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 u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) 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.