Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

Abstract

An inverse boundary value problem for the 1+1 dimensional wave equation $({\partial }_t^2-c(x{)}^2{\partial }_x^2)u(x,t)=0,x\in {\mathbb{R}}_{+}$ is considered. We give a discrete regularization strategy to recover wave speed $c(x)$ when we are given the boundary value of the wave, $u(0,t)$, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed $\tilde{c}$, satisfying a Hölder type estimate $\Vert \tilde{c}-c\Vert \le C{\epsilon }^{\gamma }$, where $\epsilon$ is the noise level.