Abstract
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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References
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)
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)
Anderson, H.L.: Metropolis, Monte Carlo and the MANIAC. Los Alamos Science (1986)
Berveiller, M.: Stochastic Finite Elements: Intrusive and Non Intrusive Methods for Reliability Analysis. Ph.D. Thesis, Universite Blaise Pascal, Clermont-Ferrand (2005)
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)
Box, G.E.P., Jenkins, G.M., Reinsel, G.C.: Time Series Analysis: Forecasting and Control, 3rd edn. Prentice Hall, Englewood Cliffs (1994)
Caflisch, R.E.: Monte Carlo and quasi-monte carlo methods. Acta Numerica 7, 1–49 (1998)
Cameron, R.H., Martin, W.T.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)
Canuto, C., Hussaini, M.Y., Quateroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domain. Springer, Berlin (2006)
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)
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)
Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., Rubin, D.: Bayesian Data Analysis, 3rd edn. Chapman and hall/CRC, London (2013)
George, D.L.: Flume problems. https://github.com/dlgeorge/flume/tree/master/GateRelease1/ (2016)
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). https://doi.org/10.1098/rspa.2013.0820
Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Boston (1989)
Haario, H., Saksman, E., Tamminen, J.: An adaptive metropolis algorithm. Bernoulli 7(2), 223–242 (2001)
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). https://doi.org/10.1016/j.cma.2015.02.006. http://www.sciencedirect.com/science/article/pii/S004578251500047X
Host, G.: Simulated Annealing - Wiley StatsRef: Statistics Reference Online. Wiley, New York (2014)
Isukapalli, S.: Uncertainty Analysis of Transport-Transformation Models. Ph.D. thesis, The State University of New Jersey, New Jersey (1999)
Iverson, R.M.: The physics of debris flows. Rev. Geophys. 35(3), 245–296 (1997)
Iverson, R.M.: Regulation of landslide motion by dilatancy and pore pressure feedback. J. Geophys. Res. 110 (F2), F02,015 (2005)
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). https://doi.org/10.1098/rspa.2013.0819
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)
Jansen, M.J.W.: Analysis of variance designs for model output. Comput. Phys. Commun. 117, 35–43 (1999)
Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. B 63, 425–464 (2001)
Kowalski, J., McElwaine, J.: Shallow two-component gravity-driven flows with vertical variation. J. Fluid Mech. 714, 434–462 (2013)
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)
Langseth, J.O., LeVeque, R.J.: A wave-propagation method for three-dimensional hyperbolic conservation laws. J. Comput. Phys. 165, 126–166 (2000)
Lawson, C.L.: Contribution to the Theory of Linear Least Maximum Approximations. Ph.D. thesis, University of California (1961)
Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer, New York (2010)
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)
LeVeque, R.J.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)
LeVeque, R.J.: Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 131, 327–353 (1997)
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002). http://www.clawpack.org/book.html
Loh, W.L.: On latin hypercube sampling. Ann. Stat. 24(5), 2058–2080 (1996)
Madras, N.: Lectures on Monte Carlo Methods. American Mathematical Society, Providence (2001)
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)
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). https://doi.org/10.7717/peerj-cs.68. https://peerj.com/articles/cs-68
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)
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)
Matlab optimization toolbox: The MathWorks, Natick, MA, USA (2016)
Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)
Polak, E.: Optimization: Algorithms and Consistent Approximations. Applied Mathematical Sciences, p. 9780387949710. Springer-Verlag, Berlin (1997)
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). https://doi.org/10.1029/2009JF001514. 29 p
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)
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)
Saltelli, A.: Making best use of model evaluations to compute sensitivity indices. Comput. Phys. Commun. 145(2), 280–297 (2002)
Scales, J.A., Gersztenkorn, A.: Robust methods in inverse theory. Inverse Probl. 4(4), 1071 (1988)
Schofield, A.N., Wroth, C.P.: Critical State Soil Mechanics. McGraw Hill, New York (1968)
Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman & Hall, London (1986)
Sobol’, I.: Sensitivity estimates for nonlinear mathematical models. Math. Modeling Comput. Exp. 1, 407–414 (1993)
Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)
Tarantola, A.: Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics (2005)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)
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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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Navarro, M., Le Maître, O.P., Hoteit, I. et al. Surrogate-based parameter inference in debris flow model. Comput Geosci 22, 1447–1463 (2018). https://doi.org/10.1007/s10596-018-9765-1
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DOI: https://doi.org/10.1007/s10596-018-9765-1