Formal derivation of an efficient parallel 2-D Gauss-Seidel method

Abstract

Presents a formal derivation of a highly efficient parallel implementation of the 2-D Gauss-Seidel method for machines based on the two-dimensional mesh of processors. A methodology is illustrated which formalizes the process of mapping and scheduling a high level algorithm onto a particular target parallel architecture. It starts from a simple initial program. Equational transformations are then applied: to partition the abstract problem onto processors; to make communication among processors explicit; to pipeline the computation by wave-front scheduling; and finally to map logical processors onto physical processors for perfect processor utilization. All the derivation steps preserve equalities so that the derived programs are equivalent to the initial program. Using this methodology, the paper develops efficient implementations for other parallel architectures.<>