On the Parallel I/O Optimality of Linear Algebra Kernels: Near-Optimal Matrix Factorizations

Authors: Grzegorz Kwasniewski (ETH Zürich); Marko Kabić (Swiss National Supercomputing Centre (CSCS), ETH Zürich); Tal Ben-Nun, Alexandros Nikolaos Ziogas, Jens Eirik Saethre, André Gaillard, Timo Schneider, and Maciej Besta (ETH Zürich); Anton Kozhevnikov and Joost VandeVondele (Swiss National Supercomputing Centre (CSCS), ETH Zürich); and Torsten Hoefler (ETH Zürich)

Abstract: Matrix factorizations are among the most important building blocks of scientific computing. State-of-the-art libraries, however, 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 N^3/(P*sqrt(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 2048 to 262,144 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.




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