A time reversal based algorithm for solving initial data inverse problems

Kazufumi Ito; Karim Ramdani; Marius Tucsnak

doi:10.3934/dcdss.2011.4.641

Article Contents

2011, Volume 4, Issue 3: 641-652. Doi: 10.3934/dcdss.2011.4.641

This issue Previous Article Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map Next Article Convergence of solutions of a non-local phase-field system

A time reversal based algorithm for solving initial data inverse problems

1.
Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695-8205
2.
INRIA Nancy Grand-Est (CORIDA), 615 rue du Jardin Botanique, 54600, Villers-lès-Nancy
3.
Institut Elie Cartan Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex

Received: March 31, 2009

Revised: October 31, 2009

Published: May 2011

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Abstract

We propose an iterative algorithm to solve initial data inverse problems for a class of linear evolution equations, including the wave, the plate, the Schrödinger and the Maxwell equations in a bounded domain $\Omega$. We assume that the only available information is a distributed observation (i.e. partial observation of the solution on a sub-domain $\omega$ during a finite time interval $(0,\tau)$). Under some quite natural assumptions (essentially : the exact observability of the system for some time $\tau_{obs}>0$, $\tau\ge \tau_{obs}$ and the existence of a time-reversal operator for the problem), an iterative algorithm based on a Neumann series expansion is proposed. Numerical examples are presented to show the efficiency of the method.

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Mathematics Subject Classification: 35L50, 35Q93, 35R30, 93B07.

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