Parallel Preconditioning Techniques for Sparse CG Solvers

Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial diierential equations. The large size and the condition of many technical or physical applications in this area result in the need for eecient par-allelization and preconditioning techniques of the CG method. In particular for very ill-conditioned matrices, sophisticated preconditioner are necessary to obtain both acceptable convergence and accuracy of CG 2] 3]. Here, we investigate variants of polynomial and incomplete Cholesky preconditioners that markedly reduce the iterations of the simply diagonally scaled CG and are shown to be well suited for massively parallel machines. The basic operations of the pure CG iteration as well as of the polynomially preconditioned method 1] 8] are matrix-vector products with the coeecient matrix and vector-vector computations. For incomplete Cholesky preconditioning, the factorization of the matrix before the CG iteration and a forward/back-substitution per iteration 6] are required in addition. To parallelize these operations on a multiprocessor system with distributed memory, we present a data distribution and a communication scheme based on the analysis of the non-zero matrix elements. By a suitable block-reordering of the matrix, the overlapped execution of computation and communication is supported to reduce waiting times 4] 5]. Factorization and forward/back-substitution of the developed incomplete Cholesky precondi-tioner are only performed on local blocks so that these operations do not 1