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

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.