On the use of low-rank arithmetic to reduce the complexity of parallel sparse linear solvers based on direct factorization techniques

Grégoire Pichon (PhD student in HiePACS team, Inria Bordeaux - Sud-Ouest)

Frank Adams Room 1,

In the first part of this talk, we present two approaches using a Block Low-Rank (BLR) compression technique to reduce the memory footprint and/or the time-to-solution of a supernodal sparse direct solver. This flat, non-hierarchical, compression method allows to take advantage of the low-rank property of the blocks appearing during the factorization of sparse linear systems. The proposed solver can be used either as a direct solver at a lower precision or as a very robust preconditioner. The first approach, called Minimal Memory, illustrates the maximum memory gain that can be obtained with the BLR compression method, while the second approach, called Just-In-Time, mainly focuses on reducing the computational complexity and thus the time-to-solution. In the second part, we present a reordering strategy that increases the block granularity to better take advantage of the locality for multicores and provide larger tasks to GPUs. This strategy relies on the block-symbolic factorization to refine the ordering produced by tools such as Metis or Scotch, but it does not impact the number of operations required to solve the problem. From this approach, we propose in the third part of this talk a new low-rank clustering technique that is designed to cluster unknowns within a separator to obtain the BLR partition, and demonstrate its assets with respect to widely used clustering strategies.

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