Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation

Laurent Bourgeois 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We consider the quasi-reversibility method to solve the Cauchy problem for Laplace's equation in a smooth bounded domain. We assume that the Cauchy data are contaminated by some noise of amplitude σ, so that we make a regular choice of ε as a function of σ, where ε is the small parameter of the quasi-reversibility method. Specifically, we present two different results concerning the convergence rate of the solution of quasi-reversibility to the exact solution when σ tends to 0. The first result is a convergence rate of type 1\big/\big(\log{\frac{1}{{{\sigma}}}}\big)^\beta in a truncated domain, the second one holds when a source condition is assumed and is a convergence rate of type {{\sigma}}^{\frac{1}{2}} in the whole domain. © 2006 IOP Publishing Ltd.
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Submitted on : Thursday, October 31, 2013 - 5:05:47 PM
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Laurent Bourgeois. Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation. Inverse Problems, IOP Publishing, 2006, 22 (2), pp.413-430. ⟨10.1088/0266-5611/22/2/002⟩. ⟨hal-00876239⟩

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