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Parallel supercomputing: Advanced methods, algorithms and software for large-scale problems

Description: Research has continued with excellent progress and new results on methodology and algorithms. We have also made supporting benchmark application studies on representative parallel computing architectures. Results from these research studies have been reported at scientific meetings, as technical reports and as journal publications. A list of pertinent presentations and publications is attached. The work on parallel element-by-element techniques and domain decomposition schemes has developed well. In particular, we have focused on the use of finite element spectral methods (or high p methods) on distributed massively parallel systems. The approach has been implemented in a prototype finite element program for solution of coupled Navier Stokes flow and transport processes. This class of problems is of fundamental interest and basic to many grand challenge'' type problems for which parallel supercomputing is warranted.
Date: April 1, 1992
Creator: Carey, G.F. & Young, D.M.
Partner: UNT Libraries Government Documents Department

Stationary second-degree iterative methods and recurrences

Description: The basic theory of stationary second-degree iterative methods is presented from the point of view of recurrences. Recurrences are encountered in the development of expressions for the spectral radii and for various norms associated with linear stationary iterative methods. We show that many of these recurrences are special cases of a single general recurrence and that its closed-form solution leads to these expressions. Citations are given showing where the expressions occur in the theory of iterative methods.
Date: February 1, 1991
Creator: Kincaid, D.R. & Young, D.M.
Partner: UNT Libraries Government Documents Department

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

Description: 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.
Date: July 1, 1988
Creator: Young, D.M.
Partner: UNT Libraries Government Documents Department

MGMRES: A generalization of GMRES for solving large sparse nonsymmetric linear systems

Description: This paper is concerned with the solution of the linear system Au = b, where A is a real square nonsingular matrix which is large, sparse and nonsymmetric. We consider the use of Krylov subspace methods. We first choose an initial approximation u{sup (0)} to the solution {bar u} = A{sup -1}b. The GMRES (Generalized Minimum Residual Algorithm for Solving Non Symmetric Linear Systems) method was developed by Saad and Schultz (1986) and used extensively for many years, for sparse systems. This paper considers a generalization of GMRES; it is similar to GMRES except that we let Z = A{sup T}Y, where Y is a nonsingular matrix which is symmetric but not necessarily SPD.
Date: November 1, 1996
Creator: Young, D.M. & Chen, Jen Yuan
Partner: UNT Libraries Government Documents Department

Parallel supercomputing: Advanced methods, algorithms, and software for large-scale linear and nonlinear problems. Final report, August 15, 1993--February 28, 1996

Description: This report gives a quick description of progress on methods, algorithms and software for large-scale parallel supercomputer applications. Focus is on large-scale applications of interest to DOE such as coupled viscous flow and heat or mass transport, and energy-related applications such as 3D petroleum and gas reservoir simulations on massively parallel systems. The interdisciplinary collaboration has been effective since it enhances the development of new iterative schemes for complex problems important to DOE: for example, significant advanced were made with modified forms of generalized gradient and multigrid methods for viscous flow and reservoir problems.
Date: July 1, 1996
Creator: Carey, G.F. & Young, D.M.
Partner: UNT Libraries Government Documents Department

Linear stationary second-degree methods for the solution of large linear systems

Description: The optimum linear stationary second-degree iterative method for solving linear systems of equations is not as good in general as the optimum semi-iterative method. However, for a suitable choice of parameters, the rate of convergence of the stationary method is very nearly as good as that of the semi-iterative method. The authors present a straightforward determination of these optimum parameter values and the asymptotic rate of convergence.
Date: July 9, 1990
Creator: Young, D.M. & Kincaid, D.R.
Partner: UNT Libraries Government Documents Department

Parallel supercomputing: Advanced methods, algorithms and software for large-scale problems. Final report, August 1, 1987--July 31, 1994

Description: The focus of the subject DOE sponsored research concerns parallel methods, algorithms, and software for complex applications such as those in coupled fluid flow and heat transfer. The research has been directed principally toward the solution of large-scale PDE problems using iterative solvers for finite differences and finite elements on advanced computer architectures. This work embraces parallel domain decomposition, element-by-element, spectral, and multilevel schemes with adaptive parameter determination, rational iteration and related issues. In addition to the fundamental questions related to developing new methods and mapping these to parallel computers, there are important software issues. The group has played a significant role in the development of software both for iterative solvers and also for finite element codes. The research in computational fluid dynamics (CFD) led to sustained multi-Gigaflop performance rates for parallel-vector computations of realistic large scale applications (not computational kernels alone). The main application areas for these performance studies have been two-dimensional problems in CFD. Over the course of this DOE sponsored research significant progress has been made. A report of the progression of the research is given and at the end of the report is a list of related publications and presentations over the entire grant period.
Date: December 31, 1994
Creator: Carey, G.F. & Young, D.M.
Partner: UNT Libraries Government Documents Department

Commentaries of three papers of Cornelius Lanczos

Description: This report contains commentaries on these three papers of Cornelius Lanczos: (A) Cornelius Lanczos [1952], ``Iterative Solution of Systems of Linear Equations by Minimized Iterations,`` Journal of Research of the National Bureau of Standards, 49, 33--53; (B) Cornelius Lanczos [1953], ``Chebyshev Polynomials in the Solution of Large-Scale Linear Systems,`` Proceedings of the ACM Conference held in Toronto, California in 1952, Sauls L. Lithograph Co., Washington, DC; (C) Cornelius Lanczos [1958], ``Iterative Solutions of Large-Scale Linear Systems,`` J. Soc. Indust. Appl. Math., 6, 91--109. These commentaries will be included in a volume of the collected published Lanczos papers which will be published as part of the Cornelius Lanczos Centenary Celebration at North Carolina State University. The volume is scheduled for publication in December 1993.
Date: October 1991
Creator: Jea, K. C. & Young, D. M.
Partner: UNT Libraries Government Documents Department