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

  • Nikolay M. EvstigneevEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 753)


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.


Arnoldi method Eigenvalue solvers Matrix free methods Krylov methods Multi GPU Fluid dynamics Linear stability analysis Floquet theory analysis 


  1. 1.
    Saad, Y.: Numerical Methods for Large Eigenvalue Problems, Revised Edition (Classics in Applied Mathematics). SIAM, Philadelphia (2011). doi: 10.1137/1.9781611970739 CrossRefGoogle Scholar
  2. 2.
    Lehoucq, R., Maschhoff, K.: Sorensen, D., Yang, C., ARPACK software, 1996–2016.
  3. 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. 4.
    Govaerts, W.J.F.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000). doi: 10.1137/1.9780898719543 CrossRefzbMATHGoogle Scholar
  5. 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 zbMATHGoogle Scholar
  6. 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 CrossRefGoogle Scholar
  7. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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. 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 CrossRefzbMATHGoogle Scholar
  10. 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/ MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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 CrossRefzbMATHGoogle Scholar
  13. 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. 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 2012Google Scholar
  15. 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. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rantalaiho, T., Weir, W., Suorsa, J.: Parallel multi-CPU/GPU(CUDA)-implementation of the Implicitly Restarted Arnoldi Method.
  18. 18.
    Bujanovic’, Z.: Krylov Type Methods for Large Scale Eigenvalue Computations. Ph.D. thesis, Zagreb (2011)Google Scholar
  19. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Breue, A.: New filtering strategies for implicitly restarted Lanczos iteration. Electron. Trans. Numer. Anal. 45, 1632 (2016)MathSciNetGoogle Scholar
  21. 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 CrossRefGoogle Scholar
  22. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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 CrossRefGoogle Scholar
  24. 24.
  25. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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 CrossRefGoogle Scholar
  27. 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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

Personalised recommendations