The Highly Parallel Incomplete Gram-Schmidt Preconditioner

In this paper we study the parallel aspects of IMGS, Incomplete Modiied Gram-Schmidt preconditioner which can be used for ef-ciently solving sparse and large linear systems and least squares problems on massively parallel distributed memory computers. The performance of this preconditioning technique on this kind of architecture is always limited because of the global communication required for the inner products, even for ParIMGS, a parallel version of IMGS where we create some possibilities such that the computation can be overlapped with the communication. We will describe a more eecient alternative, namely Improved ParIMGS (IParIMGS) which avoids the global communication of inner products and only requires local communications. Therefore, the cost of communication can be signiicantly reduced. Several numerical experiments carried out on Parsytec GC/PowerPlus are presented as well.