The Dynamical Systems Method for solving nonlinear equations with monotone operators

A review of the authors's results is given. Several methods are discussed for solving nonlinear equations $F(u)=f$, where $F$ is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These methods consist of a regularized 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 equation $F(u)=f$ is justified. New nonlinear differential inequalities are derived and applied to a study of large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established.Comment: 50p

Dynamical Systems Gradient method for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.Comment: 2 figure

Dynamical systems method for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed and justified mathematically. The results of two numerical experiments are presented. They show that the proposed version of DSM is numerically efficient. The numerical experiments consist of solving nonlinear integral equations.Comment: 19 pages, 4 figures, 4 table