Solving sparse linear systems practically dominates scientific computing, but the performance of direct sparse matrix solvers have tended to trail behind their dense matrix counterparts [7]. Parallel sparse matrix solver performance generally is less than similar dense matrix solvers even though there is more inherent parallelism in sparse matrix algorithms than dense matrix algorithms. Parallel sparse linear solvers can simultaneously factor entire groups of mutually independent contiguous columns or rows without communications; meanwhile, dense linear solvers can only update blocks of contiguous columns or rows each communication cycle. The limited success with efficient sparse matrix solvers is not surprising, because general sparse linear solvers require more complicated data structures and algorithms that must contend with irregular memory reference patterns. The irregular nature of these problems has aggravated the problems of implementing scalable sparse matrix solvers on vector or parallel architectures: efficient scalable algorithms for these classes of machines require regularity in available data vector lengths and in interprocessor communications patterns [2]. Parallel block-diagonal-bordered sparse linear solvers offer the potential for regularity often absent from other parallel sparse solvers. Nevertheless, when scalability of sparse linear solvers is examined using real irregular sparse matrices, the available parallelism in the sparse matrix can be as much the reason for poor scalability as the parallel algorithm or implementation.
Research is being performed to examine the applicability of parallel direct block-diagonal-bordered sparse solvers for electrical power system sparse matrix problems. This research focuses on real power system applications, consequently, matrix sizes are limited and available parallelism is also limited because the matrix structure defines the available parallelism for a single parallel sparse linear solver. For power system applications, however, the limited size of the matrices and load imbalance due to limited parallelism in the matrix structure significantly limits scalability for a single parallel linear solver. Our research into specialized ordering techniques has shown that it is possible to order actual power system matrices readily into block-diagonal-bordered form, but load imbalance becomes excessive beyond 16 processors, limiting scalability for a single parallel linear solver within an application. Nevertheless, other dimensions exist in electrical power system applications that can be exploited to efficiently make use of large-scale multi-processors. We believe that this research also has utility for other irregular sparse matrix applications where the data is hierarchical. Other sources of hierarchical matrices exist, for example, electrical circuits, that have the potential for larger numbers of equations than power system matrices.
In this paper we examine the potential for scalability in block-diagonal-bordered direct sparse linear solvers to be incorporated within electrical power system applications. We focus on techniques to order sparse power system matrices into block-diagonal-bordered form, and examine scalability of the sparse linear solver as a function of the available parallelism in the power system network matrix. In section 2, we introduce the electrical power system applications that are the basis for this work. In section 3 we describe the advantages of parallel block-diagonal-bordered sparse matrix solvers. Paramount to exploiting the advantages of this parallel linear solver is ordering the irregular sparse power system matrices into this form. Minimum degree ordering, recursive spectral bisection-based ordering, and node-tearing-based ordering techniques are discussed in section 4. Analysis of the performance of these ordering techniques for actual power system load flow matrices from the Boeing-Harwell series are presented. Lastly, in section 5, we discuss our conclusions concerning these ordering techniques and our conclusions concerning the scalability of block-diagonal-bordered sparse linear solvers for power system applications.