Proper Orthogonal Decomposition In Decoupling Large Dynamical Systems

  • Toan Pham
  • Damien Tromeur-Dervout
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 74)


We investigate the proper orthogonal decomposition (POD) as a powerfull tool in decoupling dynamical systems suitable for parallel computing. POD method is well known to be useful method for model reduction applied to dynamical system having slow and fast dynamics. It is based on snapshot of previous time iterate solutions that allows to generate a low dimension space for the approximation of the solution.Here we focus on the parallelism potential with decoupling the dynamical system into subsystems spread between processors. The non local to the processor sub-systems are approximated by POD leading to have a number of unknowns smaller than the original system on each processor. We provide a mathematical analysis to obtain a criterion on the error behavior in using POD for decoupling dynamical systems. Therefore, we use this result to verify when the reduced model is still appropriated for the system in order to update the basis. Several examples show the efficient gain in term of computational effort of the present method.

Key words

POD reduced-order modelling dynamical systems parallel computing 


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  1. 1.
    Moody T. Chu, Robert E. Funderlic, and Gene H. Golub. A rank-one reduction formula and its applications to matrix factorizations. SIAM Rev., 37(4):512–530, 1995.Google Scholar
  2. 2.
    S.D. Cohen and A.C. Hindmarsh. CVODE, a stiff/nonstiff ODE solver in C. Computers in Physics, 10(2):138–143, 1996.Google Scholar
  3. 3.
    F. Fang, CC Pain, IM Navon, MD Piggott, GJ Gorman, P. Allison, and AJH Goddard. Reduced order modelling of an adaptive mesh ocean model. International Journal for Numerical Methods in Fluids, sub-judice, 2007.Google Scholar
  4. 4.
    M. Garbey and D. Tromeur-Dervout. A parallel adaptive coupling algorithm for systems of differential equations. J. Comput. Phys., 161(2):401–427, 2000.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gene H. Golub and Charles F. Van Loan. Matrix computations, volume 3 of Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, second edition, 1989.Google Scholar
  6. 6.
    Ming Gu and Stanley C. Eisenstat. Downdating the singular value decomposition. SIAM J. Matrix Anal. Appl., 16(3):793–810, 1995.Google Scholar
  7. 7.
    Chris Homescu, Linda R. Petzold, and Radu Serban. Error estimation for reduced-order models of dynamical systems. SIAM Rev., 49(2):277–299, 2007.Google Scholar
  8. 8.
    Lawrence Hubert, Jacqueline Meulman, and Willem Heiser. Two purposes for matrix factorization: a historical appraisal. SIAM Rev., 42(1):68–82 (electronic), 2000.Google Scholar
  9. 9.
    K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal., 40(2):492–515 (electronic), 2002.Google Scholar
  10. 10.
    Marc Moonen, Paul Van Dooren, and Filiep Vanpoucke. On the QR algorithm and updating the SVD and the URV decomposition in parallel. Linear Algebra Appl., 188/189:549–568, 1993.Google Scholar
  11. 11.
    T. Pham and F. Oudin-Dardun. c(p, q, j) scheme with adaptive time step and asynchronous communications. volume to appear of Lecture Note in Computational Science and Engineering, Parallel CFD 2007. Lecture OPTNote in Computational Science and Engineering, 2009.Google Scholar
  12. 12.
    Stig Skelboe. Accuracy of decoupled implicit integration formulas. SIAM J. Sci. Comput., 21(6):2206–2224 (electronic), 2000.Google Scholar
  13. 13.
    Stig Skelboe. Adaptive partitioning techniques for ordinary differential equations. BIT, 46(3):617–629, 2006.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Senol Utku, Jose L. M. Clemente, and Moktar Salama. Errors in reduction methods. Comput. & Structures, 21(6):1153–1157, 1985.Google Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Université de Lyon, Université Lyon 1, CDCSP/Institute Camille Jordan UMR5208-U.Lyon1-ECL-INSA-CNRSVilleurbanne CedexFrance

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