The Dynamical Systems Method for solving nonlinear equations with monotone operators

TL;DR

This paper reviews the Dynamical Systems Method (DSM) for solving nonlinear monotone operator equations in Hilbert spaces, discussing various algorithms, stopping rules, and convergence proofs, with applications to evolution equations.

Contribution

It introduces and analyzes multiple DSM variants with stopping rules, providing convergence proofs and new nonlinear inequalities for evolution equations.

Findings

Convergence of DSM solutions to minimal norm solutions.

Justification of a discrepancy principle for noisy data.

Development of new nonlinear differential inequalities.

Abstract

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.