The Method of Fundamental Solutions for Direct Cavity Problems in EIT

The Method of Fundamental Solutions (MFS) is an effective technique for solving linear elliptic partial differential equations, such as the Laplace and Helmholtz equation. It is a form of indirect boundary integral equation method and a technique that uses boundary collocation or boundary fitting. In this paper the MFS is implemented to solve A numerically an inverse problem which consists of finding an unknown cavity within a region of interest based on given boundary Cauchy data. A range of examples are used to demonstrate that the technique is very effective at locating cavities in two-dimensional geometries for exact input data. The technique is then developed to include a regularisation parameter that enables cavities to be located accurately and stably even for noisy input data

Fast integral equation methods for the Laplace-Beltrami equation on the sphere

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple "islands" are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only O(N) operations, where $N$ is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method