Continuation Multi-level Monte Carlo

  • Michele PisaroniEmail author
  • Fabio Nobile
  • Penelope Leyland
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 140)


In this chapter, we describe the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed in Collier et al. [1] and apply it to efficiently propagate operating and geometric uncertainties in internal and external aerodynamic simulations. The key idea of MLMC, presented in the previous chapter, is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigation on a hierarchy of nested computational grids (levels). In the continuation algorithm (C-MLMC) the parameters that prescribe the number of levels and simulations per level are computed on the fly to further reduce the overall computational cost.


Multi-level Monte Carlo Sampling 


  1. 1.
    Collier, N., Haji-Ali, A.L., Nobile, F., von Schwerin, E., Tempone, R.: A continuation multilevel Monte Carlo algorithm. BIT Numer. Math. 1–34 (2014)Google Scholar
  2. 2.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Teckentrup, A., Scheichl, R., Giles, M., Ullmann, E.: Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math. 125(3), 569–600 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Pisaroni, M., Leyland, P., Nobile, F.: A Multi Level Monte Carlo algorithm for the treatment of geometrical and operational uncertainties in internal and external aerodynamics. In: 46th AIAA Fluid Dynamics Conference, 4398 (2016)Google Scholar
  5. 5.
    Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123–161 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cliffe, K., Giles, M., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3–15 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Barth, A., Lang, A., Schwab, C.: Multilevel Monte Carlo method for parabolic stochastic partial differential equations. BIT Numer. Math. 53(1), 3–27 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gatski, T.B., Bonnet, J.P.: Compressibility, Turbulence and High Speed Flow. Academic Press (2013)Google Scholar
  10. 10.
    Hirsch, C.: Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics. Butterworth-Heinemann (2007)Google Scholar
  11. 11.
    Mani, M., Babcock, D., Winkler, C., Spalart, P.: Predictions of a supersonic turbulent flow in a square duct. AIAA Paper 860 (2013)Google Scholar
  12. 12.
    Dunham, J.: CFD validation for propulsion system components (la validation CFD des organes des propulseurs). Technical report, DTIC Document (1998)Google Scholar
  13. 13.
    Reid, L., Moore, R.D.: Design and overall performance of four highly loaded, high speed inlet stages for an advanced high-pressure-ratio core compressor. NASA Technical Report (1978)Google Scholar
  14. 14.
    Loeven, G., Bijl, H.: The application of the probabilistic collocation method to a transonic axial flow compressor. In: American Institute of Aeronautics and Astronautics (AIAA) (2010)Google Scholar
  15. 15.
    Gopinathrao, N.P., Bagshaw, D., Mabilat, C., Alizadeh, S.: Non-deterministic CFD simulation of a transonic compressor rotor. In: ASME Turbo Expo 2009: Power for Land, Sea, and Air, American Society of Mechanical Engineers, pp. 1125–1134 (2009)Google Scholar
  16. 16.
    Haase, W., Brandsma, F., Elsholz, E., Leschziner, M., Schwamborn, D.: EUROVAL An European Initiative on Validation of CFD Codes: Results of the EC/BRITE-EURAM Project EUROVAL, 1990–1992, Vol. 42. Springer (2013)Google Scholar
  17. 17.
    Schmitt, V., Charpin, F.: Experimental data base for computer program assessment. Report of the Fluid Dynamics Panel Working Group. AGARD-AR-138 (1979)Google Scholar
  18. 18.
    Sobieczky, H.: Parametric airfoils and wings. In: Fujii, K., Dulikravich, G.S. (eds.) Notes on Numerical Fluid Mechanics, vol. 68, pp. 71–88 (1998)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Michele Pisaroni
    • 1
    Email author
  • Fabio Nobile
    • 1
  • Penelope Leyland
    • 1
  1. 1.Scientific Computing and Uncertainty QuantificationEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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