On the parallel I/O optimality of linear algebra kernels: Near-optimal matrix factorizations

Abstract

Matrix factorizations are among the most important building blocks of scientific computing. However, state-of-The-Art libraries are not communication-optimal, underutilizing current parallel architectures. We present novel algorithms for Cholesky and LU factorizations that utilize an asymptotically communication-optimal 2.5D decomposition. We first establish a theoretical framework for deriving parallel I/O lower bounds for linear algebra kernels, and then utilize its insights to derive Cholesky and LU schedules, both communicating N3/(P √ M) elements per processor, where M is the local memory size. The empirical results match our theoretical analysis: our implementations communicate significantly less than Intel MKL, SLATE, and the asymptotically communication-optimal CANDMC and CAPITAL libraries. Our code outperforms these state-of-The-Art libraries in almost all tested scenarios, with matrix sizes ranging from 2,048 to 524,288 on up to 512 CPU nodes of the Piz Daint supercomputer, decreasing the time-To-solution by up to three times. Our code is ScaLAPACK-compatible and available as an open-source library.