On Exponential Instability of an Inverse Problem for the Wave Equation

TL;DR

This paper demonstrates that recovering a potential in a wave equation from boundary measurements is exponentially unstable, especially when measurements are limited and the potential is in the shadow region of an obstacle.

Contribution

It provides an explicit example showing exponential instability in the inverse problem for the wave equation with obstacle constraints.

Findings

Recovery of the potential is exponentially unstable in certain measurement configurations.

Measurements in a subset and shadow region of the obstacle lead to instability.

The instability persists even with limited measurement data.

Abstract

For a time-independent potential $q\in L^\infty$, consider the source-to-solution operator that maps a source $f$ to the solution $u=u(t,x)$ of $(\Box+q)u=f$ in Euclidean space with an obstacle, where we impose on $u$ vanishing Cauchy data at $t=0$ and vanishing Dirichlet data at the boundary of the obstacle. We study the inverse problem of recovering the potential $q$ from this source-to-solution map restricted to some measurement domain. By giving an example where measurements take place in some subset and the support of $q$ lies in the `shadow region' of the obstacle, we show that recovery of $q$ is exponentially unstable.