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14
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.
Dynamical systems gradient method for solving nonlinear . . .
- ACTA APPL MATH
"... A version of the Dynamical Systems Gradient Method for solving ill-posed 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 ill-posed 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 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 and surjectivity of nonlinear maps
- COMM. NONLINEAR SCI. AND NUMER. SIMUL
, 2005
"... 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 ..."
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Cited by 3 (3 self)
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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 justification of the Dynamical Systems Method (DSM) for global homeomorphisms
"... 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), th ..."
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Cited by 2 (2 self)
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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.
On the DSM Newton-type method
"... A wide class of the operator equations F (u) = h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F ′ (u). It is assumed that F ′ (u) depends on u continuously. Ex ..."
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Cited by 2 (2 self)
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A wide class of the operator equations F (u) = h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F ′ (u). It is assumed that F ′ (u) depends on u continuously. Existence and uniqueness of the solution to evolution equation ˙u(t) = −[F ′ (u(t))] −1 (F (u(t)) − h), u(0) = u0, is proved without assuming that F ′ (u) satisfies the Lipschitz condition. The method of the inverse function theorem.
Dynamical Systems Method (DSM) for solving equations with monotone operators without smoothness assumptions on F'(u)
- JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
, 2010
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Dynamical Systems Method for ill-posed equations with monotone operators
, 2005
"... Consider an operator equation (*) B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B ′ (u) is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, whic ..."
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Cited by 1 (1 self)
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Consider an operator equation (*) B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B ′ (u) is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation (*) with monotone C2 loc operators B.
