The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium

Abstract

In this paper we consider thermoacoustic tomography as the inverse problem of determining from lateral Cauchy data the unknown initial conditions in a wave equation with spatially varying coefficients. This problem also occurs in several applications in the area of medical imaging and nondestructive testing. Using the method of quasi-reversibility, the original ill-posed problem is replaced with a boundary value problem for a fourth order partial differential equation. We find a weak $H^2$ solution of this problem and show that it is a well-posed elliptic problem. Error estimates and convergence of the approximation follow from observability estimates for the wave equation, which are proved using a Carleman estimate. We derive a numerical scheme for the solution of the quasi-reversibility problem by a $B$-spline Galerkin method, for which we give error estimates. Finally, we present numerical results supporting the robustness of this method for the reconstruction of initial conditions from full and limited boundary data.