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

An inverse boundary value problem for the 1+1 dimensional wave equation $({\mathrm{\partial }}_t^2-c(x{)}^2{\mathrm{\partial }}_x^2)u(x,t)=0,x\in {\mathbb{R}}_{+}$ 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 H"older type estimate $\mathrm{\parallel }\tilde{c}-c\mathrm{\parallel }\le C{ϵ}^{\gamma }$, where $ϵ$ is the noise level.