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

Abstract

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 $\mathcal{A}:c\mapsto \Lambda ,$ where $\Lambda$ 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 $\tilde{\Lambda }=\Lambda +\mathcal{E}$, where $\mathcal{E}$ corresponds to the measurement errors, and reconstruct an approximate wave speed $\tilde{c}$. We emphasize that $\tilde{\Lambda }$ may not not be in the range of the map $\mathcal{A}$. We show that the reconstructed wave speed $\tilde{c}$ satisfies $\Vert \tilde{c}-c{\Vert }_{L^{\infty }}<C\Vert E{\Vert }^{1/18}$. Our regularization strategy is based on a new formula to compute $c$ from $\Lambda$.

References in corpus (1)

• Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Cited by in corpus (2)

• Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data
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