Space and Time Efficient Implementations of a Parallel Direct Solver using Nested Dissection

This paper is concerned with algorithms for the efficient parallel solution of sparse, symmetric n × n linear systems via direct factorization, which eliminate groups of variables in stages. The algorithms make use of nested dissection to recursively cut data into pieces, producing orderings for direct elimination of the variables, so to reduce the storage and arithmetic requirements. Parallel Direct Solve (PDS) algorithms, which perform this variable elimination in parallel using nested dissection orderings, have been developed for various parallel machine models, including the parallel random access machine (PRAM) model and for grid architectures. In this paper, we describe the following improvements to the known DPS algorithms: 1. FAST DPS: a fast DPS algorithm for the PRAM model, which reduces the parallel time bound by a factor of O(log n), without significantly increasing the processor bound. 2. COMPACT DPS: a DPS algorithm for a mesh-connected processor array, applicable to matrices representable by a grid graph, that uses O(n) processors, takes O( √ n) parallel time, and reduces the space bounds to linear, without significant increase in time or processor bounds. 3. GENERALIZED COMPACT DPS: a DPS algorithm for a mesh-connected processor array, that uses O(n) processors, takes O( √ n) parallel time, and reduces the space bounds to linear, for the more general case of ∗A preliminary version of this paper appeared as Deganit Armon and John H. Reif, Space and time efficient implementations of parallel nested dissection, 4th Annual ACM Symposium on Parallel Algorithms and Architectures, San Diego, CA, July 1992. †Afeka Tel Aviv Academic College of Engineering ‡Department of Computer Science, Duke University, Durham, NC 27708, USA and Adjunct, Faculty of Computing and Information Technology (FCIT), King Abdulaziz University (KAU), Jeddah, Saudi Arabia. Email: [email protected]. matrices representable by a graph of constant degree and separator size √ n (as required for many 2D PDE applications).