An inverse problem for a semi-linear wave equation: a numerical study

TL;DR

This paper develops a numerical method to recover a potential in a semi-linear wave equation from boundary measurements, using higher order linearization and regularization techniques, tested on synthetic noisy data.

Contribution

It introduces a numerical scheme combining higher order linearization and Tikhonov regularization for inverse problems in semi-linear wave equations.

Findings

Successful reconstruction of potentials from noisy data

Effective estimation of derivatives via Tikhonov regularization

Validation through synthetic numerical experiments

Abstract

We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy Dirichlet-to-Neumann map on the lateral boundary. The scheme is based on the recent higher order linearization method [20]. We also present an approach to numerically estimating two-dimensional derivatives of noisy data via Tikhonov regularization. The methods are tested using synthetic noisy measurements of the Dirichlet-to-Neumann map. Various examples of reconstructions of the potential functions are given.