Advertisement

Randomized model order reduction

  • Alessandro AllaEmail author
  • J. Nathan Kutz
Article

Abstract

The singular value decomposition (SVD) has a crucial role in model order reduction. It is often utilized in the offline stage to compute basis functions that project the high-dimensional nonlinear problem into a low-dimensional model which is then evaluated cheaply. It constitutes a building block for many techniques such as the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The aim of this work is to provide an efficient computation of low-rank POD and/or DMD modes via randomized matrix decompositions. This is possible due to the randomized singular value decomposition (rSVD) which is a fast and accurate alternative of the SVD. Although this is considered an offline stage, this computation may be extremely expensive; therefore, the use of compressed techniques drastically reduce its cost. Numerical examples show the effectiveness of the method for both POD and DMD.

Keywords

Nonlinear dynamical systems Proper orthogonal decomposition Dynamic mode decomposition Randomized linear algebra 

Mathematics Subject Classification (2010)

65L02 65M02 37M05 62H25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

This study is supported by the Department of Energy (grant no. DE-SC0009324) and the U.S. Air Force Office of Scientific Research (FA9550-15-1-0385).

References

  1. 1.
    Alla, A., Nathan Kutz, J.: Nonlinear model reduction via dynamic mode decomposition. SIAM J. Sci. Comput. 39, B778–B796 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations Comptes Rendus Mathematique, 339, pp. 667–672 (2004)Google Scholar
  3. 3.
    Benner, P., Gugercin, S., Willcox, K.: A survey of Projection-Based model reduction methods for parametric dynamical systems. SIAM Rev. 57, 483–531 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brunton, S.L., Proctor, J.L., Kutz, J.N.: Compressive sampling and dynamic mode decomposition. J. Comp. Dyn. 2, 165–191 (2015)zbMATHGoogle Scholar
  5. 5.
    Chatarantabut, S., Sorensen, D.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Drineas, P., Mahoney, M.W.: RandNLA: randomized numerical linear algebra. Communications of the ACM 59.6, 80–90 (2016)CrossRefGoogle Scholar
  7. 7.
    Drmac, Z., Gugercin, S.: A new selection operator for the discrete empirical interpolation method - improved a priori error bound and extensions. SIAM J. S.i. Comput. 38, A631–A648 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duersch, J., Gu, M. (2015)Google Scholar
  9. 9.
    Erichson, N.B., Voronin, S., Brunton, S.L., Kutz, J.N.: Randomized matrix decompositions using R, arXiv:1608.02148 (2016)
  10. 10.
    Everson, R., Sirovich, L.: Karhunen-loéve procedure for gappy data. J. Opt. Soc. Am. A 12, 1657–1664 (1995)CrossRefGoogle Scholar
  11. 11.
    Frieze, A., Ravi, K., Vempala, S.: Fast Monte-Carlo algorithms for finding low-rank approximations. Journal of the ACM (JACM) 51.6, 1025–1041 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gavish, M., Donoho, D.L.: The optimal hard threshold for singular values is \(4/\sqrt {3}\). IEEE Trans Inform. Theory 60, 5040–5053 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Halko, N., Martinsson, P.-G., Tropp, J.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53, 217–288 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Isaac, T., Petra, N., Stadler, G., Ghattas, O.: Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet. J. Comp. Phys. 296, 348–368 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koopman, B.O.: Hamiltonian systems and transformation in hilbert space. PNAS 17, 315–318 (1931)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kutz, J.N., Brunton, S., Brunton, B., Proctor, J.: Dynamic mode decomposition: Data-driven modeling of complex systems. SIAM Press (2016)Google Scholar
  17. 17.
    Liberty, E., Woolfe, F., Martinsson, P.-G., Rokhlin, V.: Randomized algorithms for the low-rank approximation of matrices. Proc. Natl. Acad. Sci. 104, 20167–20172 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mahoney, M.W.: Randomized algorithms for matrices and data. Found. Trends Mach. Learn. 3.2, 123–224 (2011)zbMATHGoogle Scholar
  19. 19.
    Martinsson, P.-G.: factorizations, blocked rank-revealing QR: how randomized sampling can be used to avoid single-vector pivoting. arXiv:1505.08115 (2015)
  20. 20.
    Martinsson, P.-G., Rokhlin, V., Tygert, M.: A randomized algorithm for the decomposition of matrices. Appl. Comput. Harmon. Anal. 30, 47–68 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Martinsson, P.-G.: Randomized methods for matrix computations and analysis of high dimensional data, arXiv:1607.01649 (2016)
  22. 22.
    Martinsson, P.-G., Quintana-Orti, G., Heavner, N.: randUTV: A blocked randomized algorithm for computing a rank-revealing UTV factorization, arXiv:1703.00998 (2017)
  23. 23.
    Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D: Nonlinear Phenomena 197, 101–133 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41, 309–325 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I-II Q. Appl. Math. XVL, 561–590 (1987)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Szlam, A., Kluger, Y., Tygert, M.: An implementation of a randomized algorithm for principal component analysis, arXiv:1412.3510(2014)
  28. 28.
    Tu, J., Rowley, C., Luchtenberg, D., Brunton, S., Kutz, J.N.: On dynamic mode decomposition theory and applications. J. Comput. Dyn. 1, 391–421 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Volkwein, S.: Model Reduction Using Proper Orthogonal Decomposition. Lecture Notes, University of Konstanz (2013)Google Scholar
  30. 30.
    Voronin, S., Martinsson, P.-G.: RSVDPACK: Subroutines for computing partial singular value decompositions via randomized sampling on single core, multi core, and GPU architectures, arXiv:1502.05366 (2015)
  31. 31.
    Woolfe, F., Liberty, E., Rokhlin, V., Tygert, M.: A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25, 335–366 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zahm, O., Nouy, A.: Interpolation of inverse operators for precoditioning parameter-dependent equations. SIAM J. Sci. Comput. 38, 1004–1074 (2016)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPUC-RioRio de JaneiroBrazil
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations