Regularization of ill - conditioned linear systems

Abstract

Ill-conditioned linear systems arise naturally in many applications, especially in the numerical solution of ordinary and partial differential equations, as well as integral equations. In this dissertation, we have studied various methods for solving ill-conditioned linear systems, using the Hilbert system as a prototype. We further examined various regularization methods for obtaining meaningful approximations to such systems. Tikhonov Regularization method proved to be the method of choice for regularizing rank deficient and discrete ill-posed problems compared to the Truncated Singular Values decomposition, the Jacobi and Gauss-Seidel Preconditioner's for boundary values problems. Our results using Tikhonov method, shows that in order 1 regularization, the accuracy of the solution increases with increasing values of the regularization parameter, with optimal regularization parameter of 10°, giving an accuracy of 15 digits compared with order 2 and order 0 with 13 and 3 digits of accuracy respectively. Application of regularization to the solution of an ill-conditioned discretized Fredholm integral equation of the first kind, coupled with the LCurve method indicated that ill-conditioned linear systems can be solved using numerical regularization method. The results obtained justify that in the absence of computational errors and with proper regularization, a convergent discretization error leads to a convergent solution.