In this paper we study nonmonotone globalization techniques, in connection with finite-difference inexact Newton-GMRES methods, for solving large-scale systems of nonlinear equations in the case that the Jacobian matrix is not available. We first define a globalization scheme, which combines nonmonotone watchdog rules and nonmonotone derivative-free line searches, and we prove its global conver...

A global multi-grid/GMRES solution methodology on distributed memory machines is successfully developed in this study. To preserve the effectiveness of the multigrid scheme, the grid partitioning is based on the communication graph of the coarsest grid, so that all levels of the multi-grids are located in the same zone (processor). Each node of the graph is weighted with the total number of the...

In the Generalized Minimal Residual Method (GMRES), the global all-to-all communication required in each iteration for orthogonalization and normalization of the Krylov base vectors is becoming a performance bottleneck on massively parallel machines. Long latencies, system noise and load imbalance cause these global reductions to become very costly global synchronizations. In this work, we prop...

GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating ...

This paper presents equivalent forms of the Sylvester matrix equations. These equivalent forms allow us to use block linear methods for solving large Sylvester matrix equations. In each step of theses iterative methods we use global FOM or global GMRES algorithm for solving an auxiliary block matrix equations. Also, some numerical experiments for obtaining the numerical approximated solution of...

The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving generalized Sylvester matrix equations. The proposed methods are iterative projection methods onto matrix Kr...