Foundations of Computational Mathematics

, Volume 2, Issue 3, pp 203–245

Adaptive Wavelet Methods II—Beyond the Elliptic Case

  •  Cohen
  •  Dahmen
  •  DeVore

DOI: 10.1007/s102080010027

Cite this article as:
Cohen, Dahmen & DeVore Found. Comput. Math. (2002) 2: 203. doi:10.1007/s102080010027

Abstract.

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.

Key words. Operator equations, Indefinite problems, Adaptive methods, Convergence rates, Quasi-sparse matrices and vectors, Best N -term approximation, Fast matrix vector multiplication.

Copyright information

© 2002 Society for the Foundation of Computational Mathema tics

Authors and Affiliations

  •  Cohen
    • 1
  •  Dahmen
    • 2
  •  DeVore
    • 3
  1. 1.Laboratoire d'Analyse Numerique Universite Pierre et Marie Curie 4 Place Jussieu 75252 Paris Cedex 05, France cohen@ann.jussieu.fr http://www.ann.jussieu.fr/~cohen/FR
  2. 2.Institut f{ü}r Geometrie und Praktische Mathematik RWTH Aachen Templergraben 55 52056 Aachen, Germany dahmen@igpm.rwth-aachen.de http://www.igpm.rwth-aachen.de/~dahmen/DE
  3. 3.Department of Mathematics University of South Carolina Columbia, SC 29208, USA devore@math.sc.edu http://www.math.sc.edu/~devore/US