Reconstruction of a source domain from boundary measurements

Abstract

Inverse and ill-posed problems which consist of reconstructing the unknown support of a source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable, e.g. potential, temperature or pressure, may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz partial differential equations (PDEs). For constant coefficients, the solutions of these elliptic PDEs are sought as linear combinations of explicitly available fundamental solutions (free-space Greens functions), as in the method of fundamental solutions (MFS). Prior to this application of the MFS, the free-term inhomogeneity represented by the intensity of the source is removed by the method of particular solutions. The resulting transmission problem then recasts as that of determining the interface between composite materials. In order to ensure a unique solution, the unknown source domain is assumed to be starshaped. This in turn enables its boundary to be parametrised by the radial coordinate, as a function of the polar or, spherical angles. The problem is nonlinear and the numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin which is designed to minimize a sum of squares starting from an initial guess and with no gradient required to be supplied by the user. Simple bounds on the variables can also be prescribed. Since the inverse problem is still ill-posed with respect to small errors in the data and possibly additional ill-conditioning introduced by the spectral feature of the MFS approximation, the least-squares functional which is minimized needs to be augmented with regularizing penalty terms on the MFS coefficients and on the radial function for a stable estimation of these couple of unknowns. Thorough numerical investigations are undertaken for retrieving regular and irregular shapes of the source support from both exact and noisy input data.

Shape optimization method for an inverse geometric source problem and stability at critical shape

“…In [29] and [30], formula (11) was first derived. We think it is useful to show how, using the classical methods of shape optimization, we can prove both existence of the derivative and problem (11). Therefore, we give a rigorous proof of Theorem 3.1 and also show that material derivative is belonging to H 1 (Ω).…”

“…In [1], a direct algorithm is used to reconstruct F in the case where it is a distributed function having compact support within a finite number of small subdomains. Recently in [11], an approach based on the method of fundamental solution (MFS) was proposed where the unknown source domain is assumed to be star-shaped.…”

Reconstruction of a volumetric source domain

“…The next task is to reconstruct the source domain Ω 2 numerically. Recently, the MFS has proved, [2,3,19], easy to use in detecting cavities, rigid inclusions, as well as inhomogeneities in inverse geometric problems governed by the modified Helmholtz equation. For a recent review of the MFS, as applied to solving inverse geometric problems, [17].…”

“…In this paper, the aim is to reconstruct numerically in a stable and accurate manner the volumetric source domain of constant physical properties byemplo yingacom binedmeshless technique with nonlinear optimization, which recently has been developed by the authors, [3], in the two-dimensional planar case. This nontrivial three-dimensional extension and its computational implementation, and numerical results represent the main novelty of the current study.…”

The Method of Fundamental Solutions for Solving the Inverse Problem of Magma Source Characterization

“…MFS is a meshless boundary method applicable to certain boundary value problems and initial/boundary value problems (Golberg, 1995;Karageorghis et al 2011bKarageorghis et al , 2013Bin-Mohsin and Lesnic, 2017). Since its introduction as a numerical method, the method has become increasingly popular.…”