The search for high level parallelism for the iterative solution of large sparse linear systems

Conference · Fri Jul 01 04:00:00 EDT 1988

OSTI ID:674881

Young, D M

In this paper the author is concerned with the numerical solution, based on iterative methods, of large sparse systems of linear algebraic equations of the type which arise in the numerical solution of elliptic and parabolic partial differential equations by finite difference or finite element methods. He considers linear systems of the form Au = b where A is a given N x N matrix which is large and sparse and where b is a given N x 1 column vector. He will assumes that A is symmetric and positive definite (SPD). He considers iterative algorithms which consist of a basic iterative method, such as the Richardson, Jacobi, SSOR or incomplete Cholesky method, combined with an acceleration procedure such as Chebyshev acceleration or conjugate gradient acceleration. The object of this paper is, however, to examine some high-level methods for achieving parallelism. Such techniques involve only matrix/vector operations and do not involve working with blocks of the matrix, subdividing the region, or using different meshes. It is expected that if effective high-level methods could be developed, they could be combined with block and domain decomposition methods, and related methods, to obtain even greater speedups. It is also expected that by working at a higher level it will eventually be possible to develop general purpose software for parallel machines similar to the ITPACK software packages which have already been developed for sequential and vector machines. The discussion here is primarily devoted to describing various techniques which the author and others have considered for obtaining high-level parallelism. The author plans to continue research on these techniques and eventually to develop algorithms and programs for multiprocessors based on them.

• Research Organization: Univ. of Texas, Center for Numerical Analysis, Austin, TX (United States)
• Sponsoring Organization: USDOE Office of Energy Research, Washington, DC (United States)
• DOE Contract Number: FG05-87ER25048
• OSTI ID: 674881
• Report Number(s): DOE/ER/25048--32; CONF-8806154--; ON: DE99000920
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
ITERATIVE METHODS
MATRICES
NUMERICAL SOLUTION
PARALLEL PROCESSING
PARTIAL DIFFERENTIAL EQUATIONS