Locating Multiple Multipolar Acoustic Sources Using the Direct Sampling Method

Abstract

This work is concerned with the inverse source problem of locating multiple multipolar sources from boundary measurements for the Helmholtz equation. We develop simple and effective sampling schemes for location acquisition of the sources with a single wavenumber. Our algorithms are based on some novel indicator functions whose indicating behaviors could be used to locate multiple multipolar sources. The inversion schemes are totally "direct" in the sense that only simple integral calculations are involved in evaluating the indicator functions. Rigorous mathematical justifications are provided and extensive numerical examples are presented to demonstrate the effectiveness, robustness and efficiency of the proposed methods.

A Direct Sampling Method for Recovering Electromagnetic Dipolar Sources

“…Following the ideas of [35], we implement a two-stage scheme of the sampling method. We initially use a global coarse grid, and on the second stage a local fine grid is employed.…”

“…Recently, Zhang et al [35] investigated two-level direct sampling methods to identify multiple multipolar acoustic sources from near-field Cauchy data. The basic idea of introducing indicator functions [35] is initiated by the decaying property of oscillatory integrals. Thus indicators admit local maxima in open neighborhoods of each point source, while the sampling point coincides with the exact source location.…”

“…The current work is partially motivated by works [5,35]. Here, we are going to develop a direct sampling method for inverse point source problems in vector Maxwell systems.…”

A Fourier Method to Recover Elastic Sources with Multi-Frequency Data

“…To identify acoustic and electromagnetic sources, an eigenfunction expansion method was developed in [9,18] and to approximate such sources with near field data in the frequency domain, the Fourier expansion was used [19,22]. In [23], multiple multipolar sources are located by a sampling method.…”

A Mathematical Framework for Deep Learning in Elastic Source Imaging

“…An abundance of real-world inverse problems, for instance in biomedical imaging, non-destructive testing, geological exploration, and sensing of seismic events, is concerned with the spatial and/or temporal support localization of sources generating wave fields in acoustic, electromagnetic, or elastic media (see, e.g., [8,10,20,32,37,41,45,47,50] and references therein). Numerous applicationspecific algorithms have been proposed in the recent past to procure solutions of diverse inverse source problems from time-series or time-harmonic measurements of the generated waves (see, e.g., [2,6,7,19,36,51,54,55,58,64,65]). The inverse elastic source problems are of particular interest in this paper due to their relevence in elastography [3,10,20,50].…”