scholarly journals AN INEXACT NEWTON METHOD WITH INNER PRECONDITIONED CG FOR NON-UNIFORMLY MONOTONE ELLIPTIC PROBLEMS

2021 ◽  
Vol 26 (3) ◽  
pp. 383-394
Author(s):  
Benjámin Borsos
Keyword(s):  
Banach Space ◽  
Newton Method ◽  
Lower Bounds ◽  
Numerical Experiments ◽  
Real Life ◽  
Elliptic Problems ◽  
Upper And Lower Bounds ◽  
Preconditioned Conjugate Gradient ◽  
Inexact Newton Method ◽  
Convergence Results

The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.

scholarly journals Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
B. Borsos ◽  
János Karátson
Keyword(s):  
Finite Element ◽  
Newton Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Elliptic Problems ◽  
Iterative Solvers ◽  
Finite Element Solution ◽  
Element Solution ◽  
Nonlinear Beam ◽  
Quasi Newton

Abstract The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

scholarly journals Newton-variational solution of the Frank–Kamenetskii thermal explosion problem

1989 ◽  
Vol 67 (3) ◽  
pp. 442-445 ◽  
Author(s):  
Avygdor Moise ◽  
Huw O. Pritchard
Keyword(s):  
Thermal Explosion ◽  
Newton Method ◽  
Lower Bounds ◽  
Linear Equations ◽  
Spherical Geometry ◽  
Variational Solution ◽  
Upper And Lower Bounds ◽  
Dimensionless Temperature ◽  
Temperature Excess ◽  
Thermal Explosions

The Newton method was shown by Vatsya to be suitable for solving the generalised elliptic problem, and we show that this approach can be used to treat the Frank–Kamenetskii model of a thermal explosion, by using a variational solution of the sequence of linear equations that are encountered in the Newton method. Convergence to the desired solution is rapid in the case of spherical geometry. The method produces converging upper and lower bounds to the critical value of the dimensionless heat production rate, δc, and lower bounds to the dimensionless temperature excess distribution function θ and its critical form θc. Keywords: thermal explosions, Frank–Kamenetskii model.

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

scholarly journals A Banach Space Regularization Approach for Multifrequency Microwave Imaging

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Claudio Estatico ◽  
Alessandro Fedeli ◽  
Matteo Pastorino ◽  
Andrea Randazzo
Keyword(s):  
Banach Space ◽  
Data Processing ◽  
Numerical Simulations ◽  
Banach Spaces ◽  
Newton Method ◽  
Mathematical Formulation ◽  
Processing Technique ◽  
Microwave Imaging ◽  
Inexact Newton Method ◽  
Reconstruction Method

A method for microwave imaging of dielectric targets is proposed. It is based on a tomographic approach in which the field scattered by an unknown target (and collected in a proper observation domain) is inverted by using an inexact-Newton method developed inLpBanach spaces. In particular, the extension of the approach to multifrequency data processing is reported. The mathematical formulation of the new method is described and the results of numerical simulations are reported and discussed, analyzing the behavior of the multifrequency processing technique combined with the Banach spaces reconstruction method.

scholarly journals Keep Your Distance: Land Division With Separation

2021 ◽  
Author(s):  
Edith Elkind ◽  
Erel Segal-Halevi ◽  
Warut Suksompong
Keyword(s):  
Lower Bounds ◽  
Real Life ◽  
Fair Division ◽  
Geometric Shape ◽  
Query Complexity ◽  
Upper And Lower Bounds

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axes-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting.

scholarly journals A CONVERGENCE ANALYSIS FOR NEWTON-LIKE METHODS IN BANACH SPACE UNDER WEAK HYPOTHESES AND APPLICATIONS

1999 ◽  
Vol 30 (4) ◽  
pp. 253-261
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Banach Space ◽  
Lipschitz Continuity ◽  
Real Life ◽  
Approximate Solutions ◽  
Continuity Condition ◽  
Point Boundary ◽  
Hölder Continuous ◽  
Life Problems ◽  
Convergence Results ◽  
Bending Of Beams

In this study we use Newton-like methods to approximate solutions of nonlinear equations in a Banach space setting. Most convergence results for Newton-like methods involve some type of a Lipschitz continuity condition on the Frechet-derivative of the operator involved However there are many interesting real life problems already in the literature where the operator can only satisfy a Holder continuity condition. That is why here we chose the Frcchet-derivativc of the operator involved to be only Holder continuous, which allows us to consider a wider range of problems than before. Special choices of our parameters reduce our results to earlier ones. An error analysis is also provided for our method. At the end of our study, we provide applications to show that our results apply where earlier results do not. In paricular we solve a two point boundary value problem appearing in physics in connection with the problem of bending of beams.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Electromagnetic subsurface prospecting by a fully nonlinear multifocusing inexact Newton method

2014 ◽  
Vol 31 (12) ◽  
pp. 2618 ◽  
Author(s):  
Marco Salucci ◽  
Giacomo Oliveri ◽  
Andrea Randazzo ◽  
Matteo Pastorino ◽  
Andrea Massa
Keyword(s):  
Newton Method ◽  
Inexact Newton Method ◽  
Fully Nonlinear

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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