A safeguard approach to detect stagnation of GMRES(m) with applications in Newton-Krylov methods

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Restarting GMRES, a linear solver frequently used in numerical schemes, is known to suffer from stagnation. In this paper, a simple strategy is proposed to detect and avoid stagnation, without modifying the standard GMRES code. Numerical tests with the proposed modified GMRES(m) procedure for solving linear systems and also as part of an inexact Newton procedure, demonstrate the efficiency of this strategy.272175199Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)PRONEX-OptimizationFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

A comparative analysis of the monotone iteration method for elliptic problems

In this work, after a theoretical explanation of the monotone iteration method, there are presented several numerical experiments with this method, when applied to solve some nonlinear elliptic equations. It is shown that, in some cases, uniqueness of solution can also be verified through the numerical implementation of the method. It is also presented its application to cooperative elliptic systems. For all the examples Newton's method is also applied and a comparison between the monotone iteration method and Newton's method is made. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.364170023124

On the convergence of quasi-Newton methods for nonsmooth problems

We develop a theory of quasi-Newton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that F can be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.16419211193120

Convergence properties of the inverse column-updating method

The inverse Column-Updating method is a secant algorithm for solving nonlinear systems of equations introduced recently by Martinet and Zambaldi (Optimization Methods and Software, 1 (1992), pp. 129-140). This method is one of the less expensive reliable quasi-Newton methods for solving nonlinear simultaneous equations, in terms of linear algebra work. Since it does not belong to the well-known LCSU (least-change secant-update) class, special arguments are used for proving local convergence. In this paper we prove that, if convergence is assumed, then R-superlinear convergence takes place. Moreover, we prove local convergence for a version of the method with (not necessarily Jacobian) restarts. Finally, we prove that local and R-superlinear convergence holds without restarts in the two-dimensional case. From a practical point of view, we show that, in some cases, the numerical performance of the inverse Column-Updating method is very good.6212714

On the global convergence of Newton-like methods for nonsmooth systems

Global convergence results are proved for Newton's method and for a modified Newton method applied to nonsmooth systems of equations that satisfy some convexity conditions. Numerical experiments are presented showing that the theoretical results explain computational performance.184192195996

Inverse q-Columns Updating Methods for solving nonlinear systems of equations

In this work new quasi-Newton methods for solving large-scale nonlinear systems of equations are presented. In these methods q ( > 1) columns of the approximation of the inverse Jacobian matrix are updated in such a way that the q last secant equations are satisfied (whenever possible) at every iteration. An optimal maximum value for q that makes the method competitive is strongly suggested. The best implementation from the point of view of linear algebra and numerical stability is proposed and a local convergence result for the case q=2 is proved. Several numerical comparative tests with other quasi-Newton methods are carried out. (C) 2003 Elsevier B.V. All rights reserved.158231733

Quasi-Newton acceleration for equality-constrained minimization

Optimality (or KKT) systems arise as primal-dual stationarity conditions for constrained optimization problems. Under suitable constraint qualifications, local minimizers satisfy KKT equations but, unfortunately, many other stationary points (including, perhaps, maximizers) may solve these nonlinear systems too. For this reason, nonlinear-programming solvers make strong use of the minimization structure and the naive use of nonlinear-system solvers in optimization may lead to spurious solutions. Nevertheless, in the basin of attraction of a minimizer, nonlinear-system solvers may be quite efficient. In this paper quasi-Newton methods for solving nonlinear systems are used as accelerators of nonlinear-programming (augmented Lagrangian) algorithms, with equality constraints. A periodically-restarted memoryless symmetric rank-one (SR1) correction method is introduced for that purpose. Convergence results are given and numerical experiments that confirm that the acceleration is effective are presented.40337338

Incomplete decomposition algorithms for discrete dynamic nonlinear models

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