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Fast Matrix-Free Discontinuous Galerkin Kernels on Modern Computer Architectures

  • Martin KronbichlerEmail author
  • Katharina Kormann
  • Igor Pasichnyk
  • Momme Allalen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10266)

Abstract

This study compares the performance of high-order discontinuous Galerkin finite elements on modern hardware. The main computational kernel is the matrix-free evaluation of differential operators by sum factorization, exemplified on the symmetric interior penalty discretization of the Laplacian as a metric for a complex application code in fluid dynamics. State-of-the-art implementations of these kernels stress both arithmetics and memory transfer. The implementations of SIMD vectorization and shared-memory parallelization are detailed. Computational results are presented for dual-socket Intel Haswell CPUs at 28 cores, a 64-core Intel Knights Landing, and a 16-core IBM Power8 processor. Up to polynomial degree six, Knights Landing is approximately twice as fast as Haswell. Power8 performs similarly to Haswell, trading a higher frequency for narrower SIMD units. The performance comparison shows that simple ways to express parallelism through for loops perform better on medium and high core counts than a more elaborate task-based parallelization with dynamic scheduling according to dependency graphs, despite less memory transfer in the latter algorithm.

Keywords

Discontinuous Galerkin Quadrature Point Spectral Element Method Memory Transfer Pressure Poisson Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors acknowledge the support given by the Bayerische Kompetenznetzwerk für Technisch-Wissenschaftliches Hoch- und Höchstleistungsrechnen (KONWIHR) in the framework of the project High performance finite difference stencils for modern parallel processors. This work was supported by the German Research Foundation (DFG) under the project High-order discontinuous Galerkin for the exa-scale (ExaDG) within the priority program Software for Exascale Computing (SPPEXA). The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de) through project id pr83te.

The authors acknowledge collaboration with Benjamin Krank, Niklas Fehn, and Matthias Brehm.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute for Computational MechanicsTechnical University of MunichGarchingGermany
  2. 2.Max–Planck–Institute for Plasma PhysicsGarchingGermany
  3. 3.Zentrum MathematikTechnical University of MunichGarchingGermany
  4. 4.IBM DeutschlandGarchingGermany
  5. 5.Leibniz-Rechenzentrum der Bayerischen Akademie der WissenschaftenGarchingGermany

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