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

On The Convergence Behavior Of The Restarted Gmres Algorithm For Solving Nonsymmetric Linear Systems

The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s?0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented. Keywords: Iterative methods, nonsymmetric linear systems, GMRES, restarting. 1. Introduction We consider linear systems of the form Au = b; (1) where A 2 I C N \ThetaN is nonsingular and possibly non-Hermitian. A major class of methods for solving (1) is the class of polynomial methods, defined by u (n) = u (0) +Qn\Gamma1 (A)r (0) ..

On-line cascading event tracking and avoidance decision support tool

Cascading outages in power systems are costly events that power system operators and planners actively seek to avoid. Such events can quickly result in power outages for millions of customers. Although it is unreasonable to claim that blackouts can be completely prevented, we can nonetheless reduce the frequency and impact of such high consequence events. Power operators can take actions if they have the right information provided by tools for monitoring and managing the risk of cascading outages. Such tools are being developed in this research project by identifying contingencies that could initiate cascading outages and by determining operator actions to avoid the start of a cascade.;A key to cascading outage defense is the level of grid operator situational awareness. Severe disturbances and complex unfolding of post-disturbance phenomena, including interdependent events, demand critical actions to be taken on the part of the operators, thus making operators dependent on decision support tools and automatic controls. In other industries (e.g., airline, nuclear, process control), control operators employ computational capabilities that help them predict system response and identify corrective actions. Power system operators should have a similar capability with online simulation tools.;To create an online simulator to help operators identify the potential for and actions to avoid cascades, we developed a systematic way to identify power system initiating contingencies for operational use. The work extends the conventional contingency list by including a subset of high-order contingencies identified through topology processing. The contingencies are assessed via an online, mid-term simulator, designed to provide generalized, event-based, corrective control and decision support for operators with very high computational efficiency. Speed enhancement is obtained algorithmically by employing a multi-frontal linear solver within an implicit integration scheme. The contingency selection and simulation capabilities were illustrated on two systems: a test system with six generators and the IEEE RTS-96 with 33 generators. Comparisons with commercial grade simulators indicate the developed simulator is accurate and fast