Steps towards steady-state simulation on MIMD machines: solving almost block diagonal linear systems

Abstract

We study, in this paper, the feasibility of using preconditioned iterative methods to solve linear systems on MIMD (multiple instruction multiple data) machines when solving steady-state simulation problems. We show that, provided a good parallel preconditioner is available, the proposals in this paper give very good parallel performance. Our proposals are directly applicable to almost block diagonal linear systems, but can be extended to more general flowsheeting problems (Paloschi, 1996). This will be the subject of a future paper. A preconditioning strategy blending an incomplete LU decomposition with a polynomial preconditioner is proposed and tested with the iterative method GMRES of Saad and Schultz (1986). A parallel implementation is tested on two MIMD machines, a Fujitsu AP1000 with 128 processors and a cluster of workstations with PVM (Parallel Virtual Machine) (Geist et al., 1994). Two examples are used, a simple model of a distillation column using Antoine coefficients and the discretization of a PDE. The second example allows the variation in the size of the problem and results are presented for up to 65 536 equations on the AP1000 and 262 144 equations on the cluster. Almost perfect efficiency is observed when solving a problem of size 65 536 using 32 processors on the AP1000 (and size 16 384 using 16). Very good speed-ups also are obtained on the cluster of workstations (a speed-up of 15.3 using 16 processors) when solving a problem of size 65 536. The results on the cluster are presented for different load conditions on the network.