Performance and scalability of preconditioned conjugate gradient methods on parallel computers

This paper analyzes the performance and scalability of an iteration of the preconditioned conjugate gradient algorithm on parallel architectures with a variety of interconnection networks, such as the mesh, the hypercube, and that of the CM-5 parallel computer. It is shown that for block-tridiagonal matrices resulting from two-dimensional finite difference grids, the communication overhead due to vector inner products dominates the communication overheads of the remainder of the computation on a large number of processors. However, with a suitable mapping, the parallel formulation of a PCG iteration is highly scalable for such matrices on a machine like the CM-5 whose fast control network practically eliminates the overheads due to inner product computation. The use of the truncated Incomplete Cholesky (IC) preconditioner can lead to further improvement in scalability on the CM-5 by a constant factor,as a result, a parallel formulation of the PCG algorithm with IC preconditioner may execute faster than that with a simple diagonal preconditioner even if the latter runs faster in a serial implementation