A matrix-free conjugate gradient method for cluster computing

The conjugate gradient method is applied to a large, sparse, highly structured linear system of equations obtained from a finite difference discretization of the Poisson equation. The matrix-free implementation of the matrix-vector product is shown to be optimal with respect to both memory usage and performance. The parallel implementation of the method can give excellent performance on a cluster of work-stations, with the optimal number of processors depending on the quality of the interconnect hardware. This justifies the use of the method as computational kernel for the time-stepping in a system of reaction-diffusion equations. Key words: Poisson equation, finite difference method, matrix-free iterative method, conjugate gradient method, cluster computing 2000 MSC: 35J05, 65F10, 65F50, 65N06, 65Y05 The concentration of calcium ions (Ca2+) in human heart cells controls the beating of the heart [1–3]. The time-evolution of the concentration of the calcium ions is modeled by a system of coupled non-linear reaction-diffusion equations involving ns chemical species. The model developed in [1–3] de-scribes this evolution of the concentrations of calcium c(0) and of several indi-cator and buffer species c(1),..., c(ns−1) by ∂c(0) ∂