Sains Malaysiana 35(2): 49-54 (2006)

Splitting Strategies For Preconditioned

Explicit Group Schemes

(Strategi Pembelahan bagi Kaedah Kumpulan

Tak Tersirat Berprasyarat)

Norhashidah Hj. Mohd. Ali

Nurul Fizaham Fijasri

School of Mathematical Sciences

Universiti Sains Malaysia

11800 Minden, Pulau Pinang

ABSTRACT 

Combining iterative methods with appropriate preconditioners is a worthwhile effort in improving the performances of the methods since the reliability of these methods have been shown to improve by the use of appropriate preconditioning techniques.  However, the hardest issue is to find the suitable preconditioners which are computationally inexpensive and easy to solve for the group methods.  The aim of this paper is to study the performance of the Explicit Group (EG) method (Evans & Yousif 1986) preconditioned by a specific ‘splitting’ approach in solving the two dimensional elliptic partial differential equation.  Our goal in this work is to investigate whether the performance of this group method is affected by this preconditioner.  The experimental work performed is reported and discussed.

Keywords: Preconditioner; Explicit Group Methods; elliptic equation

ABSTRAK

Menggabungkan kaedah-kaedah lelaran dengan prasyarat yang sesuai merupakan suatu usaha yang berfaedah dalam memperbaiki prestasi kaedah tersebut kerana keberkesanan kaedah-kaedah sebegini telah dapat ditunjukkan menjadi lebih baik dengan penggunaan teknik berprasyarat yang bersesuaian. Walau bagaimanapun, isu yang paling rumit ialah dalam mencari prasyarat yang sesuai yang tidak tinggi kos pengiraannya dan mudah diselesaikan untuk kaedah berkumpulan itu. Tujuan kertas ini ialah untuk mengkaji prestasi kaedah Kumpulan Tak Tersirat (KTT) (Evans & Yousif 1986) yang diprasyaratkan oleh pendekatan ‘pemisahan’ yang spesifik dalam menyelesaikan persamaan pembezaan separa eliptik. Sasaran kerja ini adalah untuk menyelidik sama ada prestasi kaedah berkumpulan ini dipengaruhi oleh prasyarat tersebut.  Hasil kerja ujikaji yang dijalankan dilapor dan dibincangkan.

Kata kunci: Prasyarat; Kaedah Kumpulan Tak Tersirat; persamaan eliptik

RUJUKAN/REFERENCE

Ali, N.H.M., Abdullah, R., and Lee, K.J., 2003, Explicit Group Iterative Solver On A Message Passing Environment, Recent Advances in Parallel Virtual Machine and Message Passing Interface: Proceedings of the 10th European PVM/MPI 2003. Dongarra et al. (eds.) Sept. 29-Oct. 2, Venice, Italy, Lecture Notes in Computer Science, Springer-Verlag Berlin, pp. 232-236.

Evans, D.J., 1968. The Use of Preconditioning In Iterative Methods for Solving Linear Equations with Symmetric Positive Definite Matrices, J.I.M.A, 4:295-314.

Evans, D.J. 1983. Preconditioning Methods: Analysis and Applications, Gordon and Breach Science Publishers, New York.

Evans, D.J., 1994. Preconditioned Iterative Methods, Gordon and Breach Science Publishers, Switzerland.

Evans, D.J.  and Yousif, W.S. 1990. The Implementation of the explicit block  iterative methods on the balance 8000 Parallel Computer, Parallel Computing, 16: 81-97.

Yousif, W.S. and Evans, D.J. 1986. Explicit group over-relaxation methods for solving elliptic partial differential equations, Math. Computer Simulation, 28: 453-466.