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Dynamical systems method (DSM) and nonlinear problems
 IN THE BOOK: SPECTRAL THEORY AND NONLINEAR ANALYSIS, SINGAPORE, WORLD SCIENTIFIC PUBLISHERS, 2005
, 2005
"... The dynamical systems method (DSM), for solving operator equations, especially nonlinear and illposed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if th ..."
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Cited by 27 (24 self)
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The dynamical systems method (DSM), for solving operator equations, especially nonlinear and illposed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. The DSM for solving linear and nonlinear illposed problems in H consists of the construction of a dynamical system, that is, a Cauchy problem, which has the following properties: (1) it has a global solution, (2) this solution tends to a limit as time tends to infinity, (3) the limit solves the original linear or nonlinear problem. The DSM is justified for: (a) an arbitrary linear solvable equations with bounded operator, (b) for wellposed solvable nonlinear equations with twice Fréchet differentiable operator F, (c) for illposed solvable nonlinear equations with monotone operators, (d) for illposed solvable nonlinear equations with nonmonotone operators from a wide class of operators, (e) for illposed solvable nonlinear equations with operators F such that A: = F ′ (u) satisfies the spectral assumption of the type �(A+sI) −1 � ≤ c/s, where c> 0 is a constant, and s ∈ (0, s0), s0> 0 is a fixed number, arbitrarily small, c does not depend on s and u,
Stable numerical differentiation: when is it possible
 Jour. Korean SIAM
"... Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrate ..."
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Cited by 11 (8 self)
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Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrated by examples. These examples show the practical value of the new understanding of the problem of stable differentiation. 1.
Reconstructing singularities of a function from its Radon transform
 MATH. COMPUT. MODELLING
, 1993
"... We study the relation between the singularities of a function f and its Radon transform R(f). We prove that their singular loci are related via Legendre transform. Geometric properties of the singular locus of R(f) are studied. The problem of computing the Legendre transform from approximately know ..."
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Cited by 11 (5 self)
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We study the relation between the singularities of a function f and its Radon transform R(f). We prove that their singular loci are related via Legendre transform. Geometric properties of the singular locus of R(f) are studied. The problem of computing the Legendre transform from approximately known data is discussed.
Inequalities for the derivatives
 MATHEM. INEQUALITIES AND APPLICATIONS, 3, N1, (2000), PP.129132
, 2000
"... The following question is studied and answered: Is it possible to stably approximate f ′ if one knows: 1) fδ ∈ L ∞ (R) such that �f − fδ � < δ, and 2) f ∈ C ∞ (R), �f � + �f ′ � ≤ c? Here �f �: = sup x∈R f(x)  and c> 0 is a given constant. By a stable approximation one means �Lδfδ − f ′ � ..."
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Cited by 9 (4 self)
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The following question is studied and answered: Is it possible to stably approximate f ′ if one knows: 1) fδ ∈ L ∞ (R) such that �f − fδ � < δ, and 2) f ∈ C ∞ (R), �f � + �f ′ � ≤ c? Here �f �: = sup x∈R f(x)  and c> 0 is a given constant. By a stable approximation one means �Lδfδ − f ′ � ≤ η(δ) → 0 as δ → 0. By Lδfδ one denotes an estimate of f ′. The basic result of this paper is the inequality for �Lδfδ − f ′ �, a proof of the impossibility to approximate stably f ′ given the above data 1) and 2), and a derivation of the inequality η(δ) ≤ cδ a 1+a if 2) is replaced by �f�1+a ≤ m1+a, 0 < a ≤ 1. An explicit formula for the estimate Lδfδ is given.
NUMERICAL DIFFERENTIATION FROM A VIEWPOINT OF REGULARIZATION THEORY
"... Abstract. In this paper, we discuss the classical illposed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general illposed problems, we propose new rules for the choice of the st ..."
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Cited by 5 (0 self)
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Abstract. In this paper, we discuss the classical illposed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general illposed problems, we propose new rules for the choice of the stepsize in the finitedifference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methodsareshowntobeeffectiveforthedifferentiationofnoisyfunctions, and the orderoptimal convergence results for them are proved. 1.
Dynamical systems method for . . .
, 2004
"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
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Cited by 3 (1 self)
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well.
Continuous modified Newton’stype method for nonlinear operator equations
 ANN. DI MAT. PURE APPL
, 2002
"... A nonlinear operator equation F (x) = 0, F: H → H, in a Hilbert space is considered. Continuous Newton’stype procedures based on a construction of a dynamical system with the trajectory starting at some initial point x0 and becoming asymptotically close to a solution of F (x) = 0 as t → + ∞ are d ..."
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Cited by 3 (2 self)
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A nonlinear operator equation F (x) = 0, F: H → H, in a Hilbert space is considered. Continuous Newton’stype procedures based on a construction of a dynamical system with the trajectory starting at some initial point x0 and becoming asymptotically close to a solution of F (x) = 0 as t → + ∞ are discussed. Wellposed and illposed problems are investigated.