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A Survey of the Literature for Parallel Iterative Methods

Parallel implementations of Gauss-Seidel have generally been developed for regular problems such as the solution of Laplace's equations by finite differences [17,23], where red-black coloring schemes are used to provide independence in the calculations and some parallelism. This scheme has been extended to multi-coloring for additional parallelism in more complicated regular problems [23]; however, we are interested in the solution of irregular linear systems. There has been some research into applying parallel Gauss-Seidel to circuit simulation problems [48], although this work showed poor parallel speedup potential in a theoretical study. Reference [48] also extended traditional Gauss-Seidel and Gauss-Jacobi methods to waveform relaxation methods that trade overhead and convergence rate for parallelism. A theoretical discussion of parallel Gauss-Seidel methods for power system load-flow problems on an alternating sequential/parallel (ASP) multi-processor is presented in [63]. Other research with the parallel Gauss-Seidel methods for power systems applications is presented in [31], although our research differs substantially from that work. The research we present here utilizes a different matrix ordering paradigm, a different load balancing paradigm, and a different parallel implementation paradigm than that presented in [31]. Our work utilizes diakoptic-based matrix partitioning techniques developed initially for a parallel block-diagonal-bordered direct sparse linear solver [35,36,37,39,40]. In reference [37], we examined load balancing issues associated with partitioning power systems matrices for parallel Choleski factorization.



David P. Koester
Sun Oct 22 17:27:14 EDT 1995