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

TL;DR

This paper introduces a minimization algorithm leveraging boundary and time-reversed measurements to solve inverse wave problems, demonstrating that in simple manifolds, the volumes of influence domains uniquely determine the manifold.

Contribution

It generalizes the domain of influence concept and provides an efficient method to recover manifold geometry from boundary measurements.

Findings

The algorithm accurately computes influence domain volumes.

In simple manifolds, influence domain volumes determine the manifold.

The method extends to general Riemannian manifolds with boundary.

Abstract

We consider the inverse problem for the wave equation on a compact Riemannian manifold or on a bounded domain of $\R^n$, 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 $\tau(y)$.