Study of Algorithms and Applications of Approximate Algebra

課題番号：1930000

Mixed-precision AMG as linear equation solver for definite systems

AbstractThe performance of algebraic multigrid (AMG) algorithms, implemented in 4-byte floating point arithmetic, is investigated on modern cluster architecture with multi-core CPUs. The algorithmic considerations comprise the effect of preconditioning in 4-byte floating point arithmetic on Krylov solvers using standard 8-byte floating point arithmetic. The data of basic linear algebra benchmarks are used to interpret the performance of AMG algorithms employed as linear solvers in computational fluid dynamics simulation tools

Toward a modular precision ecosystem for high performance computing

[EN] With the memory bandwidth of current computer architectures being significantly slower than the (floating point) arithmetic performance, many scientific computations only leverage a fraction of the computational power in today's high-performance architectures. At the same time, memory operations are the primary energy consumer of modern architectures, heavily impacting the resource cost of large-scale applications and the battery life of mobile devices. This article tackles this mismatch between floating point arithmetic throughput and memory bandwidth by advocating a disruptive paradigm change with respect to how data are stored and processed in scientific applications. Concretely, the goal is to radically decouple the data storage format from the processing format and, ultimately, design a "modular precision ecosystem" that allows for more flexibility in terms of customized data access. For memory-bounded scientific applications, dynamically adapting the memory precision to the numerical requirements allows for attractive resource savings. In this article, we demonstrate the potential of employing a modular precision ecosystem for the block-Jacobi preconditioner and the PageRank algorithm-two applications that are popular in the communities and at the same characteristic representatives for the field of numerical linear algebra and data analytics, respectively.The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Impuls und Vernetzungsfond of the Helmholtz Association under grant VH-NG-1241. G Flegar and ES Quintana-Ortí were supported by project TIN2017-82972-R of the MINECO and FEDER and the H2020 EU FETHPC Project 732631 OPRECOMP .Anzt, H.; Flegar, G.; Gruetzmacher, T.; Quintana-Orti, ES. (2019). Toward a modular precision ecosystem for high performance computing. International Journal of High Performance Computing Applications. 33(6):1069-1078. https://doi.org/10.1177/109434201984654710691078336Anzt, H., Dongarra, J., & Quintana-Ortí, E. S. (2015). Adaptive precision solvers for sparse linear systems. Proceedings of the 3rd International Workshop on Energy Efficient Supercomputing - E2SC ’15. doi:10.1145/2834800.2834802Baboulin, M., Buttari, A., Dongarra, J., Kurzak, J., Langou, J., Langou, J., … Tomov, S. (2009). Accelerating scientific computations with mixed precision algorithms. Computer Physics Communications, 180(12), 2526-2533. doi:10.1016/j.cpc.2008.11.005Buttari, A., Dongarra, J., Langou, J., Langou, J., Luszczek, P., & Kurzak, J. (2007). Mixed Precision Iterative Refinement Techniques for the Solution of Dense Linear Systems. The International Journal of High Performance Computing Applications, 21(4), 457-466. doi:10.1177/1094342007084026Carson, E., & Higham, N. J. (2017). A New Analysis of Iterative Refinement and Its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems. SIAM Journal on Scientific Computing, 39(6), A2834-A2856. doi:10.1137/17m1122918Carson, E., & Higham, N. J. (2018). Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions. SIAM Journal on Scientific Computing, 40(2), A817-A847. doi:10.1137/17m1140819Göddeke, D., Strzodka, R., & Turek, S. (2007). Performance and accuracy of hardware-oriented native-, emulated- and mixed-precision solvers in FEM simulations. International Journal of Parallel, Emergent and Distributed Systems, 22(4), 221-256. doi:10.1080/17445760601122076Grützmacher, T., & Anzt, H. (2018). A Modular Precision Format for Decoupling Arithmetic Format and Storage Format. Euro-Par 2018: Parallel Processing Workshops, 434-443. doi:10.1007/978-3-030-10549-5_34Grutzmacher, T., Anzt, H., Scheidegger, F., & Quintana-Orti, E. S. (2018). High-Performance GPU Implementation of PageRank with Reduced Precision Based on Mantissa Segmentation. 2018 IEEE/ACM 8th Workshop on Irregular Applications: Architectures and Algorithms (IA3). doi:10.1109/ia3.2018.00015Hegland, M., & Saylor, P. E. (1992). Block jacobi preconditioning of the conjugate gradient method on a vector processor. International Journal of Computer Mathematics, 44(1-4), 71-89. doi:10.1080/00207169208804096Horowitz, M. (2014). 1.1 Computing’s energy problem (and what we can do about it). 2014 IEEE International Solid-State Circuits Conference Digest of Technical Papers (ISSCC). doi:10.1109/isscc.2014.6757323Saad, Y. (2003). Iterative Methods for Sparse Linear Systems. doi:10.1137/1.9780898718003Strzodka, R., & Goddeke, D. (2006). Pipelined Mixed Precision Algorithms on FPGAs for Fast and Accurate PDE Solvers from Low Precision Components. 2006 14th Annual IEEE Symposium on Field-Programmable Custom Computing Machines. doi:10.1109/fccm.2006.57Tadano, H., & Sakurai, T. (2008). On Single Precision Preconditioners for Krylov Subspace Iterative Methods. Lecture Notes in Computer Science, 721-728. doi:10.1007/978-3-540-78827-0_83Wulf, W. A., & McKee, S. A. (1995). Hitting the memory wall. ACM SIGARCH Computer Architecture News, 23(1), 20-24. doi:10.1145/216585.21658