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

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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.

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Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52062, Aachen, Germany

    W. Dahmen

  2. Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 5–7, D-10117, Berlin, Germany

    S. Prössdorf

  3. Fachbereich Mathematik, Technische Hochschule Darmstadt, Schloßgartenstraße 7, D-64289, Darmstadt, Germany

    R. Schneider

Authors
  1. W. Dahmen
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  2. S. Prössdorf
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  3. R. Schneider
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Dedicated to Charles A. Micchelli on the occasion of his fiftieth birthday

The third author has been supported by a grant of the Deutsche Forschungsgemeinschaft under Grant No. Ko 634/32-1.

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Dahmen, W., Prössdorf, S. & Schneider, R. Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution. Adv Comput Math 1, 259–335 (1993). https://doi.org/10.1007/BF02072014

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