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An optimal adaptive wavelet method without coarsening of the iterands


Authors: Tsogtgerel Gantumur, Helmut Harbrecht and Rob Stevenson
Journal: Math. Comp. 76 (2007), 615-629
MSC (2000): Primary 41A25, 41A46, 65F10, 65T60
DOI: https://doi.org/10.1090/S0025-5718-06-01917-X
Published electronically: November 27, 2006
MathSciNet review: 2291830
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Abstract: In this paper, an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp, 70 (2001), pp. 27-75] by Cohen, Dahmen and DeVore, in the sense that there is no recurrent coarsening of the iterands. Despite this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method.


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  • [CDD01] A. Cohen, W. Dahmen, and R. DeVore.
    Adaptive wavelet methods for elliptic operator equations - Convergence rates.
    Math. Comp, 70:27-75, 2001. MR 1803124 (2002h:65201)
  • [CDD02] A. Cohen, W. Dahmen, and R. DeVore.
    Adaptive wavelet methods II - Beyond the elliptic case.
    Found. Comput. Math., 2(3):203-245, 2002. MR 1907380 (2003f:65212)
  • [CDF92] A. Cohen, I. Daubechies, and J.C. Feauveau.
    Biorthogonal bases of compactly supported wavelets.
    Comm. Pur. Appl. Math., 45:485-560, 1992. MR 1162365 (93e:42044)
  • [CM00] A. Cohen and R. Masson.
    Wavelet adaptive method for second order elliptic problems: Boundary conditions and domain decomposition.
    Numer. Math., 86:193-238, 2000. MR 1777487 (2001j:65185)
  • [Coh03] A. Cohen.
    Numerical Analysis of Wavelet Methods.
    Elsevier, Amsterdam, 2003. MR 1990555 (2004c:65178)
  • [CTU99] C. Canuto, A. Tabacco, and K. Urban.
    The wavelet element method part I: Construction and analysis.
    Appl. Comput. Harmon. Anal., 6:1-52, 1999. MR 1664902 (99k:42055)
  • [Dah99] S. Dahlke.
    Besov regularity for elliptic boundary value problems in polygonal domains.
    Appl. Math. Lett., 12(6):31-36, 1999. MR 1751404 (2001b:35077)
  • [DD97] S. Dahlke and R. DeVore.
    Besov regularity for elliptic boundary value problems.
    Comm. Partial Differential Equations, 22(1 & 2):1-16, 1997. MR 1434135 (97k:35047)
  • [DeV98] R. DeVore.
    Nonlinear approximation.
    Acta Numer., 7:51-150, 1998. MR 1689432 (2001a:41034)
  • [DFR04] S. Dahlke, M. Fornasier, and T. Raasch.
    Adaptive frame methods for elliptic operator equations.
    Bericht Nr. 2004-3, Philipps-Universität Marburg, 2004.
    To appear in Adv. Comput. Math.
  • [DHS05] W. Dahmen, H. Harbrecht, and R. Schneider.
    Adaptive methods for boundary integral equations - complexity and convergence estimates.
    IGPM report 250, RWTH Aachen, March 2005. To appear in Math. Comp.
  • [DS99a] W. Dahmen and R. Schneider.
    Composite wavelet bases for operator equations.
    Math. Comp., 68:1533-1567, 1999. MR 1648379 (99m:65122)
  • [DS99b] W. Dahmen and R. Schneider.
    Wavelets on manifolds I: Construction and domain decomposition.
    SIAM J. Math. Anal., 31:184-230, 1999. MR 1742299 (2000k:65242)
  • [GS06a] T. Gantumur and R.P. Stevenson.
    Computation of differential operators in wavelet coordinates.
    Math. Comp. 75:697-709, 2006. MR 2196987
  • [GS06b] T. Gantumur and R.P. Stevenson.
    Computation of singular integral operators in wavelet coordinates.
    Computing, 76:77-107, 2006. MR 2174673 (2006e:65051)
  • [HS06] H. Harbrecht and R.P. Stevenson.
    Wavelets with patchwise cancellation properties.
    Math. Comp., 75(256):1871-1889, 2006.
  • [Ste03] R.P. Stevenson.
    Adaptive solution of operator equations using wavelet frames.
    SIAM J. Numer. Anal., 41(3):1074-1100, 2003. MR 2005196 (2004e:42062)
  • [Ste06] R.P. Stevenson.
    Composite wavelet bases with extended stability and cancellation properties.
    Technical Report 1345, Department of Mathematics, Utrecht University, January 2006. To appear in SIAM J. Numer. Anal.
  • [Ste04] R.P. Stevenson.
    On the compressibility of operators in wavelet coordinates.
    SIAM J. Math. Anal., 35(5):1110-1132, 2004. MR 2050194 (2005e:42128)
  • [vS04] J. van den Eshof and G.L.G. Sleijpen.
    Inexact Krylov subspace methods for linear systems.
    SIAM J. Matrix Anal. Appl., 26(1):125-153, 2004. MR 2106939 (2005i:65046)

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Additional Information

Tsogtgerel Gantumur
Affiliation: Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Email: gantumur@math.uu.nl

Helmut Harbrecht
Affiliation: Institute of Computer Science and Applied Mathematics, Christian–Albrechts–Uni- versity of Kiel, Olshausenstr. 40, 24098 Kiel, Germany
Email: hh@numerik.uni-kiel.de

Rob Stevenson
Affiliation: Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Email: stevenson@math.uu.nl

DOI: https://doi.org/10.1090/S0025-5718-06-01917-X
Keywords: Adaptive methods, operator equations, wavelets, optimal computational complexity, best $N$-term approximation
Received by editor(s): March 22, 2005
Received by editor(s) in revised form: January 25, 2006
Published electronically: November 27, 2006
Additional Notes: This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society