Solving Irregular Sparse Linear Systems on A Multicomputer Using the Cgnr Method

The efficient solution of irregular sparse linear systems on a distributed-memory parallel computer is still a major challenge. Direct methods are concerned with unbalanced load processing or data distribution and difficulties pertaining to reusing efficient sequential codes. Iterative methods of the Krylov family are well suited for parallel computing but can provide disappointing convergence for general sparse problems. Therefore, finding efficient parallel preconditioners is often required to obtain acceptable convergence rates. In this paper, we explore the use of a preconditioned conjugate gradient algorithm for the parallel solution of irregular sparse nonsymmetric systems. A first step is the choice of a high-quality algorithm for matrix partitioning. For this purpose, we have selected the Metis package, developed by Karypis and Kumar of the University of Minnesota. A second step is the choice of the preconditioner. We have selected the Block-Jacobi preconditioner for its inherent parallelism and the local dense computation it generates. The iterative method itself is the standard conjugate gradient on the normal equations (known as CGNR). Experimental results are reported involving problems from the Harwell-Boeing collection executed on the Intel Paragon, using the Aztec-distributed iterative library of the Sandia Laboratories.