Adaptive Wavelet Schemes for Nonlinear Variational Problems
Abstract
We develop and analyze wavelet based adaptive schemes for nonlinear variational problems. We derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal. Our approach is based on a new paradigm that has been put forward recently for a class of linear problems. The original problem is transformed first into an equivalent one which is well posed in the Euclidean metric $\ell_2$. Then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step. This iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances. The main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions. We prove asymptotically optimal complexity for adaptive realizations of first order iterations and of Newton's method.
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References
1.
A. Barinka, Fast Evaluation Tools for Adaptive Wavelet Schemes, Ph.D. Dissertation, Rheinisch Westfälische Technische Hochschule Aachen, Aachen, Germany, 2002, in preparation.
2.
Arne Barinka, Titus Barsch, Philippe Charton, Albert Cohen, Stephan Dahlke, Wolfgang Dahmen, Karsten Urban, Adaptive wavelet schemes for elliptic problems—implementation and numerical experiments, SIAM J. Sci. Comput., 23 (2001), 910–939
3.
P. Binev and R. DeVore, Fast computation in adaptive tree approximation, Numer. Math., to appear.
4.
P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., to appear.
5.
Claudio Canuto, Anita Tabacco, Karsten Urban, The wavelet element method. I. Construction and analysis, Appl. Comput. Harmon. Anal., 6 (1999), 1–52
6.
Claudio Canuto, Anita Tabacco, Karsten Urban, The wavelet element method. II. Realization and additional features in 2D and 3D, Appl. Comput. Harmon. Anal., 8 (2000), 123–165
7.
Albert Cohen, Numerical analysis of wavelet methods, Studies in Mathematics and its Applications, Vol. 32, North‐Holland Publishing Co., 2003xviii+336
8.
Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 70 (2001), 27–75
9.
A. Cohen, W. Dahmen, R. DeVore, Adaptive wavelet methods. II. Beyond the elliptic case, Found. Comput. Math., 2 (2002), 203–245
10.
Albert Cohen, Wolfgang Dahmen, Ronald Devore, Sparse evaluation of compositions of functions using multiscale expansions, SIAM J. Math. Anal., 35 (2003), 279–303
11.
A. Cohen, R. Masson, Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition, Numer. Math., 86 (2000), 193–238
12.
Stephan Dahlke, Wolfgang Dahmen, Reinhard Hochmuth, Reinhold Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math., 23 (1997), 21–47, Multilevel methods (Oberwolfach, 1995)
13.
Stephan Dahlke, Wolfgang Dahmen, Karsten Urban, Adaptive wavelet methods for saddle point problems—optimal convergence rates, SIAM J. Numer. Anal., 40 (2002), 1230–1262
14.
Wolfgang Dahmen, Reinhold Schneider, Composite wavelet bases for operator equations, Math. Comp., 68 (1999), 1533–1567
15.
Wolfgang Dahmen, Reinhold Schneider, Wavelets on manifolds. I. Construction and domain decomposition, SIAM J. Math. Anal., 31 (1999), 184–230
16.
Wolfgang Dahmen, Reinhold Schneider, Yuesheng Xu, Nonlinear functionals of wavelet expansions—adaptive reconstruction and fast evaluation, Numer. Math., 86 (2000), 49–101
17.
Wolfgang Dahmen, Karsten Urban, Jürgen Vorloeper, Adaptive wavelet methods—basic concepts and applications to the Stokes problem, Ser. Anal., Vol. 1, World Sci. Publishing, River Edge, NJ, 2002, 39–80
18.
P. Deuflhard and M. Weiser, Local inexact Newton multilevel FEM for nonlinear elliptic problems, in Computational Science for the 21st Century (Tours, France), M.‐O. Bristeau, G. Etgen, W. Fitzigibbon, J.‐L. Lions, J. Periaux, and M. Wheeler, eds., Wiley‐Interscience‐Europe, London, 1997, pp. 129–138.
19.
Peter Deuflhard, Martin Weiser, Global inexact Newton multilevel FEM for nonlinear elliptic problems, Lect. Notes Comput. Sci. Eng., Vol. 3, Springer, Berlin, 1998, 71–89
20.
Ronald DeVore, Nonlinear approximation, Acta Numer., Vol. 7, Cambridge Univ. Press, Cambridge, 1998, 51–150
21.
J. Pousin, J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov‐Galerkin methods applied to nonlinear problems, Numer. Math., 69 (1994), 213–231
22.
Michael Renardy, Robert Rogers, An introduction to partial differentialequations, Texts in Applied Mathematics, Vol. 13, Springer‐Verlag, 1993xiv+428
23.
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh‐Refinement Techniques, Wiley‐Teubner, New York, 1996.
24.
Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer‐Verlag, 1985xxii+662, Variational methods and optimization; Translated from the German by Leo F. Boron
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SIAM Journal on Numerical Analysis
Pages: 1785 - 1823
ISSN (online): 1095-7170
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Copyright © 2003 Society for Industrial and Applied Mathematics.
History
Published online: 25 July 2006
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