On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws

Loffeld, J.; Hittinger, J. A. F.

doi:10.1177/1094342017691876

Title: On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws

Journal Article · Wed Feb 01 00:00:00 EST 2017 · International Journal of High Performance Computing Applications

DOI:https://doi.org/10.1177/1094342017691876· OSTI ID:1860892

Loffeld, J. ^[1]; Hittinger, J. A. F. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

It has been conjectured that higher-order discretizations for partial differential equations will have advantages over the lower-order counterparts commonly used today. The reasoning is that the increase in arithmetic operations will be more than offset by the reduction in data transfers and the increase in concurrent floating-point units. To evaluate this conjecture, the arithmetic intensity of a class of high-order finite-volume discretizations for hyperbolic systems of conservation laws is theoretically analyzed for spatial discretizations from orders three through eight in arbitrary dimensions. Additionally, three cache models are considered: the limiting cases of no cache and infinite cache as well as a finite-sized cache model. Models are validated experimentally by measuring floating-point operations and data transfers on an IBM Blue Gene/Q node. Theory and experiments demonstrate that high-order finite-volume methods will be able to provide increases in arithmetic intensity that will be necessary to make better utilization of on-node floating-point capability.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1860892

Report Number(s):: LLNL-JRNL-716535; 859004

Journal Information:: International Journal of High Performance Computing Applications, Vol. 33, Issue 1; ISSN 1094-3420

Publisher:: SAGECopyright Statement

Country of Publication:: United States

Language:: English

References (35)

Improving the ratio of memory operations to floating-point operations in loops Carr, Steve; Kennedy, Ken ACM Transactions on Programming Languages and Systems, Vol. 16, Issue 6 https://doi.org/10.1145/197320.197366	journal	November 1994
A framework for hybrid parallel flow simulations with a trillion cells in complex geometries Godenschwager, Christian; Schornbaum, Florian; Bauer, Martin Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '13 https://doi.org/10.1145/2503210.2503273	conference	January 2013
11 PFLOP/s simulations of cloud cavitation collapse Rossinelli, Diego; Koumoutsakos, Petros; Hejazialhosseini, Babak Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '13 https://doi.org/10.1145/2503210.2504565	conference	January 2013
Understanding Application Performance via Micro-benchmarks on Three Large Supercomputers: Intrepid, Ranger and Jaguar Bhatelé, Abhinav; Wesolowski, Lukasz; Bohm, Eric The International Journal of High Performance Computing Applications, Vol. 24, Issue 4 https://doi.org/10.1177/1094342010370603	journal	May 2010
Estimating interlock and improving balance for pipelined architectures Callahan, David; Cocke, John; Kennedy, Ken Journal of Parallel and Distributed Computing, Vol. 5, Issue 4 https://doi.org/10.1016/0743-7315(88)90002-0	journal	August 1988
Multicore/Multi-GPU Accelerated Simulations of Multiphase Compressible Flows Using Wavelet Adapted Grids Rossinelli, Diego; Hejazialhosseini, Babak; Spampinato, Daniele G. SIAM Journal on Scientific Computing, Vol. 33, Issue 2 https://doi.org/10.1137/100795930	journal	January 2011
I/O complexity: The red-blue pebble game Jia-Wei, Hong; Kung, H. T. Proceedings of the thirteenth annual ACM symposium on Theory of computing - STOC '81 https://doi.org/10.1145/800076.802486	conference	January 1981
A Study on Balancing Parallelism, Data Locality, and Recomputation in Existing PDE Solvers Olschanowsky, Catherine; Strout, Michelle Mills; Guzik, Stephen SC14: International Conference for High Performance Computing, Networking, Storage and Analysis https://doi.org/10.1109/SC.2014.70	conference	November 2014
Performance evaluations of gyrokinetic Eulerian code GT5D on massively parallel multi-core platforms Idomura, Yasuhiro; Jolliet, Sébastien State of the Practice Reports on - SC '11 https://doi.org/10.1145/2063348.2063354	conference	January 2011
An Optimizing Code Generator for a Class of Lattice-Boltzmann Computations Pananilath, Irshad; Acharya, Aravind; Vasista, Vinay ACM Transactions on Architecture and Code Optimization, Vol. 12, Issue 2 https://doi.org/10.1145/2739047	journal	July 2015
Comparison of accurate methods for the integration of hyperbolic equations Kreiss, Heinz-Otto; Oliger, Joseph Tellus, Vol. 24, Issue 3 https://doi.org/10.3402/tellusa.v24i3.10634	journal	January 1972
Quantifying Performance Bottlenecks of Stencil Computations Using the Execution-Cache-Memory Model Stengel, Holger; Treibig, Jan; Hager, Georg Proceedings of the 29th ACM on International Conference on Supercomputing - ICS '15 https://doi.org/10.1145/2751205.2751240	conference	January 2015
Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws Shu, Chi-Wang Lecture Notes in Mathematics https://doi.org/10.1007/BFb0096355	book	January 1998
Compiler-Directed Transformation for Higher-Order Stencils Basu, Protonu; Hall, Mary; Williams, Samuel 2015 IEEE International Parallel and Distributed Processing Symposium (IPDPS) https://doi.org/10.1109/IPDPS.2015.103	conference	May 2015
Communication lower bounds and optimal algorithms for numerical linear algebra Ballard, G.; Carson, E.; Demmel, J. Acta Numerica, Vol. 23 https://doi.org/10.1017/S0962492914000038	journal	May 2014
High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids McCorquodale, P.; Dorr, M. R.; Hittinger, J. A. F. Journal of Computational Physics, Vol. 288 https://doi.org/10.1016/j.jcp.2015.01.006	journal	May 2015
On the GPU performance of cell-centered finite volume method over unstructured tetrahedral meshes Langguth, Johannes; Wu, Nan; Chai, Jun Proceedings of the 3rd Workshop on Irregular Applications Architectures and Algorithms - IA^3 '13 https://doi.org/10.1145/2535753.2535765	conference	January 2013
Weighted Essentially Non-oscillatory Schemes Liu, Xu-Dong; Osher, Stanley; Chan, Tony Journal of Computational Physics, Vol. 115, Issue 1 https://doi.org/10.1006/jcph.1994.1187	journal	November 1994
Fully multidimensional flux-corrected transport algorithms for fluids Zalesak, Steven T. Journal of Computational Physics, Vol. 31, Issue 3 https://doi.org/10.1016/0021-9991(79)90051-2	journal	June 1979
Optimization of a lattice Boltzmann computation on state-of-the-art multicore platforms Williams, Samuel; Carter, Jonathan; Oliker, Leonid Journal of Parallel and Distributed Computing, Vol. 69, Issue 9 https://doi.org/10.1016/j.jpdc.2009.04.002	journal	September 2009
Solving the compressible navier-stokes equations on up to 1.97 million cores and 4.1 trillion grid points Bermejo-Moreno, Iván; Bodart, Julien; Larsson, Johan SC13: International Conference for High Performance Computing, Networking, Storage and Analysis, Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis https://doi.org/10.1145/2503210.2503265	conference	November 2013
A 30 Year Retrospective on Dennard's MOSFET Scaling Paper Bohr, Mark IEEE Solid-State Circuits Newsletter, Vol. 12, Issue 1 https://doi.org/10.1109/N-SSC.2007.4785534	journal	January 2007
A high-order finite-volume method for conservation laws on locally refined grids McCorquodale, Peter; Colella, Phillip Communications in Applied Mathematics and Computational Science, Vol. 6, Issue 1 https://doi.org/10.2140/camcos.2011.6.1	journal	January 2011
The Piecewise Parabolic Method (PPM) for gas-dynamical simulations Colella, Phillip; Woodward, Paul R. Journal of Computational Physics, Vol. 54, Issue 1 https://doi.org/10.1016/0021-9991(84)90143-8	journal	April 1984
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
Hierarchical N-body Simulations with Autotuning for Heterogeneous Systems Yokota, Rio; Barba, Lorena Computing in Science & Engineering, Vol. 14, Issue 3 https://doi.org/10.1109/MCSE.2012.1	journal	May 2012
A performance analysis framework for identifying potential benefits in GPGPU applications Sim, Jaewoong; Dasgupta, Aniruddha; Kim, Hyesoon ACM SIGPLAN Notices, Vol. 47, Issue 8 https://doi.org/10.1145/2370036.2145819	journal	September 2012
Performance modeling of serial and parallel implementations of the fractional Adams-Bashforth-Moulton method Zhang, Wei; Wei, Wenjie; Cai, Xing Fractional Calculus and Applied Analysis, Vol. 17, Issue 3 https://doi.org/10.2478/s13540-014-0189-x	journal	June 2014
Managing application complexity in the SAMRAI object-oriented framework Hornung, Richard D.; Kohn, Scott R. Concurrency and Computation: Practice and Experience, Vol. 14, Issue 5 https://doi.org/10.1002/cpe.652	journal	January 2002
Efficient Implementation of Weighted ENO Schemes Jiang, Guang-Shan; Shu, Chi-Wang Journal of Computational Physics, Vol. 126, Issue 1 https://doi.org/10.1006/jcph.1996.0130	journal	June 1996
High-order, finite-volume methods in mapped coordinates Colella, P.; Dorr, M. R.; Hittinger, J. A. F. Journal of Computational Physics, Vol. 230, Issue 8 https://doi.org/10.1016/j.jcp.2010.12.044	journal	April 2011
High throughput software for direct numerical simulations of compressible two-phase flows Hejazialhosseini, Babak; Rossinelli, Diego; Conti, Christian 2012 SC - International Conference for High Performance Computing, Networking, Storage and Analysis, 2012 International Conference for High Performance Computing, Networking, Storage and Analysis https://doi.org/10.1109/SC.2012.66	conference	November 2012
Roofline: an insightful visual performance model for multicore architectures Williams, Samuel; Waterman, Andrew; Patterson, David Communications of the ACM, Vol. 52, Issue 4 https://doi.org/10.1145/1498765.1498785	journal	April 2009
An efficient mixed-precision, hybrid CPU–GPU implementation of a nonlinearly implicit one-dimensional particle-in-cell algorithm Chen, G.; Chacón, L.; Barnes, D. C. Journal of Computational Physics, Vol. 231, Issue 16 https://doi.org/10.1016/j.jcp.2012.04.040	journal	June 2012
Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works Boris, Jay P.; Book, David L. Journal of Computational Physics, Vol. 11, Issue 1 https://doi.org/10.1016/0021-9991(73)90147-2	journal	January 1973

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97 MATHEMATICS AND COMPUTING
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algorithmic balance

Title: On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws

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