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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 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.
AN ITERATIVE SCHEME FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS
"... An iterative scheme for solving ill-posed 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 ill-posed 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 ill-posed 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 ill-posed 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.
Dynamical Systems Method for Solving Ill-conditioned Linear Algebraic Systems
"... A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new iterative scheme for solving ICLAS is proposed. This iterative ..."
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Cited by 6 (3 self)
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A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. This paper also gives an a posteriori stopping rule for a modified iterative scheme developed in
An iterative method for solving Fredholm integral equations of the first kind
"... The purpose of this paper is to give a convergence analysis of the iterative scheme: u δ n = quδ n−1 −1 + (1 − q)Tan K∗fδ, u δ 0 = 0, where T: = K ∗ K, Ta: = T + aI, q ∈ (0, 1), an: = α0q n, α0> 0, with finite-dimensional approximations of T and K ∗ for solving stably Fredholm integral equations ..."
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Cited by 1 (0 self)
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The purpose of this paper is to give a convergence analysis of the iterative scheme: u δ n = quδ n−1 −1 + (1 − q)Tan K∗fδ, u δ 0 = 0, where T: = K ∗ K, Ta: = T + aI, q ∈ (0, 1), an: = α0q n, α0> 0, with finite-dimensional approximations of T and K ∗ for solving stably Fredholm integral equations of the first kind with noisy data.
Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces
"... Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ..."
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Cited by 1 (1 self)
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Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ‖A −1 a (u) ‖ ≤ c1 |a | b, |a |> 0, b> 0, a ∈ L. Here a may be a complex number, and L is a smooth path on the complex a-plane, joining the origin and some point on the complex a−plane, 0 < |a | < ɛ0, where ɛ0> 0 is a small fixed number, such that for any a ∈ L estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) ˙u(t) = −A −1 a(t) (u(t))[F (u(t)) + a(t)u(t) − f], du u(0) = u0, ˙u = dt, converges to y as t → +∞, where a(t) ∈ L, F (y) = f, r(t): = |a(t)|, and r(t) = c4(t + c2) −c3, where cj> 0 are some suitably chosen constants, j = 2, 3, 4. Existence of a solution y to the equation F (u) = f is assumed. It is also assumed that the equation F (wa) + awa − f = 0 is uniquely solvable for any f ∈ X, a ∈ L, and lim|a|→0,a∈L ‖wa − y ‖ = 0.
plenary speaker at the Seventh PanAfrican Congress of Mathematicians,
"... The purpose of this paper is to give a convergence analysis of the iterative scheme: uδn = qu δ n−1 + (1 − q)T−1an K∗fδ, uδ0 = 0, where T: = K∗K, Ta: = T + aI, q ∈ (0, 1), an: = α0qn, α0> 0, with finite-dimensional approximations of T and K ∗ for solving stably Fredholm integral equations of the ..."
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The purpose of this paper is to give a convergence analysis of the iterative scheme: uδn = qu δ n−1 + (1 − q)T−1an K∗fδ, uδ0 = 0, where T: = K∗K, Ta: = T + aI, q ∈ (0, 1), an: = α0qn, α0> 0, with finite-dimensional approximations of T and K ∗ for solving stably Fredholm integral equations of the first kind with noisy data.
