One dimensional inverse problem in photoacoustic. Numerical testing

TL;DR

This paper investigates a simplified numerical method for reconstructing wave data in a one-dimensional photoacoustic inverse problem, analyzing convergence and stability through numerical testing, with insights for higher dimensions.

Contribution

It adapts and simplifies a reconstruction method for 1D wave equations in photoacoustics, providing numerical results on convergence and stability.

Findings

Demonstrates convergence rate of the reconstruction method

Shows stability of the numerical procedure

Provides insights for extending to 2D and 3D cases

Abstract

We consider the problem of reconstruction of Cauchy data for the wave equation in $\mathbb{R}^1$ by the measurements of its solution on the boundary of the finite interval. This is a one-dimensional model for the multidimensional problem of photoacoustics, which was studied in \cite{BLMM}. We adapt and simplify the method for one-dimensional situation and provide the results on numerical testing to see the rate of convergence and stability of the procedure. We also give some hints on how the procedure of reconstruction can be simplified in 2d and 3d cases.