Regularization strategy for inverse problem for 1+1 dimensional wave equation

An inverse boundary value problem for a 1+1 dimensional wave equation with wave speed $c(x)$ is considered. We give a regularisation strategy for inverting the map $\mathcal{A}:c\mapsto \mathrm{\Lambda },$ where $\mathrm{\Lambda }$ is the hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed $c$. More precisely, we consider the case when we are given a perturbation of the Neumann-to-Dirichlet map $\tilde{\mathrm{\Lambda }}=\mathrm{\Lambda }+\mathcal{E}$, where $\mathcal E$ corresponds to the measurement errors, and reconstruct an approximate wave speed $\tilde{c}$. We emphasize that $\tilde{\mathrm{\Lambda }}$ may not not be in the range of the map $\mathcal{A}$. We show that the reconstructed wave speed $\tilde{c}$ satisfies $\mathrm{\parallel }\tilde{c}-c{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}<C\mathrm{\parallel }E{\mathrm{\parallel }}^{1\mathrm{/}18}$. Our regularization strategy is based on a new formula to compute $c$ from $\mathrm{\Lambda }$.