The behavior of conjugate gradient algorithms on a multivector processor with a hierarchical memory

Abstract

In this paper, an analysis of some of the tradeoffs involved in the design and efficient implementation of conjugate gradient-based algorithms for a multivector processor with a two-level memory hierarchy is presented and supplemented by experimental results obtained on an Alliant FX/8. The algorithms considered consist of the classical conjugate gradient method, preconditioning techniques that are well suited for parallel computers such as polynomial preconditioners and several versions of the incomplete Cholesky preconditioners as well as the reduced system approach. For linear systems arising from the 5-point finite difference discretization of 2-d self-adjoint elliptic P.D.E.'s, the analysis shows that conjugate gradient methods do not perform as well as algorithms for dense matrix computations on the considered architecture due to lack of data locality. By using the reduced system approach, however, a significant decrease in time could be obtained.