Abstract
High order methods for the solution of PDEs expose a trade-off between computational cost and accuracy on a per degree of freedom basis. In many cases, the cost increases due to higher arithmetic intensity while affecting data movement minimally. As architectures tend towards wider vector instructions and expect higher arithmetic intensities, the best order for a particular simulation may change.
This study highlights preferred orders by identifying the high order efficiency frontier of the spectral element method implemented in Nek5000 and NekBox: the set of orders and meshes that minimize computational cost at fixed accuracy. First, we extract Nek’s order-dependent computational kernels and demonstrate exceptional hardware utilization by hardware-aware implementations. Then, we perform production-scale calculations of the nonlinear single mode Rayleigh-Taylor instability on BlueGene/Q and Cray XC40-based supercomputers to highlight the influence of the architecture. Accuracy is defined with respect to physical observables, and computational costs are measured by the core-hour charge of the entire application. The total number of grid points needed to achieve a given accuracy is reduced by increasing the polynomial order. On the XC40 and BlueGene/Q, polynomial orders as high as 31 and 15 come at no marginal cost per timestep, respectively. Taken together, these observations lead to a strong preference for high order discretizations that use fewer degrees of freedom. From a performance point of view, we demonstrate up to 60 % full application bandwidth utilization at scale and achieve \(\approx \)1 PFlop/s of compute performance in Nek’s most flop-intense methods.
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References
LIBXSMM v1.0.2 (2015)
NekBox v2.0.0 (2015)
Bell, J.B., et al.: Direct numerical simulations of type Ia supernovae flames. II. The Rayleigh-Taylor instability. Astrophys. J. 608(2), 883–906 (2004)
Breuer, A., Heinecke, A., Rannabauer, L., Bader, M.: High-order ADER-DG minimizes energy- and time-to-solution of seissol. In: Kunkel, J.M., Ludwig, T. (eds.) ISC High Performance 2015. LNCS, vol. 9137, pp. 340–357. Springer, Heidelberg (2015)
Dimonte, G., Youngs, D.L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M.J., Ramaprabhu, P., Calder, A.C., Fryxell, B., Biello, J., Dursi, L., MacNeice, P., Olson, K., Ricker, P., Rosner, R., Timmes, F., Tufo, H., Young, Y.-N., Zingale, M.: A comparative study of the turbulent RayleighTaylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration. Phys. Fluids 16(5), 1668 (2004)
Gocharov, V., et al.: Panel 3 report: implosion hydrodynamics. LLNL report LLNLTR-562104, pp. 22–24 (2012)
Goto, K., et al.: Anatomy of high-performance matrix multiplication. ACM Trans. Math. Softw. 34(3), 12:1–12:25 (2008)
Heinecke, A., et al.: LIBXSMM: a high performance library for small matrix multiplications. In: Poster and Extended Abstract Presented at SC 2015 (2015)
Hutchinson, M.: Direct numerical simulation of single mode three-dimensional Rayleigh-Taylor experiments (2015). arXiv:1511.07254
Intel Corporation: Intel MKL 11.3 Release Notes. Introduced (S/D)GEMM_BATCH and (C/Z)GEMM3M_BATCH functions to perform multiple independent matrix-matrix multiply operations (2015)
Ivanov, I., et al.: Evaluation of parallel communication models in Nekbone, a Nek5000 mini-application. In: 2015 IEEE International Conference on Cluster Computing (CLUSTER), pp. 760–767. IEEE (2015)
Linden, P.F.: On the structure of salt fingers. Deep Sea Res. Oceanogr. Abstr. 20, 325–340 (1973)
Lottes, J.W., et al.: Hybrid multigrid/Schwarz algorithms for the spectral element method. J. Sci. Comput. 24(1), 45–78 (2005)
Markidis, S., et al.: OpenACC acceleration of the Nek5000 spectral element code. Int. J. High Perform. Comput. Appl. 29(3), 311–319 (2015)
McCalpin, J.D.: STREAM: sustainable memory bandwidth in high performance computers. Technical report, University of Virginia, Charlottesville, Virginia, 1991–2007. A continually updated technical report. http://www.cs.virginia.edu/stream/
Offermans, N., Marin, O., Schanen, M., Gong, J., Fischer, P., Schlatter, P., Obabko, A., Peplinski, A., Hutchinson, M., Merzari, E.: On the strong scaling of the spectral element solver Nek5000 on petascale systems. In: Solving Software Challenges for Exascale, pp. 57–68. Springer (2016)
Otten, M., et al.: An MPI/OpenACC implementation of a high-order electromagnetics solver with GPUDirect communication. Int. J. High Perform. Comput. Appl. (2016). http://hpc.sagepub.com/content/early/2016/02/01/1094342015626584.abstract. doi:10.1177/1094342015626584
Patera, A.T.: A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phy. 54(3), 468–488 (1984)
Shin, J., et al.: Speeding up Nek5000 with autotuning and specialization. In: Proceedings of the 24th ACM International Conference on Supercomputing, ICS 2010, pp. 253–262. ACM, New York (2010)
Tufo, H.M., et al.: Terascale spectral element algorithms and implementations. In: Proceedings of the 1999 ACM/IEEE Conference on Supercomputing, p. 68 (1999)
Wang, Z.J., et al.: High-order CFD methods: current status and perspective. Int. J. Numer. Meth. Fluids 72(8), 811–845 (2013)
Acknowledgment
This research used the resources of the Supercomputing Laboratory at the King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. We acknowledge useful conversations with Paul Fischer, James Lottes, Aleksandr Obabko, Oana Marin, Michel Schanen, Scott Parker, Vitali Morozov, Matthew Otten, and Robert Rosner.
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Hutchinson, M., Heinecke, A., Pabst, H., Henry, G., Parsani, M., Keyes, D. (2016). Efficiency of High Order Spectral Element Methods on Petascale Architectures. In: Kunkel, J., Balaji, P., Dongarra, J. (eds) High Performance Computing. ISC High Performance 2016. Lecture Notes in Computer Science(), vol 9697. Springer, Cham. https://doi.org/10.1007/978-3-319-41321-1_23
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DOI: https://doi.org/10.1007/978-3-319-41321-1_23
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