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Central European Journal of Mathematics

, Volume 11, Issue 5, pp 972–983 | Cite as

Approximate multiplication in adaptive wavelet methods

  • Dana ČernáEmail author
  • Václav Finěk
Research Article
  • 93 Downloads

Abstract

Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l 2-problem, finding the convergent iteration process for the l 2-problem and finally using its finite dimensional approximation which works with an inexact right-hand side and approximate matrix-vector multiplication. In our contribution, we pay attention to approximate matrix-vector multiplication which is enabled by an off-diagonal decay of entries of the wavelet stiffness matrices. We propose a more efficient technique which better utilizes actual decay of matrix and vector entries and we also prove that this multiplication algorithm is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined.

Keywords

Adaptive methods Wavelets Matrix-vector multiplication 

MSC

65T60 65F99 65N99 

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References

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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Didactics of MathematicsTechnical University of LiberecLiberecCzech Republic

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