Large-numbered multicolor milu preconditioning on sx-3/14

Abstract

This paper considers the vector implementation of preconditioned conjugate gradient (PCG) type methods in solving sparse linear systems of the 3D 7-point difference form. The modified incomplete LU (MILU) factorization is sought, where the system is numbered with respect to the multicolor ordering with a large number of colors (e.g. 75). The advantage in the usage of the large-numbered multicolor ordering is that the PCG-type method based on this ordering tends to require fewer iterations (to reach the same accuracy) than the same method based on a small-numbered multicolor ordering, while both orderings spend almost the same computational time per iteration if the problem is sufficiently large. Numerical experiments are carried out on the SX-3/14 supercomputer, using convection-diffusion equations discretized on a 76 × 76 × 76 grid. Results of experiments show that the large-numbered multicolor (M)ILU/Bi-CGSTAB method, which records at more than 2 GFLOPS speed, converges faster than both the small-numbered multicolor and the hyperplane (M)ILU/Bi-CGSTAB methods.

Keywords:

• Linear system
• modified incomplete LU factorization
• vectorization
• ordering
• multicolor, supercomputer

C.R. CATEGORIES::

• G.1.3.
• G.1.8.