On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR

Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms

Parallel machine architecture and compiler design facilities

The objective is to provide an integrated simulation environment for studying and evaluating various issues in designing parallel systems, including machine architectures, parallelizing compiler techniques, and parallel algorithms. The status of Delta project (which objective is to provide a facility to allow rapid prototyping of parallelized compilers that can target toward different machine architectures) is summarized. Included are the surveys of the program manipulation tools developed, the environmental software supporting Delta, and the compiler research projects in which Delta has played a role

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Research summary, January 1989 - June 1990

The Research Institute for Advanced Computer Science (RIACS) was established at NASA ARC in June of 1983. RIACS is privately operated by the Universities Space Research Association (USRA), a consortium of 62 universities with graduate programs in the aerospace sciences, under a Cooperative Agreement with NASA. RIACS serves as the representative of the USRA universities at ARC. This document reports our activities and accomplishments for the period 1 Jan. 1989 - 30 Jun. 1990. The following topics are covered: learning systems, networked systems, and parallel systems

Experiments With Elliptic Problem Solvers On The Cedar Multicluster

. The implementation and performance of algorithms for the solution of elliptic problems on the Cedar multiprocessor is examined. The algorithms considered are conjugate gradient schemes with or without preconditioning on a rectangular domain and domain decomposition techniques with overlapping (Schwarz Alternating Method) and nonoverlapping subdomains on a T-shaped domain. We present some preliminary results from our current research effort. 1. Introduction. We have presented elsewhere ([1, 2]) algorithms for solving elliptic PDEs on vector multiprocessors under a memory hierarchy, which in essence consisted of a single cluster of the Cedar architecture designed at CSRD. The major algorithmic design issues in those discussions have been centered around the exploitation of the memory hierarchy and the effective combination of vector and concurrent processing. We are currently modifying and testing algorithms for the Cedar multicluster. In this paper we will provide a brief outline of ..