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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 11, Issue 5

Issues

Volume 13 (2015)

Approximate multiplication in adaptive wavelet methods

Dana Černá
  • Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17, Liberec, Czech Republic
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/ Václav Finěk
  • Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17, Liberec, Czech Republic
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Published Online: 2013-03-14 | DOI: https://doi.org/10.2478/s11533-013-0216-x

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.

MSC: 65T60; 65F99; 65N99

Keywords: Adaptive methods; Wavelets; Matrix-vector multiplication

  • [1] Černá D., Finěk V., Construction of optimally conditioned cubic spline wavelets on the interval, Adv. Comput. Math., 2011, 34(2), 219–25 http://dx.doi.org/10.1007/s10444-010-9152-5Web of ScienceCrossrefGoogle Scholar

  • [2] Cohen A., Dahmen W., DeVore R., Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75 http://dx.doi.org/10.1090/S0025-5718-00-01252-7CrossrefGoogle Scholar

  • [3] Cohen A., Dahmen W., DeVore R., Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245 http://dx.doi.org/10.1007/s102080010027CrossrefGoogle Scholar

  • [4] Dahmen W., Wavelet and multiscale methods for operator equations, In: Acta Numer., 6, Cambridge University Press, Cambridge, 1997, 55–228 Google Scholar

  • [5] DeVore R.A., Nonlinear approximation, In: Acta Numer., 7, Cambridge University Press, Cambridge, 1998, 51–150 Google Scholar

  • [6] Dijkema T.J., Schwab Ch., Stevenson R., An adaptive wavelet method for solving high-dimensional elliptic PDEs, Constr. Approx., 2009, 30(3), 423–455 http://dx.doi.org/10.1007/s00365-009-9064-0Web of ScienceCrossrefGoogle Scholar

  • [7] Stevenson R., Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal., 2003, 41(3), 1074–1100 http://dx.doi.org/10.1137/S0036142902407988CrossrefGoogle Scholar

  • [8] Stevenson R., On the compressibility operators in wavelet coordinates, SIAM J. Math. Anal., 2004, 35(5), 1110–1132 http://dx.doi.org/10.1137/S0036141002411520CrossrefGoogle Scholar

About the article

Published Online: 2013-03-14

Published in Print: 2013-05-01


Citation Information: Open Mathematics, Volume 11, Issue 5, Pages 972–983, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0216-x.

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© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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