# Reconstruction of a Time-dependent Potential from Wave Measurements

## TL;DR

This paper addresses the challenging problem of reconstructing a time-dependent potential in a wave equation from wave measurements, providing theoretical guarantees and a practical numerical algorithm.

## Contribution

It offers a new existence and uniqueness result for the inverse problem, computes the Fréchet derivative, and develops a regularized Newton-like method for reconstruction.

## Key findings

- Successful numerical reconstructions demonstrating feasibility
- High reconstruction quality in simulated examples
- Efficient algorithm with favorable computational performance

## Abstract

We add a time-dependent potential to the inhomogeneous wave equation and consider the task of reconstructing this potential from measurements of the wave field. This dynamic inverse problem becomes more involved compared to static parameters, as, e.g. the dimensions of the parameter space do considerably increase. We give a specifically tailored existence and uniqueness result for the wave equation and compute the Fr\'echet derivative of the solution operator, for which also show the tangential cone condition. These results motivate the numerical reconstruction of the potential via successive linearization and regularized Newton-like methods. We present several numerical examples showing feasibility, reconstruction quality, and time efficiency of the resulting algorithm.