The Dynamical Systems Method for solving nonlinear equations with monotone operators
For researchers in nonlinear analysis and inverse problems, this provides a unified framework and convergence proofs for solving monotone operator equations with noisy data.
This paper reviews the Dynamical Systems Method (DSM) for solving nonlinear equations with monotone operators in Hilbert spaces, including regularized Newton-type, gradient-type, and simple iteration methods. It justifies a discrepancy principle for noisy data and proves convergence to the minimal norm solution.
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.