Uniqueness, stability and algorithm for an inverse wave-number-dependent source problems

TL;DR

This work addresses an inverse source problem for the Helmholtz equation with a wave-number-dependent source of the separated form $f(\tilde x,k)\,g(x_d)$, seeking recovery of $f$ from multi-frequency near-field data on $\partial B_R$. It introduces two non-iterative algorithms based on a Fourier-transform framework and a Dirichlet-Laplacian eigenfunction framework, and proves uniqueness of $f(\tilde x,k)$ from boundary measurements. In two dimensions, the authors establish an increasing-stability estimate showing that the reconstruction error decreases as the frequency increases, with a bound involving boundary data error $\epsilon$ and a data-norm coupling $M$. Numerical experiments in $\mathbb R^2$ validate both methods, illustrate reconstruction quality across multiple $k$ and noise levels, and highlight practical considerations such as vanishing denominators and the need for frequency-perturbed surrogates. The results provide constructive, non-iterative procedures for identifying wave-number-dependent sources with potential impact in multi-frequency acoustic imaging and related inverse problems.

Abstract

This paper is concerned with an inverse wavenumber/frequency-dependent source problem for the Helmholtz equation. In two and three dimensions, the unknown source term is supposed to be compactly supported in spatial variables but independent on one spatial variable. The dependence of the source function on wavenumber/frequency is supposed to be unknown. Based on the Dirichlet-Laplacian and Fourier-Transform methods, we develop two effcient non-iterative numerical algorithms to recover the wavenumber-dependent source. Uniqueness proof and increasing stability analysis are carried out in terms of the boundary measurement data of Dirichlet kind. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed methods.

Figures (8)

• Figure 1: The blue rectangle represents the source support, where ${\rm supp}\,\, f(\cdot,k)=[\pi/4,3\pi/4]$ and ${\rm supp}\,\, g(\cdot)=[-\pi/4,\pi/4]$. The red arsterisks show the measurement points on the circle centered at $(\pi/2,0)$ with radius of $\pi/2$.
• Figure 2: The reconstructed source by the Dirichlet- Laplacian method with a fix $k=0.5$. The four source functions $f_j(x_1,k)$ ($j=1,2,3,4$) shown in (a)-(d) are reconstructed with $N=17,\, 25,\, 26,\,26$, respectively.
• Figure 3: The reconstructed results for the source function $x_1\mapsto e^{-10k_j(x_1-\pi/2)^2}+\sin(4k_jx_1)$ supported in $[\pi/4,3\pi/4]$ and $N=26$ by the Dirichlet-Laplacian method. Here we use four different wave-numbers $k_1=0.99$, $k_2=1.99$, $k_3=2.99$ and $k_4=3.99$.
• Figure 4: The reconstructed source by the Dirichlet-Laplacian method with different noise levels $\delta$.
• Figure 5: The reconstructed source by the Fourier-Transform method with $k=0.5$. Four different sources are reconstructed with $N=9,\,12,\,12,\,12$, respectively.

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• Lemma 3.1
• proof
• Lemma 3.2