The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′ (u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linear and nonlinear ill-posed problems in H consists of the construction of a 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 non-linear problem. The DSM is justified for: (a) an arbitrary linear solvable equations with bounded operator, (b) for well-posed solvable nonlinear equations with twice Fréchet differentiable operator F, (c) for ill-posed solvable nonlinear equations with monotone operators, (d) for ill-posed solvable nonlinear equations with non-monotone operators from a wide class of operators, (e) for ill-posed solvable nonlinear equations with operators F such that A: = F ′ (u) satisfies the spectral assumption of the type �(A+sI) −1 � ≤ c/s, where c> 0 is a constant, and s ∈ (0, s0), s0> 0 is a fixed number, arbitrarily small, c does not depend on s and u,

2. Continuous methods for well posed problems 3. Discretization theorems for well-posed problems

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.

A Continuous Analog of discrete Gauss-Newton Method for numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator some regularization function is introduced. For the Continuous Analog of Gauss-Newton Method a convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization function are given. Keywords: Continuous Gauss-Newton method; iterative scheme; Fréchet derivative; regularization.

The goal of this paper is to develop a general approach to solution of ill-posed 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, Gauss-Newton and simple iteration methods is demonstrated.

Abstract not found

If F: H → H is a C2 loc−map in a real Hilbert space, supu∈B(u0,R) ||[F ′ (u)] −1 | | ≤ R m(R), and supR>0 m(R) = ∞, then F is surjective. 1

A nonlinear operator equation F (x) = 0, F: H → H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x0 and becoming asymptotically close to a solution of F (x) = 0 as t → + ∞ are discussed. Well-posed and ill-posed problems are investigated.

The Dynamical Systems Method (DSM) is justified for solving operator equations F (u) = f, where F is a nonlinear operator in a Hilbert space H. It is assumed that F is a global homeomorphism of H onto H, that F ∈ C 1 loc, that is, it has a continuous with respect to u Fréchet derivative F ′ (u), that the operator [F ′ (u)] −1 exists for all u ∈ H and is bounded, ||[F ′ (u)] −1 | | ≤ m(u), where m(u)> 0 is a constant, depending on u, and not necessarily uniformly bounded with respect to u. It is proved under these assumptions that the continuous analog of the Newton’s method ˙u = −[F ′ (u)] −1 (F (u) − f), u(0) = u0, converges strongly to the solution of the equation F (u) = f for any f ∈ H and any u0 ∈ H. The case when F is not a global homeomorphism but a monotone operator in H is also considered.

Abstract not found