On the performance of parallel normalized explicit preconditioned conjugate gradient type methods

A new class of parallel normalized preconditioned conjugate gradient type methods in conjunction with normalized approximate inverses algorithms, based on normalized approximate factorization procedures, for solving sparse linear systems of irregular structure, which are derived from the finite element method of a two dimensional boundary value problem, is introduced. Parallel normalized explicit preconditioned conjugate gradient-type methods for distributed memory systems based on the block-row distribution (for the vectors and the explicit approximate inverse), using message passing interface (MPI) communication library, is also presented with theoretical estimates on speedups and efficiency, in order to examine the parallel behavior of these methods using normalized explicit approximate inverses as the suitable preconditioner. Collective communications have been utilized at the synchronization points and non-blocking communications have been used, where the exchanging of messages can be overlapped with computations, where applicable. Application of the methods on a two dimensional boundary value problem is discussed and numerical results are given, concerning the parallel performance in terms of speedups and efficiency