Numerical methods for solving linear ill-posed problems

Abstract

A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In Chapter 1 a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. We also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme. In Chapter 2 we give a convergence analysis of the following iterative scheme: u[subscript]n[superscript]delta=q usubscript[superscript]delta+(1-q)Tsubscriptsuperscript K[superscript]f[subscript]delta, u[subscript]0[superscript]delta=0, where T:=K[superscript] K, T[subscript]a :=T+aI, q in the interval (0,1),\quad a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional approximations of T and K[superscript]* for solving stably Fredholm integral equations of the first kind with noisy data. In Chapter 3 a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f(t) is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f(t), are proposed in this chapter.