Meshless Methods For Some Inverse Problems Associated With Elliptic Partial Differential Equations

Inverse problems arise in many engineering and scientific contexts, and they are now playing a central role there. In practical applications, the problem under investigation is often governed by partial differential equations. Hence the solution of practical inverse problems often leads to solving inverse problems associated with partial differential equations.This thesis focuses on the class of inverse problems that can be formulated as Cauchy problems with elliptic partial differential equations, which are notorious for their ill-posedness. In particular, we will investigate the Cauchy problem associated with the Laplace equation for the steady-state heat conduction, nondestructive testing (NDT) and geophysics, the Helmholtz equations for acoustic analysis, electro-magnetic field, vibration of membranes and other structures, and heat conduction in a fin, and the Navier system in linear elasticity.In the thesis, several novel numerical schemes based on meshless methods are proposed for the solution of the inverse problems associated with elliptic partial differential equations. These meshless methods can be formulated in the general framework of radial basis function collocation techniques. Preliminary numerical results for the Laplace equation and Navier system are presented, and tricky issues associated with these methods, such as the ill-conditioning with the interpolation matrix and effect of shape parameter, are investigated. It is found that using regularization methods, such as the truncated singular value decomposition and Tikhonov regularization, to solve the resulting system of linear equations, not only mitigates the ill-conditioning with the matrix equation, but also alleviates the difficulty with determining an appropriate shape parameter in the radial basis functions. It is concluded that the radial basis function collocation techniques coupled with regularization methods could be competitive alternatives to existing methods for these problems.However, for specific equations, such as the Laplace equation, the Helmholtz equation and the Navier system, we can improve the computational efficiency significantly by employing differential operator-geared radial basis functions. To be more specific, we may employ the fundamental and general solutions to the differential operators as the radial basis functions, whichnaturally give rise to the method of fundamental solutions and the boundary knot method. In the thesis, the controversial ill-conditioning associated with the method of fundamental solutions is reinterpreted in terms of the theory for the Fredholm integral equation of the first kind. A remedy for the ill-conditioning is proposed based on regularization methods. The boundary knot method is extended to inhomogeneous problems in a unified manner.Numerical experiments were performed to demonstrate the efficiency and efficacy of the proposed methods, for both smooth and piecewise smooth domains with exact and noisy data. The numerical verifications show that the methods are computationally very efficient, highly accurate, stable with respect to the data noise, and convergent with respect to decreasing the amount of noise in the data. They could be competitive alternatives to existing methods for this class of problems.