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18
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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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.
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.
On stable numerical differentiation
 Australian J. Math. Anal. Appl
"... ABSTRACT. Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. N ..."
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Cited by 2 (1 self)
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ABSTRACT. Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete favorably with the variational regularization method for stable calculating the derivatives of noisy functions. Key words and phrases: illposed problems, numerical differentiation.
On deconvolution methods
, 2003
"... Several methods for solving efficiently the onedimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ku:=R t 0 k(t \Gamma s)u(s)ds = g(t); 0 ^ t ^ T. The data, g(t), are noisy. Of special practical interest is the case when the data are noisy and known at a ..."
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Cited by 1 (0 self)
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Several methods for solving efficiently the onedimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ku:=R t 0 k(t \Gamma s)u(s)ds = g(t); 0 ^ t ^ T. The data, g(t), are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent k = A(I + S), where a method for a stable inversion of A is known, S is a compact operator, and I + S is injective. This method is illustrated by examples: smooth kernels k(t), and weakly singular kernels, corresponding to Abeltype of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method.
An essay on some problems of approximation theory
, 2005
"... Several questions of approximation theory are discussed: 1) can one approximate stably in L ∞ norm f ′ given approximation fδ, � fδ −f �L∞< δ, of an unknown smooth function f(x), such that � f ′ (x) �L∞ ≤ m1? 2) can one approximate an arbitrary f ∈ L2 (D), D ⊂ Rn, n ≥ 3, is a bounded domain, by lin ..."
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Several questions of approximation theory are discussed: 1) can one approximate stably in L ∞ norm f ′ given approximation fδ, � fδ −f �L∞< δ, of an unknown smooth function f(x), such that � f ′ (x) �L∞ ≤ m1? 2) can one approximate an arbitrary f ∈ L2 (D), D ⊂ Rn, n ≥ 3, is a bounded domain, by linear combinations of the products u1u2, where um ∈ N(Lm), m = 1, 2, Lm is a formal linear partial differential operator and N(Lm) is the nullspace of Lm in D, N(Lm): = {w: Lmw = 0 in D}? 3) can one approximate an arbitrary L2 (D) function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation?