Multilevel Adaptive Stochastic Collocation with Dimensionality Reduction

  • Ionuţ-Gabriel Farcaş
  • Paul Cristian Sârbu
  • Hans-Joachim BungartzEmail author
  • Tobias Neckel
  • Benjamin Uekermann
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 123)


We present a multilevel stochastic collocation (MLSC) with a dimensionality reduction approach to quantify the uncertainty in computationally intensive applications. Standard MLSC typically employs grids with predetermined resolutions. Even more, stochastic dimensionality reduction has not been considered in previous MLSC formulations. In this paper, we design an MLSC approach in terms of adaptive sparse grids for stochastic discretization and compare two sparse grid variants, one with spatial and the other with dimension adaptivity. In addition, while performing the uncertainty propagation, we analyze, based on sensitivity information, whether the stochastic dimensionality can be reduced. We test our approach in two problems. The first one is a linear oscillator with five or six stochastic inputs. The dimensionality is reduced from five to two and from six to three. Furthermore, the dimension-adaptive interpolants proved superior in terms of accuracy and required computational cost. The second test case is a fluid-structure interaction problem with five stochastic inputs, in which we quantify the uncertainty at two instances in the time domain. The dimensionality is reduced from five to two and from five to four.



We thank David Holzmueller for developing the dimension-adaptive interpolation module in SG++, employed in this paper. Moreover, we thankfully acknowledge the financial support of the German Academic Exchange Service (, of the German Research Foundation through the TUM International Graduate School of Science and Engineering (IGSSE) within the project 10.02 BAYES (, and the financial support of the priority program 1648 - Software for Exascale Computing of the German Research Foundation (


  1. 1.
    H.-J. Bungartz, Finite elements of higher order on sparse grids. Habilitationsschrift, Fakultät für Informatik, Technische Universität München, Shaker Verlag, Aachen, 1998Google Scholar
  2. 2.
    H.-J. Bungartz, S. Dirnstorfer, Multivariate quadrature on adaptive sparse grids. Computing 71(1), 89–114 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    H.-J. Bungartz, M. Griebel, Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    H.-J. Bungartz, F. Lindner, B. Gatzhammer, M. Mehl, K. Scheufele, A. Shukaev, B. Uekermann, preCICE – a fully parallel library for multi-physics surface coupling. Comput. Fluids (2016).
  5. 5.
    P. Chen, A. Quarteroni, A new algorithm for high-dimensional uncertainty quantification based on dimension-adaptive sparse grid approximation and reduced basis methods. J. Comput. Phys. 298(Supplement C), 176–193 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P.R. Conrad, Y.M. Marzouk, Adaptive smolyak pseudospectral approximations. SIAM J. Sci. Comput. 35(6), A2643–A2670 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    P.G. Constantine, M.S. Eldred, E.T. Phipps, Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229–232, 1–12 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Donea, A. Huerta, J.-P. Ponthot, A. Rodriguez-Ferran, Arbitrary lagrangian-eulerian methods. Encycl. Comput. Mech. 1, 413–437 (2004)Google Scholar
  9. 9.
    I.-G. Farcas, B. Uekermann, T. Neckel, H.-J. Bungartz, Nonintrusive uncertainty analysis of fluid-structure interaction with spatially adaptive sparse grids and polynomial chaos expansion. SIAM J. Sci. Comput. 40(2), B457–B482. MathSciNetCrossRefGoogle Scholar
  10. 10.
    F. Franzelin, P. Diehl, D. Pflüger, Non-intrusive uncertainty quantification with sparse grids for multivariate peridynamic simulations, in Meshfree Methods for Partial Differential Equations VII, ed. by M. Griebel, M.A. Schweitzer. Lecture Notes in Computational Science and Engineering, vol. 100 (Springer International Publishing, Cham, 2015), pp. 115–143Google Scholar
  11. 11.
    T. Gerstner, M. Griebel, Numerical integration using sparse grids. Numer. Algorithms 18(3), 209–232 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. Gerstner, M. Griebel, Dimension–adaptive tensor–product quadrature. Computing 71(1), 65–87 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Griebel, M. Schneider, C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, ed. by P. de Groen, R. Beauwens (IMACS, Elsevier, North Holland, 1992), pp. 263–281Google Scholar
  14. 14.
    F. Heiss, V. Winschel, Likelihood approximation by numerical integration on sparse grids. J. Econometrics 144(1), 62–80 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Klimke, Uncertainty modeling using fuzzy arithmetic and sparse grids. PhD thesis, Universität Stuttgart, Shaker Verlag, Aachen, 2006Google Scholar
  16. 16.
    F. Leja, Sur certaines suites liées aux ensemble plan et leur application à la representation conforme. Ann. Polon. Math. 5, 8–13 (1957)CrossRefGoogle Scholar
  17. 17.
    X. Ma, N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228(8), 3084–3113 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S.D. Marchi, On leja sequences: some results and applications. Appl. Math. Comput. 152(3), 621–647 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    M. Mehl, B. Uekermann, H. Bijl, D. Blom, B. Gatzhammer, A. van Zuijlen, Parallel coupling numerics for partitioned fluid-structure interaction simulations. Comput. Math. Appl. 71(4), 869–891 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Narayan, J.D. Jakeman, Adaptive leja sparse grid constructions for stochastic collocation and high-dimensional approximation. SIAM J. Sci. Comput. 36(6), A2952–A2983 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    N.M. Newmark, A method of computation for structural dynamics. J .Eng. Mech. Div. 85(3), 67–94 (1959)Google Scholar
  22. 22.
    F. Nobile, R. Tempone, C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309–2345 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    A.B. Owen, Sobol’ indices and shapley value. SIAM/ASA J. Uncertain. Quantif. 2(1), 245–251 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    B. Peherstorfer, Model order reduction of parametrized systems with sparse grid learning techniques. Dissertation, Department of Informatics, Technische Universität München, 2013Google Scholar
  25. 25.
    D. Pflüger, Spatially Adaptive Sparse Grids for High-Dimensional Problems (Verlag Dr. Hut, München, 2010)zbMATHGoogle Scholar
  26. 26.
    K. Sargsyan, C. Safta, H.N. Najm, B.J. Debusschere, D. Ricciuto, P. Thornton, Dimensionality reduction for complex models via bayesian compressive sensing. Int. J. Uncertain. Quantif. 4(1), 63–93 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4, 240–243 (1963)zbMATHGoogle Scholar
  28. 28.
    I. Sobol, Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Math. Comput. Simul. 55(1–3), 271–280 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    D. Stirzaker, Elementary Probability (Cambridge University Press, Cambridge, 2003)CrossRefGoogle Scholar
  30. 30.
    B. Sudret, Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)CrossRefGoogle Scholar
  31. 31.
    A.L. Teckentrup, P. Jantsch, C.G. Webster, M. Gunzburger, A multilevel stochastic collocation method for partial differential equations with random input data.SIAM/ASA J. Uncertain. Quantif. 3(1), 1046–1074 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    U. Trottenberg, A. Schuller, Multigrid (Academic, Orlando, 2001)zbMATHGoogle Scholar
  33. 33.
    B. Uekermann, J.C. Cajas, B. Gatzhammer, G. Houzeaux, M. Mehl, M. Vazquez, Towards partitioned fluid-structure interaction on massively parallel systems, in 11th World Congress on Computational Mechanics (WCCM XI), ed. by E. Oñate, J. Oliver, A. Huerta (2014), pp. 1–12Google Scholar
  34. 34.
    M. Vázquez, G. Houzeaux, S. Koric, A. Artigues, J. Aguado-Sierra, R. Arís, D. Mira, H. Calmet, F. Cucchietti, H. Owen, A. Taha, E.D. Burness, J.M. Cela, M. Valero, Alya: multiphysics engineering simulation toward exascale. J. Comput. Sci. 14, 15–27 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach (Princeton University Press, Princeton, 2010)zbMATHGoogle Scholar
  36. 36.
    D. Xiu, G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)MathSciNetCrossRefGoogle Scholar
  37. 37.
    C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, ed. by W. Hackbusch. Notes on Numerical Fluid Mechanics, vol. 31 (Vieweg Verlag, Braunschweig, 1991), pp. 241–251Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ionuţ-Gabriel Farcaş
    • 1
  • Paul Cristian Sârbu
    • 1
  • Hans-Joachim Bungartz
    • 1
    Email author
  • Tobias Neckel
    • 1
  • Benjamin Uekermann
    • 1
  1. 1.Technical University of MunichGarchingGermany

Personalised recommendations