We have developed the parallel block-diagonal-bordered sparse Gauss-Seidel algorithm in order to examine the performance of parallel iterative methods and compare performance with parallel direct methods for power systems networks. In this section, we discuss the performance of the parallel sparse Gauss-Seidel linear solver implementation, and in section 7.3, we compare the performance of parallel direct and parallel iterative methods.
Overall performance of our parallel Gauss-Seidel linear solver is dependent on both the performance of the preprocessing phase to order the matrix and the performance of the parallel Gauss-Seidel implementation. Because these two components of the parallel Gauss-Seidel algorithm are inextricably related, the best way to assess the potential of this parallel iterative algorithm is to measure the empirical performance using matrices from real power systems networks. But first, in section 7.2.1, we illustrate the ordering capabilities of the node-tearing nodal analysis for the Gauss-Seidel algorithm by presenting pseudo-images of selected sparse power systems network matrices after we have applied both our node-tearing algorithm to partition the matrices into block-diagonal-bordered form and our pigeon-hole load-balancing algorithm. We next describe the performance of the parallel block-diagonal-bordered sparse Gauss-Seidel method in section 7.2.2, and we also present data to illustrate the performance of the pigeon-hole load-balancing performed in the preprocessing phase.
Iterative linear solvers must be concerned with the rate of convergence for the intended applications, because iterative solutions only converge in the limit to the solution [23]. We have chosen the Gauss-Seidel method because it converges better than similar iterative techniques such as the Gauss-Jacobi, and we have been able to develop a parallel sparse Gauss-Seidel algorithm that can maintain the strict precedence relations while having no inherently sequential calculations. We discuss the convergence of Gauss-Seidel techniques for power systems network applications in section 7.2.3, we compare the performance of low-latency, active message-based implementations and buffered communications-based implementations in section 7.2.4, and in section 7.2.5, we present our conclusions concerning the performance of our parallel iterative implementations.