Parallel Preconditioned Iterative Equation Solving in Boundary Element Analysis

Nickolas S. Jovanovic

Department of Engineering Technology
University of Arkansas at Little Rock
Little Rock, AR 72204-1099


Abstract

We demonstrate the effectiveness of preconditioned Krylov subspace iterative methods, relative to direct methods based on Gauss elimination, when applied to the dense systems of linear equations arising in direct boundary element analysis (BEA), on sequential and parallel computers, using performance models and numerical experiments. The numerical experiments include potential and elastostatic problems in two and three dimensions, discretized with low-order collocation. We use the iterative solvers GMRES, Bi-CGSTAB, CGS, and TFQMR, preconditioned with Jacobi, Block Jacobi, and ILU. The methods are implemented using the PETSc and ScaLAPACK software packages.

Sequential and parallel performance models are developed for Jacobi-GMRES and compared with models for a block LU direct method. The sequential models predict that, on a single processor, Jacobi-GMRES will be faster than block LU when the discrete problem size is larger than 225 unknowns. Sequential numerical experiments demonstrate a crossover point very near to this - approximately 250 unknowns.

The parallel performance models predict that Jacobi-GMRES will have scalable parallel performance, as measured by parallel self-efficiency, when N/p is held constant, while block LU will scale with N(N/p), with N unknowns and p processors. Parallel numerical experiments confirm these scalings. The parallel models also predict that, for a given number of processors, Jacobi-GMRES will be faster than block LU when the problem size is large enough, with crossover points ranging from about 150 unknowns on two processors to 4500 unknowns on 128 processors. Parallel numerical experiments confirm the existence of these crossover points. _____________________________________________________________ Nickolas S. Jovanovic Assistant Professor of Mechanical Engineering Technology Department of Engineering Technology University of Arkansas at Little Rock 2801 South University Avenue Little Rock, AR 72204-1099 office: (501) 569-8226 secretary: (501) 569-8200 fax: (501) 569-8002 email: nsjovanovic@ualr.edu web: http://www.ualr.edu/~nsjovanovic/ _____________________________________________________________