is the number of unknowns andb ≥ 0 is some real number.

Keywords

Periodic pseudodifferential equations pre-wavelets biorthogonal wavelets generalized Petrov-Galerkin schemes wavelet representation atomic decomposition Calderón-Zygmund operators matrix compression error analysis 

Subject classification AMS

65F35 65J10 65N30 65N35 65R20 47A20 47G30 45P05 41A25 

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • W. Dahmen
    • 1
  • S. Prössdorf
    • 2
  • R. Schneider
    • 3
  1. 1.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  2. 2.Institut für Angewandte Analysis und StochastikBerlinGermany
  3. 3.Fachbereich MathematikTechnische Hochschule DarmstadtDarmstadtGermany

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