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AN ITERATIVE SCHEME FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS
"... An iterative scheme for solving illposed 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 illposed 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 illposed 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 illposed 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.
2011): Iterative solution of a system of nonlinear algebraic equations F(x) = 0, using ẋ= λ [αR+βP] or λ [αF+βP∗], R is a normal to a hypersurface function of F, P normal to R, and P∗ normal to F. CMES: Computer Modeling in Engineering & Sciences
"... Function of F, P Normal to R, and P ∗ Normal to F ..."
An Iterative Method Using an Optimal Descent Vector, for Solving an IllConditioned System Bx = b, Better and Faster than the Conjugate Gradient Method
"... Abstract: To solve an illconditioned system of linear algebraic equations (LAEs): Bx−b = 0, we define an invariantmanifold in terms of r: = Bx−b, and a monotonically increasing function Q(t) of a timelike variable t. Using this, we derive an evolution equation for dx/dt, which is a system of Non ..."
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Cited by 2 (2 self)
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Abstract: To solve an illconditioned system of linear algebraic equations (LAEs): Bx−b = 0, we define an invariantmanifold in terms of r: = Bx−b, and a monotonically increasing function Q(t) of a timelike variable t. Using this, we derive an evolution equation for dx/dt, which is a system of Nonlinear Ordinary Differential Equations (NODEs) for x in terms of t. Using the concept of discrete dynamics evolving on the invariant manifold, we arrive at a purely iterative algorithm for solving x, which we label as an Optimal Iterative Algorithm (OIA) involving an Optimal Descent Vector (ODV). The presently used ODV is a modification of the Descent Vector used in the wellknown and widely used Conjugate Gradient Method (CGM). The presently proposed OIA/ODV is shown, through several examples, to converge faster, with better accuracy, than the CGM. The proposed method has the potential for a wideapplicability in solving the LAEs arising out of the spatialdiscretization (using FEM, BEM, Trefftz, Meshless, and other methods)
Comparative Studies for Different Image Restoration Methods
, 2015
"... Abstract: Image restoration refers to the problem of removal or reduction of degradation in blurred noisy images. The image degradation is usually modeled by a linear blur and an additive white noise process. The linear blur involved is always an illconditioned which makes image restoration proble ..."
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Abstract: Image restoration refers to the problem of removal or reduction of degradation in blurred noisy images. The image degradation is usually modeled by a linear blur and an additive white noise process. The linear blur involved is always an illconditioned which makes image restoration problem an illposed problem for which the solutions are unstable. Procedures adopted to stabilize the inversion of illposed problem are called regularization, so the selection of regularization parameter is very important to the effect of image restoration. In this paper, we study some numerical techniques for solving this illposed problem. Dynamical systems method (DSM), Tikhonov regularization method, Lcurve method and generalized cross validation (GCV) are presented for solving this illposed problems. Some test examples and comparative study are presented. From the numerical results it is clear that DSM showed improved restored images compared to Lcurve and GCV.