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

2023/08/07 by Seongyeon Kim, Kim, Seongyeon, Sung‐Hwan Moon +3
Engineering · #Advanced X-ray and CT Imaging #Analysis of PDEs (math.AP) #Electrical and Bioimpedance Tomography #FOS: Mathematics #Photoacoustic and Ultrasonic Imaging

paper · pdf · doi:10.48550/arxiv.2308.03362

openalex publication_date 2023/08/07 · openalex created_date 2023/08/09 · openalex updated_date 2026/07/28

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.

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