Thermoacoustic tomography with an arbitrary elliptic operator

TL;DR

This paper establishes logarithmic stability estimates and convergence results for thermoacoustic tomography involving arbitrary elliptic operators, extending previous work that required restrictive conditions on the operator.

Contribution

It provides the first stability and convergence analysis for thermoacoustic tomography with a general variable principal elliptic operator.

Findings

Logarithmic stability estimates derived for arbitrary elliptic operators.

Convergence of the Quasi-Reversibility Method proven for this general case.

Results applicable to both complete and incomplete data scenarios.

Abstract

Thermoacoustic tomography is a term for the inverse problem of determining of one of initial conditions of a hyperbolic equation from boundary measurements. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stability estimates are obatined for an arbitrary variable principal part of that operator. Convergence of the Quasi-Reversibility Method to the exact solution is also established for this case. Both complete and incomplete data collection cases are considered.