One dimensional inverse problem in photoacoustic. Numerical testing

TL;DR

The paper addresses the inverse problem of recovering the initial data $a(x)$ (with $b(x)$ fixed) for the 1D wave equation from boundary measurements $F(x,t)=u(x,t)$ on $|x|=1$, $0<t<T$, as a tractable model for photoacoustic tomography. It develops a Fourier-based forward representation with $u(x,t)=\sum_k X_k(x)T_k(t)$, where $X_k$ and $\lambda_k$ have explicit closed forms in 1D and analogous expressions in higher dimensions via Bessel-type functions, enabling spectral reconstruction. For the 1D IP, it derives a practical coefficient recovery scheme, identifies and mitigates singular modes where denominators vanish, and provides a corrective reconstruction using $A(x)$ and parity-dependent scaling to recover $a(x)$. Extensions to $n=2,3$ are discussed through 2D/3D angular-Bessel expansions and bi-orthogonal frameworks, highlighting stability considerations linked to denominators like $J_k(\nu_k^l/(1+T))$. Numerical tests with $T=2$ validate the approach, showing feasible recovery of $a(x)$ (up to modal exclusions) and offering insight into the role of sampling and noise, with practical implications for faster, higher-dimensional reconstructions.

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.

Figures (5)

• Figure 1: Space-time cylinder.
• Figure 2: Observation.
• Figure 3: Minimizing least-square-error.
• Figure 4: Smooth wave.
• Figure 5: Step function.

Theorems & Definitions (3)

• Lemma 1
• proof
• Proposition 1