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A new discrepancy principle
 J. Math. Anal. Appl
, 2005
"... The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem (see problem (2) below) approximately, rather than exactly, and in the proof of a stability result. To explain this in m ..."
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The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem (see problem (2) below) approximately, rather than exactly, and in the proof of a stability result. To explain this in more detail, let us recall the usual discrepancy principle, which can be stated as follows. Consider an operator eqution Au = f, (1) where A: H → H is a bounded linear operator on a Hilbert space H, and assume that the range R(A) is not closed, so that problem (1) is illposed. Assume that f = Ay where y is the minimalnorm solution to (1), and that noisy data fδ are given, such that fδ − f  ≤ δ. One wants to construct a stable approximation to y, given fδ. The variational regularization method for solving this problem consists of solving the minimization problem F (u): = Au − fδ  2 + ɛu  2 = min. (2) It is well known that problem (2) has a solution and this solution is unique (see e.g. [1]). Let uδ,ɛ solve (2). Consider the equation for finding ɛ = ɛ(δ):
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 (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 aplane, 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 lima→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), 355
, 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.
Inequalities for solutions to some nonlinear equations
"... Let F be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation F(u) = 0 is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suit ..."
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Let F be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation F(u) = 0 is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suitable assumptions.
On deconvolution problems: numerical aspects
, 2004
"... An optimal algorithm is described for solving the deconvolution problem of the form ku: = ∫ t 0 k(t − s)u(s)ds = f(t) given the noisy data fδ, f − fδ  ≤ δ. The idea of the method consists of the representation k = A(I +S), where S is a compact operator, I + S is injective, I is the identity o ..."
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An optimal algorithm is described for solving the deconvolution problem of the form ku: = ∫ t 0 k(t − s)u(s)ds = f(t) given the noisy data fδ, f − fδ  ≤ δ. The idea of the method consists of the representation k = A(I +S), where S is a compact operator, I + S is injective, I is the identity operator, A is not boundedly invertible, and an optimal regularizer is constructed for A. The optimal regularizer is constructed using the results of the paper MR 40#5130.