Stability of the inverse source problem for the Helmholtz equation in R³

Abstract

We consider the reconstruction of a compactly supported source term in the constantcoefficient Helmholtz equation in R 3 , from the measurement of the outgoing solution at a source-enclosing sphere. The measurement is taken at a finite number of frequencies. We explicitly characterize certain finite-dimensional spaces of sources that can be stably reconstructed from such measurements. The characterization involves only the measurement frequencies and the problem geometry parameters. We derive a singular value decomposition of the measurement operator, and prove a lower bound for the spectral bandwidth of this operator. By relating the singular value decomposition and the eigenvalue problem for the Dirichlet-Laplacian on the source support, we devise a fast and stable numerical method for the source reconstruction. We do numerical experiments to validate the stability and efficiency of the numerical method.

Fundamental solutions to elliptic multipliers with real-analytic symbol in $\mathbb{R}^n$

“…The components of f that can be reconstructed stably, when the actual measurement contains noise, are dictated by the spectral properties of the "forward operator" F : f → u| S . For details, see the development in [10,12]. If F is compact with singular system (σ j , ψ j , φ j ), then by [6, Theorem 4.8], we have…”

Statistical inverse estimation of source terms in the Helmholtz equation

“…Uniqueness, stability and inversion methods of the single-frequency and multifrequency source identification were treated, for example, in previous studies. [13][14][15][16] Recent work on inverse scattering for Helmholtz equation has centered on iterative methods based either on the Volterra scattering series with both reflection and transmission data 17 or on optimization using a strictly convex cost functional. 18 Generally, regularization parameters in classical regularization methods always need to be selected by a priori information of the unknown solution.…”

“…Based on Tikhonov regularization in conjunction with the Morozov discrepancy principle, Yao and Bo 12 examined the data completion problem of Helmholtz equation in two and three dimensions. Uniqueness, stability and inversion methods of the single‐frequency and multifrequency source identification were treated, for example, in previous studies 13–16 . Recent work on inverse scattering for Helmholtz equation has centered on iterative methods based either on the Volterra scattering series with both reflection and transmission data 17 or on optimization using a strictly convex cost functional 18 …”

Dimension and Sampling of the Near-Field and Its Intensity Over Curves

“…Field sampling is a relevant research topic arising in antenna characterization [1]- [5], inverse source [6]- [9], imaging problems [10]- [14], etc. In particular, sampling methods employing a non-redundant number of field measurements are gaining an ever-increasing interest since they allow to save time in the field probing which is dominated by the mechanical scanning.…”