Parallel Solution of Linear Systems

2016 ◽  
Vol 6 (3) ◽  
pp. 278-289
Author(s):  
Sidi-Mahmoud Kaber ◽  
Amine Loumi ◽  
Philippe Parnaudeau
Keyword(s):  
Linear Systems ◽  
Numerical Algorithms ◽  
Parallel Architecture ◽  
Decomposition Methods ◽  
Partial Fraction ◽  
Domain Decomposition Methods ◽  
Simple Method ◽  
Parallel Solution ◽  
Partial Fraction Decomposition ◽  
Programming Paradigms

AbstractComputational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.

scholarly journals A parallel algorithm for direct solution of large sparse linear systems, well suitable to domain decomposition methods

2009 ◽  
pp. 589-605
Author(s):  
Ibrahima Gueye ◽  
Xavier Juvigny ◽  
Frédéric Feyel ◽  
François-Xavier Roux ◽  
Georges Cailletaud
Keyword(s):  
Domain Decomposition ◽  
Parallel Algorithm ◽  
Linear Systems ◽  
Decomposition Methods ◽  
Computational Effort ◽  
Direct Solution ◽  
Sparse Linear Systems ◽  
Domain Decomposition Methods ◽  
Direct Solver ◽  
Upper Level

The goal of this paper is to develop a parallel algorithm for the direct solution of large sparse linear systems and integrate it into domain decomposition methods. The computational effort for these linear systems, often encountered in numerical simulation of structural mechanics problems by finite element codes, is very significant in terms of run-time and memory requirements.In this paper, a two-level parallelism is exploited. The exploitation of the lower level of parallelism is based on the development of a parallel direct solver with a nested dissection algorithm and to introduce it into the FETI methods. This direct solver has the advantage of handling zero-energy modes in floating structures automatically and properly. The upper level of parallelism is a coarse-grain parallelism between substructures of FETI. Some numerical tests are carried out to evaluate the performance of the direct solver.

scholarly journals Finite Element Tearing and Interconnecting Method and its Algorithms for Parallel Solution of Magnetic Field Problems

2013 ◽  
Vol 3 (1) ◽  
pp. 25-30
Author(s):  
Dániel Marcsa ◽  
Miklós Kuczmann
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Domain Decomposition ◽  
Data Exchange ◽  
Domain Decomposition Method ◽  
Decomposition Methods ◽  
Engineering Practice ◽  
Domain Decomposition Methods ◽  
Parallel Solution ◽  
Solution Algorithms

Abstract Because of the exponential increase of computational resource requirement for numerical field simulations of more and more complex physical phenomena and more and more complex large problems in science and engineering practice, parallel processing appears to be an essential tool to handle the resulting large-scale numerical problems. One way of parallelization of sequential (singleprocessor) finite element simulations is the use of domain decomposition methods. Domain decomposition methods (DDMs) for parallel solution of linear systems of equations are based on the partitioning of the analyzed domain into sub-domains which are calculated in parallel while doing appropriate data exchange between those sub-domains. In this case, the non-overlapping domain decomposition method is the Lagrange multiplier based Finite Element Tearing and Interconnecting (FETI) method. This paper describes one direct solver and two parallel solution algorithms of FETI method. Finally, comparative numerical tests demonstrate the differences in the parallel running performance of the solvers of FETI method. We use a single-phase transformer and a three-phase induction motor as twodimensional static magnetic field test problems to compare the solvers

A simple method to test the existence of 2-D partial fraction decomposition

1984 ◽  
Vol 6 (2) ◽  
pp. 153-160 ◽  
Author(s):  
D.Raghu Rami Reddy ◽  
P.S. Reddy ◽  
M.N.S. Swamy
Keyword(s):  
Partial Fraction ◽  
Simple Method ◽  
Partial Fraction Decomposition

scholarly journals h-Adic Polynomials and Partial Fraction Decomposition of Proper Rational Functions over R or C

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Kwang Hyun Kim ◽  
Xin Zhang
Keyword(s):  
Commutative Algebra ◽  
Computer Algebra System ◽  
Rational Functions ◽  
Partial Fraction ◽  
Decomposition Technique ◽  
Taylor Polynomial ◽  
Simple Method ◽  
Recursive Computation ◽  
Algebraic Operations ◽  
Partial Fraction Decomposition

The partial fraction decomposition technique is very useful in many areas including mathematics and engineering. In this paper we present a new and simple method on the partial fraction decomposition of proper rational functions which have completely factored denominators over R or C. The method is based on a recursive computation of the h-adic polynomial in commutative algebra which is a generalization of the Taylor polynomial. Since its computation requires only simple algebraic operations, it does not require a computer algebra system to be programmed.

Sign in / Sign up

Export Citation Format

Share Document