The Method of Fundamental Solutions for Helmholtz-type Problems

The purpose of this thesis is to extend the range of application of the method fundamental solutions (MFS) to solve direct and inverse geometric problems associated with twoor three-dimensional Helmholtz-type equations. Inverse problems have become more and more important in various fields of sicence and technology, and have certainly been one of the fastest growing areas in applied mathematics over the last three decades. However, as inverse geometric problems typically lead to mathematical models which are ill-posed, their solutions are unstable under data perturbations and classical numerical techniques fail to provide accurate and stable solutions. The novel contribution of this thesis involves the developement of the MFS combined with standard techniques for composite bi-material problems, the determination of inner boundaries, inverse shape and heat transfer coefficient identification, identification of a corroded boundary and its Robin coefficient, as well as the numerical reconstruction of an inhomogeneity. Based on the MFS, unknows are determined by imposing the available boundary conditions, this allows to obtain a system of linear/nonlinear algebraic equations. A well-conditioned system of linear algebraic equations is solved by using the Gaussian elimination method, whilst a highly ill-conditioned system of equations is solved by the regularised least-squares method using a standard NAG routine E04FCF. The accuracy and convergence of the MFS numerical technique used in this thesis is investigated using certain test examples for various geometry domains. The stability of the numerical solutions is investigated by introducing random noise into the input data, this yields unstable results if no regularisation is used. The Tikhonov regularisation method is employed in order to reduce the influence of the measurement errors on the numerical results. The inverse numerical solutions are compared with their known analytical solution, where available, and with the corresponding direct numerical solution where no analytical solution is available.