Preconditioned CG methods for sparse matrices on massively parallel machines

Abstract

Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial differential equations. The large size and the condition of many technical or physical applications in this area result in the need for efficient parallelization and preconditioning techniques of the CG method, in particular on massively parallel machines. Here, the data distribution and the communication scheme for the sparse matrix operations of the preconditioned CG are based on the analysis of the indices of the non-zero elements. Polynomial preconditioning is shown to reduce global synchronizations considerably, and a fully local incomplete Cholesky preconditioner is presented. On a PARAGON XP/S 10 with 138 processors, the developed parallel methods outperform diagonally scaled CG markedly with respect to both scaling behavior and execution time for many matrices from real finite element applications.