Abstract
In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in a least-square sense and minimisation of the objective function. This approach can also be considered with noisy boundary Cauchy data. The simplicity and efficiency of this method is illustrated in several numerical examples.
Highlights
The purpose of this paper is to extend the aforementioned current approach to the threedimensional Laplace equation based on the method of fundamental solutions
In the classic method of fundamental solutions (MFS), the solution of a homogeneous linear partial differential equation (PDE) is approximated by a linear combination of the fundamental solutions with the set of sources located outside the problem domain and a set of points on the domain boundary
The coefficients vector c = (c j ) j=1,M+ N in linear combination (6) and the radial vector r = (r j ) j=1,N can be determined by imposing the boundary conditions (2)â(4) in a least-square sense, which recasts into minimising the objective function
Summary
The inverse geometry problems, as an important subclass of inverse problems, can be subdivided into two subclasses, depending on the location of the unknown boundary. The portion of the outer boundary of the solution domain is unknown, whilst in the second kind, the inner boundary is unknown. There are many methods for solving the inverse geometry problems, such as the boundary element regularisation method by Lesnic et al [1], the method of fundamental solutions and moving pseudo-boundary method by Karageorghis et al [2,3,4], the boundary function method by Wang et al [5], the conjugate gradient method (CGM) and the boundary element technique by Huang et al [6,7]. Bin-Mohsin and Lesnic in 2012 utilised the method of fundamental solutions (MFS) to the modified Helmholtz inverse geometry problem on an annular domain [8]. Two examples are presented to show the simplicity and efficiency of this method
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