Detection of cavities using the method of fundamental solutions. (English) Zbl 1175.65130

Karageorghis, A.; Lesnic, D.

Inverse Probl. Sci. Eng. 17, No. 6, 803-820 (2009).

Summary: The determination of the boundary of a cavity, defined here as a perfectly insulated inclusion, within a conducting medium from a single voltage and current flux measurements on the accessible boundary of the medium, can be modelled as an inverse boundary value problem for harmonic functions. We propose a novel numerical solution method for this inverse problem based on the method of fundamental solutions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown cavity shape in two dimensions (or spherical parametrization in three dimensions). This discretization yields an ill-conditioned system of highly non-linear equations. The system is recast as a non-linear least-squares problem with penalty regularizing terms included in order to improve the stability of the numerical solution with respect to random noise introduced in the measured error-contaminated input data. The feasibility of this new method is illustrated by some numerical examples.

MSC:

65N21	Numerical methods for inverse problems for boundary value problems involving PDEs
65N80	Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65H10	Numerical computation of solutions to systems of equations
78A30	Electro- and magnetostatics
78M25	Numerical methods in optics (MSC2010)

Keywords:

cavity detection; inverse problem; method of fundamental solutions; Laplace equation; ill-conditioned system of highly nonlinear equations; electrostatic imaging methods; non-linear least-squares problem; stability; numerical examples