Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements

Lauri Oksanen

doi:10.3934/ipi.2011.5.731

Article Contents

2011, Volume 5, Issue 3: 731-744. Doi: 10.3934/ipi.2011.5.731

This issue Previous Article Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice Next Article

Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements

Lauri Oksanen^1,

1.
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki

Received: December 31, 2010

Revised: March 31, 2011

Published: July 2011

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Abstract

We consider the inverse problem for the wave equation on a compact Riemannian manifold or on a bounded domain of $\mathbb{R}^n$, and generalize the concept of domain of influence. We present an efficient minimization algorithm to compute the volume of a domain of influence using boundary measurements and time-reversed boundary measurements. Moreover, we show that if the manifold is simple, then the volumes of the domains of influence determine the manifold. For a continuous real valued function $\tau$ on the boundary of the manifold, the domain of influence is the set of those points on the manifold from which the travel time to some boundary point $y$ is less than $\tau(y)$.

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Mathematics Subject Classification: Primary: 35R30.

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