Identifying point sources for biharmonic wave equation from the scattered fields at sparse sensors

TL;DR

The paper addresses the inverse problem of identifying $M$, $\{z_m\}$, and $\{\\tau_m\\}$ for the biharmonic wave equation from multifrequency scattered fields measured at sparse sensors. It develops uniqueness results in $\\mathbb{R}^2$ and $\\mathbb{R}^3$, employing a novel 2D approach based on the Fourier transform and the Funk–Hecke formula, and extends the results to Helmholtz-type settings. Building on constructive uniqueness proofs, it proposes three numerical algorithms: (i) a single-source identification method in $\\mathbb{R}^3$ using four sensors and three frequencies; (ii) direct sampling-based localization for multiple sources in $\\mathbb{R}^2$ and $\\mathbb{R}^3$; (iii) per-source strength recovery from multi-frequency data. Numerical experiments with noisy data validate robustness and demonstrate accurate localization and strength estimation. The methods are efficient and applicable to sparse-sensor scenarios, with potential extensions to related wave equations.

Abstract

This work is dedicated to uniqueness and numerical algorithms for determining the point sources of the biharmonic wave equation using scattered fields at sparse sensors. We first show that the point sources in both $\mathbb{R}^2$ and $\mathbb{R}^3$ can be uniquely determined from the multifrequency sparse scattered fields. In particular, to deal with the challenges arising from the fundamental solution of the biharmonic wave equation in $\mathbb{R}^2$, we present an innovative approach that leverages the Fourier transform and Funk-Hecke formula. Such a technique can also be applied for identifying the point sources of the Helmholtz equation. Moreover, we present the uniqueness results for identifying multiple point sources in $\mathbb{R}^3$ from the scattered fields at sparse sensors with finitely many frequencies. Based on the constructive uniqueness proofs, we propose three numerical algorithms for identifying the point sources by using multifrequency sparse scattered fields. The numerical experiments are presented to verify the effectiveness and robustness of the algorithms.

Figures (4)

• Figure 1: Locating the position $(2,2,2)$ by $I_{{\mathbb R}^3-\text{single}}(z)$.
• Figure 2: Locating 4 point sources in ${\mathbb R}^2$ by $I_{{\mathbb R}^2-\text{multiple}}^{\text{real}}(z)$ with a few sensors. $k_{+}=50$.
• Figure 3: Locating 11 $\pi$-shaped point sources in ${\mathbb R}^2$ by $I_{{\mathbb R}^2-\text{multiple}}^{\text{complex}}(z)$. $k_{+}=50$.
• Figure 4: Locating 4 point sources in ${\mathbb R}^3$ by $I_{{\mathbb R}^3-\text{multiple}}(z)$.

Theorems & Definitions (12)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof