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Adaptive Wavelet Methods for Linear and Nonlinear Least-Squares Problems

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Abstract

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27–75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations \(G(u)=0\), where \(G:H \rightarrow K'\), is studied. For formulations of partial differential equations as first-order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient.

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Acknowledgments

The author would like to thank Dr. Nabi Chegini for helpful discussions and for performing the numerical calculations.

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Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE , Amsterdam, The Netherlands

    Rob Stevenson

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  1. Rob Stevenson
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Correspondence to Rob Stevenson.

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Communicated by Wolfgang Dahmen.

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Stevenson, R. Adaptive Wavelet Methods for Linear and Nonlinear Least-Squares Problems. Found Comput Math 14, 237–283 (2014). https://doi.org/10.1007/s10208-013-9184-6

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  • DOI: https://doi.org/10.1007/s10208-013-9184-6

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