Reconstruction of the initial data from the trace of the solutions on an infinite time cylinder of damped wave equations

TL;DR

This paper addresses the inverse problem of reconstructing initial data for damped wave equations from solution traces, with applications to Photoacoustic Tomography, using spectral methods and spherical harmonics.

Contribution

It introduces a method to recover initial velocity in damped wave equations from boundary measurements, linking mathematical theory to medical imaging applications.

Findings

Successful recovery of initial velocity using spectral methods.

Application of spherical harmonics to inverse wave problems.

Relevance to improving Photoacoustic Tomography techniques.

Abstract

In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation. We recover the initial velocity from the trace of the solution on a space-time cylinder. This inverse problem is related to Photoacoustic Tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.