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

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 {\em 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 $\tau (y)$.