Computational identification of the source domain in an inverse problem of potential theory

TL;DR

This paper addresses localizing the source domain $D$ in an inverse potential problem for the 2D Laplace operator using exterior measurements on $\Gamma$. It reframes the continuation toward the sources by approximating the volume potential with a boundary single-layer potential on $S$ of a candidate domain $\Omega$, and imposes a nonnegativity constraint on the layer density, solving the resulting Fredholm first-kind problem via regularized nonnegative least squares. The authors compare standard least-squares, Tikhonov regularization, and NNLS, and introduce the Nonnegative Density Domain (NNDD) algorithm to identify the source support by scanning window positions and using residuals or the total layer mass as localization criteria. The approach yields stable, localized estimates in 2D with analytic test data, and demonstrates robustness to noise, offering a practical framework for source localization in ill-posed potential problems. The work provides a concrete, nonnegative-density methodology with clear computational steps and a mechanism for diagnosing localization via residual behavior and cumulative density.

Abstract

The inverse potential problem consists in determining the density of the volume potential from measurements outside the sources. Its ill-posedness is due both to the non-uniqueness of the solution and to the instability of the solution with respect to measurement errors. The inverse problem is solved under additional assumptions about the sources using regularizing algorithms. In this work, an inverse problem is posed for identifying the domain that contains the sources. The computational algorithm is based on approximating the volume potential by the single-layer potential on the boundary of the domain containing the sources. The inverse problem is considered in the class of a priori constraints of nonnegativity of the potential density. Residual minimization in the class of nonnegative solutions is performed using the classical Nonnegative Least Squares algorithm. The capabilities of the proposed approach are illustrated by numerical experiments for a two-dimensional test problem with an analytically prescribed potential on the observation surface.

Figures (26)

• Figure 1: To the problem statement.
• Figure 2: Localization domain of the sources.
• Figure 3: Discretization.
• Figure 4: Solution of the problem for $N_1 = N_2 = 50$.
• Figure 5: Solution of the problem for $N_1 = N_2 = 100$.