Mixed precision low-rank approximations and their application to block low-rank LU factorization

Abstract

We introduce a novel approach to exploit mixed precision arithmetic for low-rank approximations. Our approach is based on the observation that singular vectors associated with small singular values can be stored in lower precisions while preserving high accuracy overall. We provide an explicit criterion to determine which level of precision is needed for each singular vector. We apply this approach to block low-rank (BLR) matrices, most of whose off-diagonal blocks have low rank. We propose a new BLR LU factorization algorithm that exploits the mixed precision representation of the blocks. We carry out the rounding error analysis of this algorithm and prove that the use of mixed precision arithmetic does not compromise the numerical stability of the BLR LU factorization. Moreover, our analysis determines which level of precision is needed for each floating-point operation (flop), and therefore guides us toward an implementation that is both robust and efficient. We evaluate the potential of this new algorithm on a range of matrices coming from real-life problems in industrial and academic applications. We show that a large fraction of the entries in the LU factors and flops to perform the BLR LU factorization can be safely switched to lower precisions, leading to significant reductions of the storage and expected time costs, of up to a factor three using fp64, fp32, and bfloat16 arithmetics.

Communication Avoiding Block Low-Rank Parallel Multifrontal Triangular Solve with Many Right-Hand Sides

“…However, we mention as an important perspective of this work that the new hybrid variants may be even more successful in a mixed precision context. Indeed, in recent work, we have shown that the BLR LU factors can be stored in mixed precision while preserving the same accuracy [2]. As a result, the mixed precision BLR LU factors are further compressed and can reduce the triangular solve time.…”

“…This storage reduction is translated into a comparable RL forward solve time reduction with a single RHS (1.3× speedup), but not with multiple ones (only 1.1× speedup). In future work it could therefore be promising to combine the use of mixed precision proposed in [2] and the new hybrid variants proposed in this article.…”