The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation

This paper investigates the applications of the method of fundamental solutions together with the conditional number analysis to solve various inverse 2D Laplace problems involving under-specified and/or over-specified boundary conditions. Through the method of fundamental solutions and the condition number analysis, it is numerically found that solutions of inverse Laplace problems can be obtained without iteration or regularization for small noise levels, since the method of fundamental solutions is a boundary-type meshless numerical method that can automatically satisfy the governing equation. However for larger values of noise levels regularization is still necessary to obtain promising result. The present paper mainly focuses on the two types of numerical predictions of inverse 2D Laplace problems: (1) Cauchy problem, and (2) shape identification problem. Good quantitative agreement with the analytical solutions and other numerical methods for small perturbed boundary data is observed by using present meshless numerical scheme.