Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation

TL;DR

This paper develops a fractional time stepping method and an adjoint-based gradient computation scheme for an inverse problem involving a fractionally damped wave equation, with applications in photoacoustic tomography.

Contribution

It introduces a novel numerical scheme for the forward problem and an efficient adjoint-based approach for gradient computation in the inverse problem.

Findings

Numerical reconstructions demonstrate the effectiveness of the proposed methods.

The methods improve the accuracy and efficiency of inverse problem solutions.

The approach is validated with two-dimensional numerical experiments.

Abstract

In this paper we consider the inverse problem of identifying the initial data in a fractionally damped wave equation from time trace measurements on a surface, as relevant in photoacoustic or thermoacoustic tomography. We derive and analyze a time stepping method for the numerical solution of the corresponding forward problem. Moreover, to efficiently obtain reconstructions by minimizing a Tikhonov regularization functional (or alternatively, by computing the MAP estimator in a Bayesian approach), we develop an adjoint based scheme for gradient computation. Numerical reconstructions in two space dimensions illustrate the performance of the devised methods.