Preconditioned Conjugate Gradient Square Research Articles

Changing over stopping criterion for stable solving nonsymmetric linear equations by preconditioned conjugate gradient squared method

An improved preconditioned conjugate gradient squared (PCGS) algorithm has recently been proposed. This algorithm is more accurate and efficient than the conventional PCGS algorithm, and retains the advantages of the left-PCGS in terms of a solution structure. In this paper, we propose a changing over stopping criterion for the improved PCGS, that results in a higher accuracy than the conventional and the left-PCGS. A series of numerical results illustrate the stable solving statuses and the enhanced effectiveness of the improved PCGS with changing over stopping criterion.

Structure of the preconditioned system in various preconditioned conjugate gradient squared algorithms

An improved preconditioned conjugate gradient squared (PCGS) algorithm has recently been proposed, and it performs much better than the conventional PCGS algorithm. In this paper, the improved PCGS algorithm is verified as a coordinative to the left-preconditioned system, and it has the advantages of both the conventional and the left-PCGS; this is done by comparing, analyzing, and executing numerical examinations of various PCGS algorithms, including another improved one. We show that the direction of the preconditioned system for the CGS method is determined by the operations of αk and βk in the PCGS algorithm. By comparing the logical structures of these algorithms, we show that the direction of the preconditioned system can be switched by the construction and setting of the initial shadow residual vector.

High performance finite element approximate inverse preconditioning

Abstract A new parallel normalized optimized approximate inverse algorithm, based on the concept of the “fish bone” computational approach satisfying an antidiagonal data dependency, for computing classes of explicit approximate inverses, is introduced for symmetric multiprocessor systems. The parallel normalized explicit approximate inverses are used in conjunction with parallel normalized explicit preconditioned conjugate gradient square schemes, for the efficient solution of finite element sparse linear systems. The parallel design and implementation issues of the new proposed algorithms are discussed and the parallel performance is presented, using OpenMP.

Parallel Preconditioned Conjugate Gradient Square Method Based on Normalized Approximate Inverses

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.

Iterative Methods for Queuing Systems with Batch Arrivals and Negative Customers

In this paper, we are interested in solving the stationary probability distributions of Markovian queuing systems having batch arrivals and negative customers by using the Preconditioned Conjugate Gradient Squared (PCGS) method. The preconditioner is constructed by exploiting the near-Toeplitz structure of the generator matrix of the system. We proved that under some mild conditions the preconditioned linear systems have singular values clustered around one when the size of the queue tends to infinity. Numerical results indicated that the convergence rate of the proposed method is very fast.

An efficient solution method for incompressible N-S equations using non-orthogonal collocated grid

An efficient strategy for the solution of N-S Equations using collocated, non-orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co-ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non-orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non-orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid-driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti-symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.

Domain decomposition and parallel processing of a finite element model of the shallow water equations

We present nonoverlapping domain decomposition techniques applied to a two-stage Numerov-Galerkin finite element model of the shallow water equations over a limited-area domain. The Schur complement matrix formulation is employed and a modified interface matrix approach is proposed to handle the coupling between subdomains. The resulting nonsymmetric Schur complement matrices, modified interface matrices as well as the subdomain coefficient matrices are solved using Preconditioned Conjugate Gradient Squared (PCGS) non-symmetric iterative solvers. Various stages of the finite element solution are parallelized and the code is implemented ona four processor CRAY Y-MP supercomputer applying multitasking techniques in a dedicated environment.

The application of conjugate gradient methods to NLTE rate matrix equations

This paper brings to attention recent developments in algorithms to efficiently solve linear systems resulting from the discretization of systems of population rate equations which arise in X-ray laser kinetics modeling. Specifically, we report the success of two variants of the preconditioned gradient method, namely the preconditioned conjugate gradient square (PCGS) algorithm and the method of generalized conjugate residuals (GCR) in solving non-LTE rate equations.

SIERRA: a 3-D device simulator for reliability modeling

SIERRA is a 3-D general-purpose semiconductor-device simulation program which serves as a foundation for investigating integrated-circuit (IC) device and reliability issues. This program solves the Poisson and continuity equations in silicon under DC, transient, and small-signal conditions. Executing on a vector/parallel minisupercomputer, SIERRA utilizes a matrix solver which uses an incomplete LU (ILU) preconditioned conjugate gradient square (CGS, BCG) method. The ILU-CGS method provides a good compromise between memory size and convergence rate. The authors have observed a 5* to 7* speedup over standard direct methods in simulations of transient problems containing highly coupled Poisson and continuity equations such as those found in reliability-oriented simulations. The application of SIERRA to parasitic CMOS latchup and DRAM (dynamic random-access memory) single-event-upset studies is described. >