Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software

© ACM, 2021. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM Transactions on Mathematical Software, Volume 47, Issue , June 2021, http://doi.acm.org/10.1145/3441850[EN] The use of mixed precision in numerical algorithms is a promising strategy for accelerating scientific applications. In particular, the adoption of specialized hardware and data formats for low-precision arithmetic in high-end GPUs (graphics processing units) has motivated numerous efforts aiming at carefully reducing the working precision in order to speed up the computations. For algorithms whose performance is bound by the memory bandwidth, the idea of compressing its data before (and after) memory accesses has received considerable attention. One idea is to store an approximate operator-like a preconditioner-in lower than working precision hopefully without impacting the algorithm output. We realize the first high-performance implementation of an adaptive precision block-Jacobi preconditioner which selects the precision format used to store the preconditioner data on-the-fly, taking into account the numerical properties of the individual preconditioner blocks. We implement the adaptive block-Jacobi preconditioner as production-ready functionality in the Ginkgo linear algebra library, considering not only the precision formats that are part of the IEEE standard, but also customized formats which optimize the length of the exponent and significand to the characteristics of the preconditioner blocks. Experiments run on a state-of-the-art GPU accelerator show that our implementation offers attractive runtime savings.H. Anzt and T. Cojean were supported by the "Impuls und Vernetzungsfond of the Helmholtz Association" under grant VH-NG-1241. G. Flegar and E. S. Quintana-Orti were supported by project TIN2017-82972-R of the MINECO and FEDER and the H2020 EU FETHPC Project 732631 "OPRECOMP". This research was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration. The authors want to acknowledge the access to the Piz Daint supercomputer at the Swiss National Supercomputing Centre (CSCS) granted under the project #d100 and the Summit supercomputer at the Oak Ridge National Lab (ORNL).Flegar, G.; Anzt, H.; Cojean, T.; Quintana-Ortí, ES. (2021). Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software. ACM Transactions on Mathematical Software. 47(2):1-28. https://doi.org/10.1145/344185012847