Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

Jérémi  Dardé

doi:10.3934/ipi.2016005

Article Contents

2016, Volume 10, Issue 2: 379-407. Doi: 10.3934/ipi.2016005

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Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

Jérémi Dardé^1,

1.
Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9

Received: April 2015

Published: May 2016

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Abstract

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method proves to be efficient even with highly corrupted data.

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Mathematics Subject Classification: Primary: 35A15, 35R25, 35R30; Secondary: 35N25.

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