Several methods for a stable solution to the equation $F(u)=f$ have been developed.

Here $F:H\to H$ is an operator in a Hilbert space $H$,

and we assume that noisy data $f_\delta$, $\|f_\delta-f\|\le \delta$, are given in place of the exact data $f$.

When $F$ is a linear bounded operator, two versions of the Dynamical Systems Method (DSM) with stopping rules of Discrepancy Principle type are proposed and justified mathematically.

When $F$ is a non-linear monotone operator, various versions of the DSM are studied.

A Discrepancy

Principle for solving the equation is formulated and justified. Several

versions of the DSM for solving the equation

are

formulated. These methods consist of a Newton-type method, a

gradient-type method, and a simple iteration method. A priori and a

posteriori choices of stopping rules for these methods are proposed and

justified. Convergence of the solutions, obtained by these methods, to

the minimal norm solution to the equation $F(u)=f$ is proved. Iterative

schemes with a posteriori choices of stopping rule corresponding to the

proposed DSM are formulated. Convergence of these iterative schemes to a

solution to the equation $F(u)=f$ is proved.

This dissertation consists of six chapters which are based on joint papers by the author and his advisor Prof. Alexander G. Ramm.

These papers are published in different journals.

The first two chapters deal with equations with linear and bounded operators and the last four chapters deal with non-linear equations with monotone operators.