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Implementation of Implicitly Restarted Arnoldi Method on MultiGPU Architecture with Application to Fluid Dynamics Problems

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Parallel Computational Technologies (PCT 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 753))

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Abstract

The parallel CPU+multiGPU implementation of the Implicitly Restarted Arnoldi method (IRA) is presented in the paper. We focus on the problem of implementing an efficient method for large scale non-symmetric eigenvalue problems arising in linear stability and Floquet theory analysis in fluid dynamics problems. We give brief details about the problem in both cases. We use and cross-compare different methods to implement QR shifts with polynomial filtering. Then we conduct a benchmark on standard non-symmetric matrices. The presented results give insight on the best choice of the method of polynomial filters. It turns out that a polynomial filter is essential to accelerate computations in fluid dynamics stability problems as well as to increase the Krylov space dimension. However, some knowledge about the spectral radius of the problem is required. Further, we investigate the parallel efficiency of the method for single-GPU and multi-GPU modes using the problems previously considered. It is shown that the implementation of Givens rotations on one GPU for QR computations (of a Hessenberg matrix) is essential in order to achieve high speed for a considerable Krylov subspace dimension. In the final part we present some results regarding the application of the IRA polynomial filtered method to linear stability and Floquet analysis for large fluid dynamics problem with parallel efficiency comparison on multi-GPU architecture.

The work is supported by the Russian Foundation for Basic Research (grant RFBR 14-07-00123).

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References

  1. Saad, Y.: Numerical Methods for Large Eigenvalue Problems, Revised Edition (Classics in Applied Mathematics). SIAM, Philadelphia (2011). doi:10.1137/1.9781611970739

    Book  Google Scholar 

  2. Lehoucq, R., Maschhoff, K.: Sorensen, D., Yang, C., ARPACK software, 1996–2016. http://www.caam.rice.edu/software/ARPACK

  3. Sorensen, D.C.: Implicitly Restarted Arnoldi/Lanczos Methods For Large Scale Eigenvalue Calculations. NASA Contractor Report 198342 (1996). doi:10.1007/978-94-011-5412-3_5

  4. Govaerts, W.J.F.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000). doi:10.1137/1.9780898719543

    Book  MATH  Google Scholar 

  5. Ericsson, T., Ruhe, A.: The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput. 35, 1251–1268 (1980). doi:10.2307/2006390

    MATH  Google Scholar 

  6. Lehoucq, R.B., Scott, J.A.: Implicitly restarted Arnoldi methods and eigenvalues of the discretized Navier-Stokes equations. SIAM J. Matrix Anal. Appl. 23, 551–562 (1997). doi:10.1137/s0895479899358595

    Article  Google Scholar 

  7. Cliffe, K.A., Hall, E.J.C., Houston, P.: Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. SIAM J. Sci. Comput. 31(6), 4607–4632 (2010). doi:10.1137/080731918

    Article  MathSciNet  MATH  Google Scholar 

  8. Burroughs, E.A., Romero, L.A., Lehoucq, R.B., Salinger, A.G.: Large scale eigenvalue calculations for computing the stability of buoyancy driven flows. Sandia Report SAND2001-0113 Unlimited Release (2001). doi:10.2172/782594

  9. Henningson, D.S., Akervik, E.: The use of global modes to understand transition and perform flow control. Phys. Fluids 20, 031302 (2008). doi:10.1063/1.2832773

    Article  MATH  Google Scholar 

  10. Garnaud, X., Lesshafft, L., Schmid, P., Chomaz, J.-M.: A relaxation method for large eigenvalue problems, with an application to flow stability analysis. J. Comput. Phys. 231, 3912–3927 (2012). doi:10.1016/j.jcp.2012.01.038

    Article  MathSciNet  MATH  Google Scholar 

  11. Tiesinga, G., Wubs, F.W., Veldman, A.E.P.: Bifurcation analysis of incompressible flow in a driven cavity by the Newton - Picard method. J. Comput. Appl. Math. 140(1–2), 751–772 (2002). doi:10.1016/s0377-0427(01)00515-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Barkley, D., Henderson, R.D.: Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241 (1996). doi:10.1017/s0022112096002777

    Article  MATH  Google Scholar 

  13. Bres, G.A., Colonius, T.: Three-dimensional linear stability analysis of cavity flows. In: AIAA 2007–1126 45th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 January 2007, Reno, Nevada (2007). doi:10.2514/6.2007-1126

  14. Nebauer, J.R.A., Blackburn, H.M.: Floquet stability of time periodic pipe flow. In: Proceedings of International Conference on CFD in the Minerals and Process Industries, CSIRO, Melbourne, Australia, 10–12 December 2012

    Google Scholar 

  15. Baranyi, L., Darczy, L.: Floquet stability analysis of the wake of a circular cylinder in low Reynolds number flow. In: Proceedings of XXVI Conference in MicroCAD, University of Miskolc, Hungary (2012)

    Google Scholar 

  16. Dubois, J., Calvin, C., Petiton, S.G.: Accelerating The explicitly restarted arnoldi method with GPUs using an autotuned matrix vector product. SIAM J. Sci. Comput. 33(5), 3010–3019 (2011). doi:10.1137/10079906x

    Article  MathSciNet  MATH  Google Scholar 

  17. Rantalaiho, T., Weir, W., Suorsa, J.: Parallel multi-CPU/GPU(CUDA)-implementation of the Implicitly Restarted Arnoldi Method. https://github.com/trantalaiho/Cuda-Arnoldi

  18. Bujanovic’, Z.: Krylov Type Methods for Large Scale Eigenvalue Computations. Ph.D. thesis, Zagreb (2011)

    Google Scholar 

  19. Embree, M.: The Arnoldi eigenvalue iteration with exact shifts can fail. SIAM J. Matrix Anal. Appl. 31(1), 110 (2009). doi:10.1137/060669097

    Article  MathSciNet  MATH  Google Scholar 

  20. Breue, A.: New filtering strategies for implicitly restarted Lanczos iteration. Electron. Trans. Numer. Anal. 45, 1632 (2016)

    MathSciNet  Google Scholar 

  21. Fitzgibbon, A.W., Pilum, M., Fisher, R.B.: Direct least-squares fitting of ellipses. IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1996). doi:10.1109/34.765658

    Article  Google Scholar 

  22. Bos, L., De Marchi, S., Sommariva, A., Vianello, M.: Computing multivariate Fekete and Leja points by numerical linear algebra. SIAM J. Numer. Anal. 48(5), 1984–1999 (2010). doi:10.1137/090779024

    Article  MathSciNet  MATH  Google Scholar 

  23. Akervik, E., Brandt, L., Henningson, D.S., Hapffner, J., Marxen, O., Schlatter, P.: Steady solutions of the Navier-Stokes equations by selective frequency damping. Phys. Fluids 18, 068102 (2006). doi:10.1063/1.2211705

    Article  Google Scholar 

  24. SuiteSparse Matrix Collection. http://www.cise.ufl.edu/research/sparse/matrices/

  25. Evstigneev, N.M., Magnitskii, N.A., Silaev, D.A.: Qualitative analysis of dynamics in Kolmogorovs problem on a flow of a viscous incompressible fluid. Differ. Equ. 51(10), 1292–1305 (2015). doi:10.1134/s0012266115100055

    Article  MathSciNet  MATH  Google Scholar 

  26. Evstigneev, N.M.: Laminar-turbulent bifurcation scenario in 3D Rayleigh-Benard convection problem. Open J. Fluid Dyn. 6, 496–539 (2016). doi:10.4236/ojfd.2016.64035

    Article  Google Scholar 

  27. Evstigneev, N.M., Magnitskii, N.A.: Nonlinear dynamics of laminar-turbulent transition in the generalized 3D Kolmogorov problem for the incompressible viscous fluid at symmetric solution subset. J. Appl. Nonlinear Dyn. (2017, accepted, to appear)

    Google Scholar 

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Authors and Affiliations

  1. Federal Research Center “Informatics and Control”, Institute for System Analysis, Russian Academy of Science, Moscow, Russia

    Nikolay M. Evstigneev

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  1. Nikolay M. Evstigneev
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Correspondence to Nikolay M. Evstigneev .

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Editors and Affiliations

  1. South Ural State University, Chelyabinsk, Russia

    Leonid Sokolinsky

  2. South Ural State University, Chelyabinsk, Russia

    Mikhail Zymbler

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Evstigneev, N.M. (2017). Implementation of Implicitly Restarted Arnoldi Method on MultiGPU Architecture with Application to Fluid Dynamics Problems. In: Sokolinsky, L., Zymbler, M. (eds) Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol 753. Springer, Cham. https://doi.org/10.1007/978-3-319-67035-5_22

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  • DOI: https://doi.org/10.1007/978-3-319-67035-5_22

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67034-8

  • Online ISBN: 978-3-319-67035-5

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