Regularized continuous analog of the newton method for monotone equations in a Hilbert space

2016 ◽  
Vol 60 (11) ◽  
pp. 45-57
Author(s):  
I. P. Ryazantseva
Keyword(s):  
Hilbert Space ◽  
Newton Method ◽  
Continuous Analog ◽  
Monotone Equations

scholarly journals A global convergent quasi-Newton method for systems of monotone equations

2013 ◽  
Vol 44 (1-2) ◽  
pp. 455-465 ◽  
Author(s):  
Zixin Chen ◽  
Wanyou Cheng ◽  
Xiaoliang Li
Keyword(s):  
Newton Method ◽  
Monotone Equations ◽  
Quasi Newton

scholarly journals Continuous method of second order with constant coefficients for monotone equations in Hilbert space

2020 ◽  
Vol 22 (4) ◽  
pp. 449-455
Author(s):  
Irina P. Ryazantseva
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Difference ◽  
Monotone Equations ◽  
Sufficient Conditions For Convergence ◽  
Second Order Difference Equation ◽  
Constant Coefficients

Convergence of an implicit second-order iterative method with constant coefficients for nonlinear monotone equations in Hilbert space is investigated. For non-negative solutions of a second-order difference numerical inequality, a top-down estimate is established. This estimate is used to prove the convergence of the iterative method under study. The convergence of the iterative method is established under the assumption that the operator of the equation on a Hilbert space is monotone and satisfies the Lipschitz condition. Sufficient conditions for convergence of proposed method also include some relations connecting parameters that determine the specified properties of the operator in the equation to be solved and coefficients of the second-order difference equation that defines the method to be studied. The parametric support of the proposed method is confirmed by an example. The proposed second-order method with constant coefficients has a better upper estimate of the convergence rate compared to the same method with variable coefficients that was studied earlier.

NUMERICAL INVESTIGATION OF BIFURCATIONS AND TRANSITIONS OF JOSEPHSON VORTICES IN INHOMOGENEOUS JUNCTIONS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350192
Author(s):  
E. G. SEMERDJIEVA ◽  
M. D. TODOROV
Keyword(s):  
Magnetic Field ◽  
Numerical Investigation ◽  
Newton Method ◽  
Josephson Junctions ◽  
Eigenvalue Problems ◽  
Continuous Analog ◽  
Critical Curves ◽  
Static Solutions ◽  
Josephson Vortices ◽  
Current Magnetic Field

We consider in-line and overlap geometry models of Josephson junctions with point or rectangular inhomogeneity and investigate the effect of their location on the Josephson vortices and the current. We analyze numerically the critical dependencies "current-magnetic field" caused by one- and two-point current injections. The obtained results elucidate the relation between these critical curves and the fractions of the injection current at the ends of the junction. We also find out similarities between the exponentially shaped junctions, and those with inhomogeneity at the end when a two-point current injection is present. We juxtapose the critical curves of the distinct junctions with inner inhomogeneity and discuss the similarity between them and the Josephson junctions with phase shifts. The transitions of Josephson junctions from a superconducting mode to a resistive one as bifurcations of the static solutions of appropriately posed multiparametric compound boundary- and eigenvalue problems are interpreted and solved using the continuous analog of Newton method.

scholarly journals CONTINUOUS ANALOG OF THE GAUSS–NEWTON METHOD

1999 ◽  
Vol 09 (03) ◽  
pp. 463-474 ◽  
Author(s):  
R. G. AIRAPETYAN ◽  
A. G. RAMM ◽  
A. B. SMIRNOVA
Keyword(s):  
Inverse Problem ◽  
Newton Method ◽  
Convergence Theorem ◽  
Numerical Experiments ◽  
Nonlinear Problems ◽  
Nonlinear Inverse Problem ◽  
Derivative Operator ◽  
Continuous Analog ◽  
Ill Posed ◽  
Practical Recommendations

A Continuous Analog of discrete Gauss–Newton Method (CAGNM) for numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization function is introduced. For the CAGNM, a convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments, practical recommendations for the choice of the regularization function are given.

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