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Dynamical systems method for . . .
, 2004
"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
Abstract

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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well.
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"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
Abstract
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well. 1