On Solving L-SR1 Trust-Region Subproblems

In this article, we consider solvers for large-scale trust-region subproblems when the quadratic model is defined by a limited-memory symmetric rank-one (L-SR1) quasi-Newton matrix. We propose a solver that exploits the compact representation of L-SR1 matrices. Our approach makes use of both an orthonormal basis for the eigenspace of the L-SR1 matrix and the Sherman-Morrison-Woodbury formula to compute global solutions to trust-region subproblems. To compute the optimal Lagrange multiplier for the trust-region constraint, we use Newton's method with a judicious initial guess that does not require safeguarding. A crucial property of this solver is that it is able to compute high-accuracy solutions even in the so-called hard case. Additionally, the optimal solution is determined directly by formula, not iteratively. Numerical experiments demonstrate the effectiveness of this solver.Comment: 2015-0

A symmetric rank-one Quasi-Newton line-search method using negative curvature directions

We propose a quasi-Newton line-search method that uses negative curvature directions for solving unconstrained optimization problems. In this method, the symmetric rank-one (SR1) rule is used to update the Hessian approximation. The SR1 update rule is known to have a good numerical performance; however, it does not guarantee positive definiteness of the updated matrix. We first discuss the details of the proposed algorithm and then concentrate on its numerical efficiency. Our extensive computational study shows the potential of the proposed method from different angles, such as; its second order convergence behavior, its exceeding performance when compared to two other existing packages, and its computation profile illustrating the possible bottlenecks in the execution time. We then conclude the paper with the convergence analysis of the proposed method

Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition

This paper provides a detailed review of the global/local non-intrusive coupling algorithm. Such method allows to alter a global finite element model, without actually modifying its corresponding numerical operator. We also look into improvements of the initial algorithm (Quasi- Newton and dynamic relaxation), and provide comparisons based on several relevant test cases. Innovative examples and advanced applications of the non-intrusive coupling algorithm are provided, granting a handy framework for both researchers and engineers willing to make use of such process. Finally, a novel nonlinear domain decomposition method is derived from the global/local non-intrusive cou- pling strategy, without the need to use a parallel code or software. Such method being intended to large scale analysis, we show its scalability. Jointly, an efficient high level Message Passing Interface coupling framework is also proposed, granting an universal and flexible way for easy software coupling. A sample code is also given

Local convergence of the symmetric rank-one iteration

SIGLEAvailable from TIB Hannover: RR 1843(95-07) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Local convergence of the symmetric rank-one iteration

SIGLEAvailable from TIB Hannover: RR 1843(95-07) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Local Convergence Of The Symmetric Rank-One Iteration

We consider conditions under which the SR1 iteration is locally convergent. We apply the result to a pointwise structured SR1 method that has been used in optimal control