scholarly journals Parallel Preconditioned Conjugate Gradient Square Method Based on Normalized Approximate Inverses

2005 ◽  
Vol 13 (2) ◽  
pp. 79-91 ◽  
Author(s):  
George A. Gravvanis ◽  
Konstantinos M. Giannoutakis
Keyword(s):  
Linear Systems ◽  
Conjugate Gradient ◽  
Message Passing ◽  
Message Passing Interface ◽  
Distributed Memory ◽  
Inverse Matrix ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Approximate Inverse Matrix Techniques

A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.

On the multigrid cycle strategy with approximate inverse smoothing

2014 ◽  
Vol 31 (1) ◽  
pp. 110-122 ◽  
Author(s):  
George A. Gravvanis ◽  
Christos K. Filelis-Papadopoulos
Keyword(s):  
Finite Difference ◽  
Linear Systems ◽  
Multigrid Method ◽  
Multigrid Methods ◽  
Inverse Matrix ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Content Type ◽  
Sparsity Pattern ◽  
Approximate Inverses

Purpose – The purpose of this paper is to propose multigrid methods in conjunction with explicit approximate inverses with various cycles strategies and comparison with the other smoothers. Design/methodology/approach – The main motive for the derivation of the various multigrid schemes lies in the efficiency of the multigrid methods as well as the explicit approximate inverses. The combination of the various multigrid cycles with the explicit approximate inverses as smoothers in conjunction with the dynamic over/under relaxation (DOUR) algorithm results in efficient schemes for solving large sparse linear systems derived from the discretization of partial differential equations (PDE). Findings – Application of the proposed multigrid methods on two-dimensional boundary value problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than the V-cycle multigrid schemes presented in a recent report (Filelis-Papadopoulos and Gravvanis). Research limitations/implications – The limitations of the proposed scheme lie in the fact that the explicit finite difference approximate inverse matrix used as smoother in the multigrid method is a preconditioner for specific sparsity pattern. Further research is carried out in order to derive a generic explicit approximate inverse for any type of sparsity pattern. Originality/value – A novel smoother for the geometric multigrid method is proposed, based on optimized banded approximate inverse matrix preconditioner, the Richardson method in conjunction with the DOUR scheme, for solving large sparse linear systems derived from finite difference discretization of PDEs. Moreover, the applicability and convergence behavior of the proposed scheme is examined based on various cycles and comparative results are given against the damped Jacobi smoother.

ON THE RATE OF CONVERGENCE AND COMPLEXITY OF NORMALIZED IMPLICIT PRECONDITIONING FOR SOLVING FINITE DIFFERENCE EQUATIONS IN THREE SPACE VARIABLES

2004 ◽  
Vol 01 (02) ◽  
pp. 367-386 ◽  
Author(s):  
GEORGE A. GRAVVANIS ◽  
KONSTANTINOS M. GIANNOUTAKIS
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Linear Systems ◽  
Conjugate Gradient ◽  
Three Dimensional ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems ◽  
Approximate Factorization ◽  
Dimensional Boundary ◽  
Three Space

Normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference method of partial differential equations in three space variables, are presented. Normalized implicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate factorization procedures are presented for the efficient solution of sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized implicit preconditioned conjugate gradient method are also given. Application of the proposed method on characteristic three dimensional boundary value problems is discussed and numerical results are given.

scholarly journals Reducing communication in algebraic multigrid with multi-step node aware communication

2020 ◽  
Vol 34 (5) ◽  
pp. 547-561
Author(s):  
Amanda Bienz ◽  
William D Gropp ◽  
Luke N Olson
Keyword(s):  
Message Passing ◽  
Message Passing Interface ◽  
Parallel Implementation ◽  
Algebraic Multigrid ◽  
Sparse Linear Systems ◽  
Parallel Scalability ◽  
Strong Scaling ◽  
The Cost ◽  
Communication Schedule ◽  
Inter Process Communication

Algebraic multigrid (AMG) is often viewed as a scalable [Formula: see text] solver for sparse linear systems. Yet, AMG lacks parallel scalability due to increasingly large costs associated with communication, both in the initial construction of a multigrid hierarchy and in the iterative solve phase. This work introduces a parallel implementation of AMG that reduces the cost of communication, yielding improved parallel scalability. It is common in Message Passing Interface (MPI), particularly in the MPI-everywhere approach, to arrange inter-process communication, so that communication is transported regardless of the location of the send and receive processes. Performance tests show notable differences in the cost of intra- and internode communication, motivating a restructuring of communication. In this case, the communication schedule takes advantage of the less costly intra-node communication, reducing both the number and the size of internode messages. Node-centric communication extends to the range of components in both the setup and solve phase of AMG, yielding an increase in the weak and strong scaling of the entire method.

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