Stability estimate for hyperbolic inverse problem with time dependent coefficient

TL;DR

This paper establishes a log-type stability estimate for determining a time-dependent coefficient in a wave equation from boundary data, advancing inverse problem analysis in higher dimensions.

Contribution

It provides new stability estimates for the inverse problem of identifying a time-dependent coefficient in the wave equation, including extensions to larger regions with more data.

Findings

Log-type stability estimate in dimension n≥2

Extension of stability to larger regions with additional data

Applicable to inverse problems with boundary observations

Abstract

We study the stability in the inverse problem of determining the time dependent zeroth-order coefficient $q(t,x)$ arising in the wave equation, from boundary observations. We derive, in dimension $n\geq 2$, a log-type stability estimate in the determination of $q$ from the Dirichlet-to-Neumann map, in a subset of our domain assuming that it is known outside this subset. Moreover, we prove that we can extend this result to the determination of $q$ in a larger region, and then in the whole domain provided that we have much more data.