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Surrogate-based parameter inference in debris flow model


This work tackles the problem of calibrating the unknown parameters of a debris flow model with the drawback that the information regarding the experimental data treatment and processing is not available. In particular, we focus on the evolution over time of the flow thickness of the debris with dam-break initial conditions. The proposed methodology consists of establishing an approximation of the numerical model using a polynomial chaos expansion that is used in place of the original model, saving computational burden. The values of the parameters are then inferred through a Bayesian approach with a particular focus on inference discrepancies that some of the important features predicted by the model exhibit. We build the model approximation using a preconditioned non-intrusive method and show that a suitable prior parameter distribution is critical to the construction of an accurate surrogate model. The results of the Bayesian inference suggest that utilizing directly the available experimental data could lead to incorrect conclusions, including the over-determination of parameters. To avoid such drawbacks, we propose to base the inference on few significant features extracted from the original data. Our experiments confirm the validity of this approach, and show that it does not lead to significant loss of information. It is further computationally more efficient than the direct approach, and can avoid the construction of an elaborate error model.

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  1. 1.

    Alexanderian, A., Maître, O.P.L., Najm, H., Iskandarani, M., Knio, O.: Multiscale stochastic preconditioners in non-intrusive spectral projection. J. Sci. Comput. 50(2), 306–340 (2012)

    Article  Google Scholar 

  2. 2.

    Alexanderian, A., Rizzi, F., Rathinam, M., Maître, O.L., Knio, O.: Preconditioned Bayesian regression for stochastic chemical kinetics. J. Sci. Comput. 58(3), 592–626 (2014)

    Article  Google Scholar 

  3. 3.

    Anderson, H.L.: Metropolis, Monte Carlo and the MANIAC. Los Alamos Science (1986)

  4. 4.

    Berveiller, M.: Stochastic Finite Elements: Intrusive and Non Intrusive Methods for Reliability Analysis. Ph.D. Thesis, Universite Blaise Pascal, Clermont-Ferrand (2005)

  5. 5.

    Bouchut, F., Fernández-Nieto, E., Mangeney, A., Narbona-Reina, G.: A two-phase two-layer model for fluidized granular flows with dilatancy effects. J. Fluid Mech. 801, 166–221 (2016)

    Article  Google Scholar 

  6. 6.

    Box, G.E.P., Jenkins, G.M., Reinsel, G.C.: Time Series Analysis: Forecasting and Control, 3rd edn. Prentice Hall, Englewood Cliffs (1994)

    Google Scholar 

  7. 7.

    Caflisch, R.E.: Monte Carlo and quasi-monte carlo methods. Acta Numerica 7, 1–49 (1998)

    Article  Google Scholar 

  8. 8.

    Cameron, R.H., Martin, W.T.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)

    Article  Google Scholar 

  9. 9.

    Canuto, C., Hussaini, M.Y., Quateroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domain. Springer, Berlin (2006)

    Google Scholar 

  10. 10.

    Conn, A.R., Gould, N.I.M., Toint, P.L.: A globally convergent augmented lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28(2), 545–572 (1991)

    Article  Google Scholar 

  11. 11.

    Crestaux, T., Le maître, O.P., Martinez, J.M.: Polynomial chaos expansion of sensitivity analysis. J. Rel. Eng. Syst. Saf. 94(7), 1161–1182 (2009)

    Article  Google Scholar 

  12. 12.

    Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., Rubin, D.: Bayesian Data Analysis, 3rd edn. Chapman and hall/CRC, London (2013)

    Google Scholar 

  13. 13.

    George, D.L.: Flume problems. (2016)

  14. 14.

    George, D.L., Iverson, R.M.: A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests. Proc. R. Soc. 470, 20130820 (2014).

    Article  Google Scholar 

  15. 15.

    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

    Book  Google Scholar 

  16. 16.

    Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Boston (1989)

    Google Scholar 

  17. 17.

    Haario, H., Saksman, E., Tamminen, J.: An adaptive metropolis algorithm. Bernoulli 7(2), 223–242 (2001)

    Article  Google Scholar 

  18. 18.

    Hampton, J., Doostan, A.: Coherence motivated sampling and convergence analysis of least squares polynomial chaos regression. Comput. Methods Appl. Mech. Eng. 290, 73 – 97 (2015).

    Article  Google Scholar 

  19. 19.

    Host, G.: Simulated Annealing - Wiley StatsRef: Statistics Reference Online. Wiley, New York (2014)

    Google Scholar 

  20. 20.

    Isukapalli, S.: Uncertainty Analysis of Transport-Transformation Models. Ph.D. thesis, The State University of New Jersey, New Jersey (1999)

    Google Scholar 

  21. 21.

    Iverson, R.M.: The physics of debris flows. Rev. Geophys. 35(3), 245–296 (1997)

    Article  Google Scholar 

  22. 22.

    Iverson, R.M.: Regulation of landslide motion by dilatancy and pore pressure feedback. J. Geophys. Res. 110 (F2), F02,015 (2005)

    Article  Google Scholar 

  23. 23.

    Iverson, R.M., George, D.L.: A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. Physical basis. Proc. R. Soc. A 470, 20130819 (2014).

    Article  Google Scholar 

  24. 24.

    Iverson, R.M., Reid, M.E., Iverson, N.R., LaHusen, R.G., Logan, M., Mann, J.E., Brien, D.L.: Acute sensitivity of landslide rates to initial soil porosity. Science 290(5491), 513–516 (2000)

    Article  Google Scholar 

  25. 25.

    Jansen, M.J.W.: Analysis of variance designs for model output. Comput. Phys. Commun. 117, 35–43 (1999)

    Article  Google Scholar 

  26. 26.

    Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. B 63, 425–464 (2001)

    Article  Google Scholar 

  27. 27.

    Kowalski, J., McElwaine, J.: Shallow two-component gravity-driven flows with vertical variation. J. Fluid Mech. 714, 434–462 (2013)

    Article  Google Scholar 

  28. 28.

    Langenhove, J.V., Lucor, D., Belme, A.: Robust uncertainty quantification using preconditioned least-squares polynomial approximations with l1-regularizations. Int. J. Uncertain. Quantif. 6, 57–77 (2016)

    Article  Google Scholar 

  29. 29.

    Langseth, J.O., LeVeque, R.J.: A wave-propagation method for three-dimensional hyperbolic conservation laws. J. Comput. Phys. 165, 126–166 (2000)

    Article  Google Scholar 

  30. 30.

    Lawson, C.L.: Contribution to the Theory of Linear Least Maximum Approximations. Ph.D. thesis, University of California (1961)

  31. 31.

    Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer, New York (2010)

    Book  Google Scholar 

  32. 32.

    Le Maître, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M.: A stochastic projection method for fluid flow. II. Random process. J. Comput. Phys. 181, 9–44 (2002)

    Article  Google Scholar 

  33. 33.

    LeVeque, R.J.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)

    Article  Google Scholar 

  34. 34.

    LeVeque, R.J.: Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 131, 327–353 (1997)

    Article  Google Scholar 

  35. 35.

    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002).

    Book  Google Scholar 

  36. 36.

    Loh, W.L.: On latin hypercube sampling. Ann. Stat. 24(5), 2058–2080 (1996)

    Article  Google Scholar 

  37. 37.

    Madras, N.: Lectures on Monte Carlo Methods. American Mathematical Society, Providence (2001)

    Book  Google Scholar 

  38. 38.

    Mai, C.V., Sudret, B.: Surrogate models for oscillatory systems using sparse polynomial chaos expansions and stochastic time warping. SIAM/ASA J. Uncertain. Quantif. 5(1), 540–571 (2017)

    Article  Google Scholar 

  39. 39.

    Mandli, K.T., Ahmadia, A.J., Berger, M., Calhoun, D., George, D.L., Hadjimichael, Y., Ketcheson, D.I., Lemoine, G.I., LeVeque, R.J.: Clawpack: building an open source ecosystem for solving hyperbolic PDEs. PeerJ Comput. Sci. 2(3), e68 (2016).

    Article  Google Scholar 

  40. 40.

    Marzouk, Y.M., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228(6), 1862–1902 (2009)

    Article  Google Scholar 

  41. 41.

    Marzouk, Y.M., Najm, H.N., Rahn, L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224(2), 560–586 (2007)

    Article  Google Scholar 

  42. 42.

    Matlab optimization toolbox: The MathWorks, Natick, MA, USA (2016)

  43. 43.

    Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)

    Article  Google Scholar 

  44. 44.

    Polak, E.: Optimization: Algorithms and Consistent Approximations. Applied Mathematical Sciences, p. 9780387949710. Springer-Verlag, Berlin (1997)

    Book  Google Scholar 

  45. 45.

    Iverson, R.M. , Logan, M., LaHusen, R.G., Berti, M.: The perfect debris flow? Aggregated results from 28 large-scale experiments. J. Geophys. Res. 115, F03005 (2010). 29 p

    Article  Google Scholar 

  46. 46.

    Reagan, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M.: Uncertainty quantification in reacting flow simulations through non-intrusive spectral projection. Combust. Flame 132, 545–555 (2003)

    Article  Google Scholar 

  47. 47.

    Rowe, P.W.: The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Roy. Soc. Lond. A269, 500–527 (1962)

    Article  Google Scholar 

  48. 48.

    Saltelli, A.: Making best use of model evaluations to compute sensitivity indices. Comput. Phys. Commun. 145(2), 280–297 (2002)

    Article  Google Scholar 

  49. 49.

    Scales, J.A., Gersztenkorn, A.: Robust methods in inverse theory. Inverse Probl. 4(4), 1071 (1988)

    Article  Google Scholar 

  50. 50.

    Schofield, A.N., Wroth, C.P.: Critical State Soil Mechanics. McGraw Hill, New York (1968)

    Google Scholar 

  51. 51.

    Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman & Hall, London (1986)

    Book  Google Scholar 

  52. 52.

    Sobol’, I.: Sensitivity estimates for nonlinear mathematical models. Math. Modeling Comput. Exp. 1, 407–414 (1993)

    Google Scholar 

  53. 53.

    Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)

    Article  Google Scholar 

  54. 54.

    Tarantola, A.: Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics (2005)

  55. 55.

    Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)

    Article  Google Scholar 

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Research reported in this publication was supported by research funding from King Abdullah University of Science and Technology (KAUST).

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Correspondence to Maria Navarro.

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Navarro, M., Le Maître, O.P., Hoteit, I. et al. Surrogate-based parameter inference in debris flow model. Comput Geosci 22, 1447–1463 (2018).

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  • Bayesian inference
  • Polynomial chaos expansion
  • Debris flow
  • Uncertainty quantification