  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.
