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The Dynamical Systems Method for solving nonlinear equations with monotone operators
"... 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. V ..."
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Cited by 14 (10 self)
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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 Newtontype method, a gradienttype 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 largetime behavior of solutions to evolution equations. Discrete versions of these inequalities are established.
Asymptotic stability of solutions to abstract differential equations
, 2010
"... An evolution problem for abstract differential equations is studied. The typical problem is: ˙u = A(t)u + F(t,u), t ≥ 0; u(0) = u0; ˙u = du dt Here A(t) is a linear bounded operator in a Hilbert space H, and F is a nonlinear operator, ‖F(t,u) ‖ ≤ c0‖u ‖ p, p> 1, c0, p = const> 0. It is assumed tha ..."
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Cited by 13 (12 self)
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An evolution problem for abstract differential equations is studied. The typical problem is: ˙u = A(t)u + F(t,u), t ≥ 0; u(0) = u0; ˙u = du dt Here A(t) is a linear bounded operator in a Hilbert space H, and F is a nonlinear operator, ‖F(t,u) ‖ ≤ c0‖u ‖ p, p> 1, c0, p = const> 0. It is assumed that Re(A(t)u,u) ≤ −γ(t)‖u ‖ 2 ∀u ∈ H, where γ(t)> 0, and the case when limt→ ∞ γ(t) = 0 is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality.
Global convergence for illposed equations with monotone operators: the dynamical systems method
, 2003
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Solving illconditioned linear algebraic systems by the dynamical systems method (DSM)
 INVERSE PROBLEMS IN SCI. AND ENGINEERING
, 2008
"... An iterative scheme for the Dynamical Systems Method (DSM) is given such that one does not have to solve the Cauchy problem occuring in the application of the DSM for solving illconditioned linear algebraic systems. The novelty of the algorithm is that the algorithm does not have to find the regular ..."
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Cited by 11 (7 self)
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An iterative scheme for the Dynamical Systems Method (DSM) is given such that one does not have to solve the Cauchy problem occuring in the application of the DSM for solving illconditioned linear algebraic systems. The novelty of the algorithm is that the algorithm does not have to find the regularization parameter a by solving a nonlinear equation. Numerical experiments show that DSM competes favorably with the Variational Regularization.
Dynamical systems gradient method for solving nonlinear . . .
 ACTA APPL MATH
"... A version of the Dynamical Systems Gradient Method for solving illposed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed ..."
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Cited by 11 (7 self)
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A version of the Dynamical Systems Gradient Method for solving illposed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.
Dynamical Systems Method for solving nonlinear operator equations
, 2004
"... Consider an operator equation (*) B(u) + ɛu = 0 in a real Hilbert space, where ɛ> 0 is a small constant. The DSM (Dynamical Systems Method) for solving equation (*) consists of finding and solving a Cauchy problem: ˙u = Φ(t, u), u(0) = u0, t ≥ 0, which has the following properties: 1) it has a glo ..."
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Cited by 10 (8 self)
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Consider an operator equation (*) B(u) + ɛu = 0 in a real Hilbert space, where ɛ> 0 is a small constant. The DSM (Dynamical Systems Method) for solving equation (*) consists of finding and solving a Cauchy problem: ˙u = Φ(t, u), u(0) = u0, t ≥ 0, which has the following properties: 1) it has a global solution u(t), 2) this solution tends to a limit as time tends to infinity, i.e., u(∞) exists, 3) this limit solves the equation B(u) = 0, i.e., B(u(∞)) = 0. Existence of the unique solution is proved by the DSM for equation (*) with operators B defined on all of H and satisfying a spectral assumption: [B ′ (u) + ɛI] −1   ≤ c/ɛ for any u ∈ H, where c> 0 is a constant independent of u and ɛ ∈ (0, ɛ0). If ɛ = 0 and equation (**) B(u) = 0 is solvable, the DSM yields a solution to (**). The case when B is a monotone, hemicontinuous, defined on all of H operator is also studied, and DSM is justified for this case, that is, above properties 1),2), and 3) are proved. A sufficient condition for surjectivity of a nonlinear map is given. Meyer’s generalization of the Hadamard theorem about global homeomorphisms is proved by the DSM. The DSM method is justified for nondifferentiable, hemicontinuous, monotone, defined on all of H operators.