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Dynamical systems and discrete methods for solving nonlinear illposed problems
 Appl.Math.Reviews
, 2000
"... 2. Continuous methods for well posed problems 3. Discretization theorems for wellposed problems ..."
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Cited by 23 (18 self)
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2. Continuous methods for well posed problems 3. Discretization theorems for wellposed problems
A numerical method for solving nonlinear illposed problems
, 2000
"... The goal of this paper is to develop a general approach to solution of illposed nonlinear problems in a Hilbert space based on continuous processes with a regularization procedure. To avoid the illposed inversion of the Fréchet derivative operator a regularizing oneparametric family of operators i ..."
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Cited by 17 (13 self)
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The goal of this paper is to develop a general approach to solution of illposed nonlinear problems in a Hilbert space based on continuous processes with a regularization procedure. To avoid the illposed inversion of the Fréchet derivative operator a regularizing oneparametric family of operators is introduced. Under certain assumptions on the regularizing family a general convergence theorem is proved. The proof is based on a lemma describing asymptotic behavior of solutions of a new nonlinear integral inequality. Then the applicability of the theorem to the continuous analogs of the Newton, GaussNewton and simple iteration methods is demonstrated.
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 ..."
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Cited by 3 (1 self)
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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 ..."
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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
Communications in Nonlinear Science and Numerical Simulation, 9, N2, (2003) Dynamical systems method for solving operator equations ∗
"... 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