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Journal Articles Inverse Problems Year : 2006

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

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.

Dates and versions

hal-00876239 , version 1 (31-10-2013)

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Laurent Bourgeois. Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation. Inverse Problems, 2006, 22 (2), pp.413-430. ⟨10.1088/0266-5611/22/2/002⟩. ⟨hal-00876239⟩
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