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 non-destructive 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 proven 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.

A globally convergent numerical method for a coefficient inverse problem with backscattering data

“…Indeed, the FDM has allowed in [14] to image sharp peaks. On the other hand, the FEM of [9] did not let to image such peaks. So, we have written both terms under signs of norms of (3.26) in the FDM form.…”

“…we have not included the mixed second derivative ∂ z ∂ x 1 u. Based on the experience of some earlier works on the QRM [9,14,15] for linear illposed Cauchy problems, we have implemented the QRM via the FDM. Indeed, the FDM has allowed in [14] to image sharp peaks.…”

“…So, the resulting over-determined boundary value problem is solved via the Quasi-Reversibility Method (QRM). The QRM was first proposed in [19] and was developed further in [6,7,9,13,14,15]. It has proven to be effective for solving boundary value problems with over-determined boundary conditions.…”

Exact series reconstruction in photoacoustic tomography with circular integrating detectors

“…A possible reconstruction method is based of time reversal (back-propagation) similar to [4,5,14]. However, the degeneration of r −1 ∂ r r∂ r at r = 0 and the open detector set may cause difficulties in such procedures.…”

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…However, the normal derivative is not measured by transducers and thus needs to be either found mathematically or measured in a different experiment. Thus, feasibility of techniques [13,28] relying on full Cauchy data requires further mathematical and experimental study.…”