Compressed threshold pivoting for sparse symmetric indefinite systems

TL;DR

This paper introduces two new pivoting strategies for sparse symmetric indefinite matrix factorization that significantly reduce communication costs and improve parallel efficiency while maintaining numerical stability.

Contribution

The paper proposes compressed pivoting methods that decrease communication complexity from O(p log P) to O(log P) in parallel sparse matrix factorizations, with proven stability.

Findings

Achieves O(log P) message complexity in parallel pivoting

Maintains numerical stability with theoretical proof

Demonstrates practical speedup on multicore systems

Abstract

A key technique for controlling numerical stability in sparse direct solvers is threshold partial pivoting. When selecting a pivot, the entire candidate pivot column below the diagonal must be up-to-date and must be scanned. If the factorization is parallelized across a large number of cores, communication latencies can be the dominant computational cost. In this paper, we propose two alternative pivoting strategies for sparse symmetric indefinite matrices that significantly reduce communication by compressing the necessary data into a small matrix that can be used to select pivots. Once pivots have been chosen, they can be applied in a communication-efficient fashion. For an n x p submatrix on P processors, we show our methods perform a factorization using O(log P) messages instead of the O(p log P) for threshold partial pivoting. The additional costs in terms of operations and communication bandwidth are relatively small. A stability proof is given and numerical results using a range of symmetric indefinite matrices arising from practical problems are used to demonstrate the practical robustness. Timing results on large random examples illustrate the potential speedup on current multicore machines.