José Penadés

Block Two‐stage Methods for Singular Systems and Markov Chains

Research Report 95-121, Department of Mathematics, Temple University, December 1995. This paper appeared, in revised form, in Numerical Linear Algebra with Applications, vol. 3 (1996) 413-426.

Block and asynchronous two-stage methods for mildly nonlinear systems

Abstract. Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form

Convergence of non-stationary parallel multisplitting methods for hermitian positive definite matrices

Ax=\Phi(x)

Nonstationary Multisplittings with General Weighting Matrices

are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems includes cases not treated before in the literature.

Nonstationary parallel relaxed multisplitting methods

Non-stationary multisplitting algorithms for the solution of linear systems are studied. Convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite. Asynchronous versions of these algorithms are considered and their convergence investigated.

GPU-based parallel algorithms for sparse nonlinear systems

In the convergence theory of multisplittings for symmetric positive definite (s.p.d.) matrices it is usually assumed that the weighting matrices are scalar matrices, i.e., multiples of the identity. In this paper, this restrictive condition is eliminated. In its place it is assumed that more than one (inner) iteration is performed in each processor (or block). The theory developed here is applied to nonstationary multisplittings for s.p. d. matrices, as well as to two-stage multisplittings for symmetric positive semidefinite matrices.

Alternating two-stage methods for consistent linear systems with applications to the parallel solution of Markov chains

Abstract Relaxed nonstationary multisplitting methods are studied for the parallel solution of nonsingular linear systems. Convergence results of the synchronous and asynchronous versions for systems with H -matrices are presented. Computational results of these methods on a shared memory multiprocessor vector computer are reported. These results show that nonstationary methods (synchronous and asynchronous) are better than the standard ones, especially when the matrix of the linear system has a relatively small bandwidth. Moreover, asynchronous versions always behave better than the synchronous ones.

Non-stationary Parallel Newton Iterative Methods for Nonlinear Problems

In this work we describe some parallel algorithms for solving nonlinear systems using CUDA (Compute Unified Device Architecture) over a GPU (Graphics Processing Unit). The proposed algorithms are based on both the Fletcher-Reeves version of the nonlinear conjugate gradient method and a polynomial preconditioner type based on block two-stage methods. Several strategies of parallelization and different storage formats for sparse matrices are discussed. The reported numerical experiments analyze the behavior of these algorithms working in a fine grain parallel environment compared with a thread-based environment.

Chaotic methods for the papallel solution of linear systems

Two-stage methods in which the inner iterations are accomplished by an alternating method are developed. Convergence of these methods is shown in the context of solving singular and nonsingular linear systems. These methods are suitable for parallel computation. Experiments related to finding stationary probability distribution of Markov chains are performed. These experiments demonstrate that the parallel implementation of these methods can solve singular systems of linear equations in substantially less time than the sequential counterparts.

A heuristic relaxed extrapolated algorithm for accelerating PageRank

Parallel algorithms for solving nonlinear systems are studied. Non-stationary parallel algorithms based on the Newton method are considered. Convergence properties of these methods are studied when the matrix in question is either monotone or an H-matrix. In order to illustrate the behavior of these methods, we implemented these algorithms on two distributed memory multiprocessors. The first platform is an Ethernet network of five 120 MHz Pentiums. The second platform is an IBM RS/6000 with 8 nodes. Several versions of these algorithms are tested. Experiments show that these algorithms can solve the nonlinear system in substantially less time that the current (stationary or non-stationary) parallel nonlinear algorithms based on the multisplitting technique.