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

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 employed in [23]. 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.