# American Institute of Mathematical Sciences

August  2011, 5(3): 731-744. doi: 10.3934/ipi.2011.5.731

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

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

Received  January 2011 Revised  April 2011 Published  August 2011

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 $\tau(y)$.
Citation: Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems & Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731
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