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Introduction

Solving sparse linear systems practically dominates scientific computing [15], but the performance of sparse matrix solvers have tended to trail behind their dense matrix counterparts [12]. 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. The limited success with efficient sparse matrix solvers is not surprising, because general sparse matrix solvers require more complicated algorithms and significantly more complicated data structures that require irregular memory reference patterns. The irregular nature of these problems has aggravated the problems of implementing sparse matrix solvers on vector or parallel architectures: efficient algorithms for these classes of machines require regularity in available data vector lengths and in interprocessor communications patterns. The greater complexity and irregularity of sparse matrix computations make this field an area with active research as parallel sparse matrix solvers are developed and optimized for particular applications.

Our research has focused on applications from the electrical power systems community where portions of the sparse matrix are naturally in block-diagonal-bordered form and the remaining portions of the sparse matrix can be ordered to yield additional independent diagonal blocks and reduce the number of equations in the borders. These naturally block-bordered-diagonal sparse matrices occur when simulating power systems in order to examine the stability of electrical power system generators when a transient-type event occurs [1,2,6]. Given adequate computing power, electrical power system transient stability analysis could be incorporated into real-time control software for electrical power utilities, and a detailed view of an unfolding power grid failure could be provided to operators so that they could rapidly make informed, proactive decisions to minimize the extend of cascading power system network and generator failures. As a real-time grand challenge computing application, transient stability analysis would require a fast, efficient parallel sparse matrix solver.

These matrix forms remain static for extended periods of time because the sparse matrices represent actual electrical power distribution networks and electrical generators. Modifications to the electrical distribution networks are costly and represent the addition or removal of actual high voltage electrical distribution lines. Meanwhile, generators usually have high startup and shutdown costs, so they are brought on-line for periods usually lasting a minimum of several hours. Consequently, sparse matrices representing electrical power systems remain static for a sufficiently long period of time to justify the additional effort for special matrix ordering to minimize the number of fillin and the number of calculations and also justify the additional effort for balancing the number of calculations on individual processors in order to ensure efficient use of parallel computing resources.

The work presented in this paper could significantly improve performance of electrical power systems transient stability analysis and also electrical power systems load flow analysis. Moreover, we believe that this research also has utility beyond this application and the techniques developed in this work can readily be extended to develop efficient parallel direct solvers for other irregular, hierarchical sparse matrices encountered in other scientific and engineering applications.





next up previous
Next: Parallel Direct Block-Diagonal-Bordered Up: Parallel LU Factorization of Previous: Parallel LU Factorization of



David P. Koester
Sun Oct 22 16:27:33 EDT 1995