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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 15 (12 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.
Global convergence for illposed equations with monotone operators: the dynamical systems method
, 2003
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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 12 (8 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.
AN ITERATIVE SCHEME FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS
"... An iterative scheme for solving illposed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of illposed operator equations with monotone operators is proposed and its c ..."
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Cited by 11 (6 self)
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An iterative scheme for solving illposed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of illposed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified.
On unbounded operators and applications
, 2007
"... Assume that Au = f (1) is a solvable linear equation in a Hilbert space H, A is a linear, closed, densely defined, unbounded operator in H, which is not boundedly invertible, so problem (1) is illposed. It is proved that the closure of the operator (A*A + ffI)1A*, with the domain D(A*), where ff&g ..."
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Cited by 8 (8 self)
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Assume that Au = f (1) is a solvable linear equation in a Hilbert space H, A is a linear, closed, densely defined, unbounded operator in H, which is not boundedly invertible, so problem (1) is illposed. It is proved that the closure of the operator (A*A + ffI)1A*, with the domain D(A*), where ff> 0 is a constant, is a linear bounded everywhere defined operator with norm<= 1 2pff. This result is applied to the variational problem F (u): = Au f  2 + ffu2 =
How large is the class of operator equations solvable by a DSM Newtontype method?
"... It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newtontype method ˙u = −[F ′ (u) + a(t)I] −1 [F u(t) + a(t)u − f], u(0) = u0, (∗) is large. Here F: X → X is a continuously Fréchet differentiable operator in a Banach space X, a(t) : [0, ∞) ..."
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Cited by 6 (6 self)
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It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newtontype method ˙u = −[F ′ (u) + a(t)I] −1 [F u(t) + a(t)u − f], u(0) = u0, (∗) is large. Here F: X → X is a continuously Fréchet differentiable operator in a Banach space X, a(t) : [0, ∞) → C is a function, limt→ ∞ a(t)  = 0, and there exists a y ∈ X such that F (y) = f. Under weak assumptions on F and a it is proved that This justifies the DSM (*). ∃!u(t) ∀t ≥ 0; ∃u(∞); F (u(∞)) = f.