REGULARIZATION OF ILL-CONDITIONED LINEAR SYSTEMS

ABSTRACT The numerical solution of the linear system Ax = b, arises in many branches of applied mathematics, sciences, engineering and statistics. The most common source of these problems is in the numerical solutions of ordinary and partial differential equations, as well as integral equations. The process of discretization by means of finite differences often leads to the solution of linear systems, whose solution is an approximation to the solution of the original differential equation. If the coefficient matrix is ill-conditioned or rank-deficient, then the computed solution is often a meaningless approximation to the unknown solution. Regularization methods are often used to obtain reasonable approximations to ill-conditioned systems. However, the methods for choosing an optimal regularization parameter is not always clearly defined. In this dissertation, we have studied various methods for solving ill-conditioned linear systems, using the Hilbert system as a prototype. These systems are highly ill-conditioned. We have examined various regularization methods for obtaining meaningfully 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 and the Jacobi and Gauss-Seidel Preconditioner’s for boundary values problems. The truncated singular value decomposition truncate the harmful effect of the small singular values on the solution by replacing them with exact zero. The truncation improves the solution to an extent and the solution deteriorates again. The maximum error in the solution occurs at the cut-off level λ = 2.2520×10−10 . The optimal solution was obtained at λ = 2.2520 × 10−10 . The Jacobi and the Gauss-Seidel preconditioner’s for sparse systems also gave an optimal solution