Wavelet moment method for the Cauchy problem for the Helmholtz equation

Authors:
Teresa Regińska;Andrzej Wakulicz
Affiliations:
Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland;Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Ten lectures on wavelets

Ten lectures on wavelets
Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation

SIAM Journal on Numerical Analysis

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Abstract

The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from inexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the moment problem that is equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of the Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of the inexact Cauchy data, allows us to prove the convergence and stability of the method.