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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.
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.
A discrepancy principle for equations with monotone continuous operators
- NONLINEAR ANALYSIS
, 2008
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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
How large is the class of operator equations solvable by a DSM Newton-type method?
"... It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newton-type 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) Newton-type 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.
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.
DSM of Newton type for solving operator equations F(u) = f with minimal smoothness assumptions on F
- JOURN. COMP. SCI AND MATH., 3, N1/2, (2010), 3-55
, 2010
"... This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation (*) F (u) = f. It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fréchet diffe ..."
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This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation (*) F (u) = f. It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fréchet differentiable, but no smoothness assumptions on F ′ (u) are imposed. The DSM for solving equation (*) is developed. Under weak assumptions global existence of the solution u(t) is proved, the existence of u(∞) is established, and the relation F(u(∞)) = f is obtained. The DSM is developed for a stable solution of equation (*) when noisy data fδ are given, ‖‖f − fδ‖‖ ≤ δ.
