Parallel conjugate gradient-like algorithms for solving sparse nonsymmetric linear systems on a vector multiprocessor

Abstract

We describe the parallel implementation on a vector multiprocessor of two extensions of the preconditioned conjugate gradient algorithm for nonsymmetric systems: the conjugate gradient squared algorithm (CGS) and the generalized minimal residual algorithm GMRES(κ). For both methods, we consider preconditioning by a diagonal matrix and by an incomplete LU factorization. The uniprocessor implementation of CGS and GMRES(κ) is based on a general sparse matrix representation to deal with matrices with an irregular sparsity structure (the ITPACKV format). The parallelization of the non-preconditioned versions is straightforward and leads to very good speedups. The parallelization of the ILU preconditioned versions is more challenging. We describe two parallel preconditioners: • - we compute the global ILU factorization and partition it into one block per processor, • - we compute in parallel local partial factorizations for each block of the matrix. The blocks can overlap, and this partitioning is not restricted to matrices with a special sparsity structure. In the iteration loop the information between blocks is exchanged at each synchronization point, through the matrix-vector product and through the overlap between blocks. We discuss how the size of the overlap influences the efficiency. Our experiments, using up to 6 processors, show that the best strategy is to compute in parallel a local ILU factorization on slightly overlapping blocks.