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Mathematics of Computation

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Space-time adaptive wavelet methods for parabolic evolution problems

Authors: Christoph Schwab and Rob Stevenson
Journal: Math. Comp. 78 (2009), 1293-1318
MSC (2000): Primary 35K10, 41A25, 46B28, 65N99, 65T60
Published electronically: November 25, 2008
MathSciNet review: 2501051
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Abstract | References | Similar Articles | Additional Information

Abstract: With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.

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Additional Information

Christoph Schwab
Affiliation: Department of Mathematics, ETH Zürich, ETH Zentrum, HG G58.1, CH 8092 Zürich, Switzerland
MR Author ID: 305221

Rob Stevenson
Affiliation: Korteweg-de Vries Institute for Mathematics, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
MR Author ID: 310898

Keywords: Parabolic differential equations, wavelets, adaptivity, optimal computational complexity, best $N$-term approximation, matrix compression
Received by editor(s): January 3, 2008
Received by editor(s) in revised form: July 23, 2008
Published electronically: November 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.