Memory Aspects and Performance of Iterative Solvers

Abstract

In many scientific computing problems, the overall execution time is dominated by the time to solve very large linear systems. Quite often, the matrices are unsymmetric and ill conditioned, with an irregular sparsity structure reflecting the irregular refinement in the discretization grid. With increasing problem size and problem dimension, direct solvers cannot be used because of the huge memory requirements. The performance of preconditioned iterative solvers is largely dominated by memory-related aspects like size, bandwidth, and indirect addressing speed. This article summarizes the experience of the authors on the relationship between memory aspects and performance in real applications in the domain of very large scale integration (VLSI) device simulation. The authors analyze storage requirements of direct and iterative solvers on a statistical data set, and demonstrate performance variations due to memory-related architectural features on a number of computers ranging from workstations to Cray, NEC, and Fujitsu supercomputers for typical and ill-conditioned linear systems, using different iterative methods and preconditioners. The experiments are done using PILS, a package of iterative linear solvers. PILS implements a large number of iterative methods and preconditioners and allows them to be combined in a flexible way.