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 \begin{document}$ (\partial_t^2 - c(x)^2 \partial_x^2)u(x,t) = 0,\quad x\in\mathbb{R}_+ $\end{document} is considered. We give a discrete regularization strategy to recover wave speed \begin{document}$ c(x) $\end{document} when we are given the boundary value of the wave, \begin{document}$ u(0,t) $\end{document} , that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed \begin{document}$ \widetilde c $\end{document} , satisfying a Holder type estimate \begin{document}$ \| \widetilde c-c\|\leq C \epsilon^{\gamma} $\end{document} , where \begin{document}$ \epsilon $\end{document} is the noise level.