Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T19:04:36.909Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive Bayesian Inference for Discontinuous Inverse Problems, Application to Hyperbolic Conservation Laws

Published online by Cambridge University Press:  03 June 2015

Alexandre Birolleau ,
Gaël Poëtte  and
Didier Lucor
Alexandre Birolleau*
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
Gaël Poëtte*
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France
Didier Lucor*
Affiliation:
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

Various works from the literature aimed at accelerating Bayesian inference in inverse problems. Stochastic spectral methods have been recently proposed as surrogate approximations of the forward uncertainty propagation model over the support of the prior distribution. These representations are efficient because they allow affordable simulation of a large number of samples from the posterior distribution. Unfortunately, they do not perform well when the forward model exhibits strong nonlinear behavior with respect to its input.

In this work, we first relate the fast (exponential) L2-convergence of the forward approximation to the fast (exponential) convergence (in terms of Kullback-Leibler divergence) of the approximate posterior. In particular, we prove that in case the prior distribution is uniform, the posterior is at least twice as fast as the convergence rate of the forwardmodel in those norms. The Bayesian inference strategy is developed in the framework of a stochastic spectral projectionmethod. The predicted convergence rates are then demonstrated for simple nonlinear inverse problems of varying smoothness.

We then propose an efficient numerical approach for the Bayesian solution of inverse problems presenting strongly nonlinear or discontinuous systemresponses. This comes with the improvement of the forward model that is adaptively approximated by an iterative generalized Polynomial Chaos-based representation. The numerical approximations and predicted convergence rates of the former approach are compared to the new iterative numericalmethod for nonlinear time-dependent test cases of varying dimension and complexity, which are relevant regarding our hydrodynamics motivations and therefore regarding hyperbolic conservation laws and the apparition of discontinuities in finite time.

Keywords

52B1065D1868U0568U07Stochastic inverse problemsBayesian inferenceiterative generalized Polynomial Chaoscompressible gas dynamics.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 1 - 34
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Archibald, R., Gelb, A., Saxena, R. and Xiu, D., Discontinuity detection in multivariate space for stochastic simulations, J. Comput. Phys., 228(7) (2009), 2676–2689.Google Scholar
[2]Arnst, M. and Ghanem, R., Probabilistic equivalence and stochastic model reduction in multiscale analysis, Comput. Methods Appl. Mech. Eng., 197(43-44) (2008), 3584–3592.Google Scholar
[3]Berger, J. O., Bernardo, J. M. and Sun, D., The formal definition of Reference priors, The Annals Stat., 37(2) (2009), 905–938.CrossRefGoogle Scholar
[4]Rosic, Bojana V., Kucerovâ, Anna, Sykora, Jan, Pajonk, Oliver, Litvinenko, Alexander and Matthies, Her-mann G., Parameter identification in a probabilistic setting, Eng. Struct., 50 (2013), 179–196.Google Scholar
[5]Cameron, R. H. and Martin, W. T., The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annals Math., 48 (1947), 385–392.Google Scholar
[6]Constantine, Paul G., Eldred, Michael S. and Phipps, Eric T., Sparse pseudospectral approximation method, Comput. Methods Appl. Mech. Eng., 229-232(0) (2012), 1–12.Google Scholar
[7]Das, Sonjoy, Ghanem, Roger and Spall, James C., Asymptotic sampling distribution for polynomial chaos representation from data: a maximum entropy and fisher information approach, 30(5) (2008), 2207–2234.Google Scholar
[8]Desceliers, C., Ghanem, R. and Soize, C., Maximum likelihood estimation of stochastic chaos representations from experimental data, Int. J. Numer. Methods Eng., 66(6) (2006), 978–1001.Google Scholar
[9]Després, B. et al., Lois de Conservations Eulériennes, Lagrangiennes, et Méthodes Numériques, Volume 68, Springer Verlag, 2010.Google Scholar
[10]Galbally, D., Fidkowski, K., Willcox, K. and Ghattas, O., Non-linear model reduction for uncertainty quantification in large-scale inverse problems, Int. J. Numer. Methods Eng., 81(12) (2010), 1581–1608.Google Scholar
[11]Gautschi, Walter, Orthogonal Polynomials: Applications and Computation, Volume 5, Oxford University Press, 1996.Google Scholar
[12]Gelfand, Alan E. and Smith, Adrian F. M., Sampling-based approaches to calculating marginal densities, J. Am. Stat. Association, 85(410) (1990), 398–409.Google Scholar
[13]Geman, Stuart and Geman, Donald, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, PAMI, 6 (1984), 721–741.Google Scholar
[14]Gerritsma, M. I., van der Steen, J.-B., Vos, P. and Karniadakis, G. E., Time-dependent generalized polynomial chaos, J. Comput. Phys., (2010), 8333–8363.Google Scholar
[15]Ghanem, R. G. and Spanos, P., Stochastic Finite Elements: a Spectral Approach (revised edition), Springer-Verlag, 2003.Google Scholar
[16]Ghanem, Roger G. and Doostan, Alireza, On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data, J. Comput. Phys., 217(1) (2006), 63–81.Google Scholar
[17]Ghanem, Roger G., Alireza Doostan and John Red-Horse, A probabilistic construction of model validation, Comput. Methods Appl. Mech. Eng., 197(29-32) (2008), 2585–2595.Google Scholar
[18]Gilks, W. R., Richardson, S. and Spiegelhalter, D. J., Markov Chain Monte Carlo in Practice, Chapman & Hall, 1995.Google Scholar
[19]Gottlieb, S., Jung, J. H. and Kim, S., A review of David Gottlieb’s work on the resolution of the Gibbs phenomenon, Commun. Comput. Phys., 9 (2011), 497–519.Google Scholar
[20]Higdon, D., Kennedy, M., Cavendish, J. C., Cafeo, J. A. and Ryne, R. D., Combining field data and computer simulations for calibration and prediction, SIAM J. Sci. Comput., 26 (2004), 448.Google Scholar
[21]Jakeman, J. D., Narayan, A. and Dongbin, X., Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions, J. Comput. Phys., 242 (2013), 790–808.Google Scholar
[22]Jaynes, E. T., Prior probabilities, Systems Science and Cybernetics, IEEE Transactions on, 4(3) (1968), 227–241.Google Scholar
[23]Jaynes, E. T., Marginalization and Prior Probabilities, Bayesian Analysis in Econometrics and Statistics, Zellner, A., ed., North-Holland Publishing Company, Amsterdam, 1980.Google Scholar
[24]Jaynes, E. T., Papers on Probability, Statistics and Statistical Physics, Volume 50, Springer, 1989.Google Scholar
[25]Jeffreys, H., An invariant form for the prior probability in estimation problems, Proc. Roy. Soc. London Math. Phys. Sci., 186 (1946), 453–461.Google Scholar
[26]Junk, M., Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci., 10(7) (2000), 1001–1026.CrossRefGoogle Scholar
[27]Kaipio, J. and Somersalo, E., Statistical and Computational Inverse Problems, Volume 160 of Applied Mathematical Sciences, Springer, 2005.Google Scholar
[28]Keese, Andreas, Numerical Solution of Systems with Stochastic Uncertainties: a General Purpose Framework for Stochastic Finite Elements, PhD thesis, Technische Universität Braun-schweig, Mechanik-Zentrum, 2005.Google Scholar
[29]Kennedy, M. C. and O’Hagan, A., Bayesian calibration of computer models, J. Royal Stat. Soc. Ser. B, (Statistical Methodology), 63(3) (2001), 425–464.Google Scholar
[30]Kullback, S. and Leibler, R. A., On information and sufficiency, Annals Math. Stat., 22(1) (1951), 79–86.Google Scholar
[31]Le Maitre, O., Knio, O., Najm, H. and Ghanem, R., Uncertainty propagation using Wiener-Haar expansions, J. Comput. Phys., 197 (2004), 28–57.Google Scholar
[32]Le Maitre, O., Reagan, M., Najm, H., Ghanem, R. and Knio, O., Multi-resolution analysis of Wiener-type uncertainty propagation schemes, J. Comput. Phys., 197 (2004), 502–531.Google Scholar
[33]Le Maître, O. P. and Knio, Omar M., Spectral Methods for Uncertainty Quantification with applications to Computational Fluid Dynamics, Scientific Computation, Springer Netherlands, 2010.Google Scholar
[34]Le Meitour, J., Lucor, D. and Chassaing, J.-C., Adaptive piecewise polynomial chaos approaches: application to nonlinear aeroelastic flutter, in ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting: Volume 1, Symposia-Parts A, B, and C, pages 29572968, August 1-5 2010, Montreal, Canada, 2010.Google Scholar
[35]Liese, F. and Vajda, I., On divergences and informations in statistics and information theory, Information Theory, IEEE Transactions on, 52(10) (2006), 4394–4412.Google Scholar
[36]Lucor, D., Enaux, C., Jourdren, H. and Sagaut, P., Stochastic design optimization: application to reacting flows and detonation, Comput. Meth. Appl. Mech. Eng., 196(49-52) (2007), 5047–5062.Google Scholar
[37]Lucor, D., Meyers, J. and Sagaut, P., Sensitivity analysis of LES to subgrid-scale-model parametric uncertainty using Polynomial Chaos, J. Fluid Mech., 585 (2007), 255–279.Google Scholar
[38]Lucor, D., Witteveen, J., Constantine, P., Schiavazzi, D. and Iaccarino, G., Comparison of adaptive uncertainty quantification approaches for shock wave-dominated flows, in Proceedings of the Summer Program, Center for Turbulence Research, Stanford University, 2013.Google Scholar
[39]Ma, Xiang and Zabaras, Nicholas, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228(8) (2009), 3084–3113.Google Scholar
[40]Marzouk, Y. and Xiu, D., A stochastic collocation approach to bayesian inference in inverse problems, Commun. Comput. Phys., 2009.CrossRefGoogle Scholar
[41]Marzouk, Y. M. and Najm, H. N., Dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems, J. Comput. Phys., 228(6) (2009), 1862–1902.CrossRefGoogle Scholar
[42]Marzouk, Y. M., Najm, H. N. and Rahn, L. A., Stochastic spectral methods for efficient bayesian solution of inverse problems, J. Comput. Phys., 224(2) (2007), 560–586.Google Scholar
[43]Marzouk, Youssef and Xiu, Dongbin, A stochastic collocation approach to Bayesian inference in inverse problems, Commun. Comput. Phys., 6(4) (2009), 826–847.Google Scholar
[44]Marzouk, Youssef M. and Najm, Habib N., Dimensionality reduction and Polynomial Chaos acceleration of Bayesian inference in inverse problems, J. Comput. Phys., 228(6) (2009), 1862–1902.CrossRefGoogle Scholar
[45]Marzouk, Youssef M., Najm, Habib N. and Rahn, Larry A., Stochastic spectral methods for efficient Bayesian solution of inverse problems, J. Comput. Phys., 224(2) (2007), 560–586.Google Scholar
[46]Mathelin, L. and Le Maître, O. P., A posteriori error analysis for stochastic finite element solutions of fluid flows with parametric uncertainties, ECCOMAS CFD, 2006.Google Scholar
[47]Matthies, H. and Keese, A., Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Meth. Appl. Mech. Eng., 192(12-16) (2005), 1295–1331.Google Scholar
[48]Mead, L. R. and Papanicolaou, N., Maximum entropy in the problem of moments, J. Math. Phys., 25(8) (1984).Google Scholar
[49]Meldi, M., Sagaut, P. and Lucor, D., A stochastic view of isotropic turbulence decay, J. Fluid Mech., 668 (2011), 351–362.Google Scholar
[50]Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E., Equation of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 1087–1092.Google Scholar
[51]Najm, H. N.Le Maître, O.P., Knio, O. M. and Ghanem, R. G., A stochastic projection method for fluid flow i: basic formulation, J. Comput. Phys., 173 (2001), 481–511.Google Scholar
[52]Pinsker, M. S., Information and information stability of random variables and processes, Holden-Day Series in Time Series Analysis, 1960.Google Scholar
[53]Poëtte, G., Birolleau, A. and Lucor, D., Iterative generalized Polynomial Chaos approxima-tions, SIAM J. Sci. Comput., submitted, 2012.Google Scholar
[54]Poëtte, G., Després, B. and Lucor, D., Uncertainty quantification for systems of conservation laws, J. Comput. Phys., 228(7) (2009), 2443–2467.Google Scholar
[55]Poëtte, G., Després, B. and Lucor, D., Adaptive hybrid spectral methods for stochastic systems of conservation laws, in Sequeira, A.Pereira, J. C. F. and Pereira, J. M. C., editors, ECCOMAS CFD 2010 Proceedings of the V European Conference on Computational Fluid Dynamics, Lisbon, Portugal, 2010.Google Scholar
[56]Poëtte, G., Després, B. and Lucor, D., Treatment of uncertain interfaces in compressible flows, Comput. Meth. Appl. Math. Eng., 200(1-4) (2011), 284–308.Google Scholar
[57]Poëtte, G., Després, B. and Lucor, D., Uncertainty Propagation for Systems of Conservation Laws, High-Order Stochastic Spectral Methods, in Hesthaven, Jan S. and Renquist, Einar M., editors, Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, pages 293–307, ICOSAHOM ’09 conference, June 22-26 2009, Trondheim, Norway, 2011.Google Scholar
[58]Poëtte, G. and Lucor, D., Non intrusive iterative stochastic spectral representation with application to compressible gas dynamics, J. Comput. Phys., 2011, DOI information: 10.1016/j.jcp.2011.12.038.Google Scholar
[59]Thorembey, Poggi and Rodriguez, , Velocity Measurements in turbulent gaseous mixtures iduced by Richtmyer-Meshkov instability, Phys. Fluids, 10 (1998), 11.Google Scholar
[60]Robert, C. P. and Casella, G., Monte Carlo Statistical Methods, Springer-Verlag, New York, 2nd edition, 2004.Google Scholar
[61]Robert, C. P., The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, Springer Verlag, 2007.Google Scholar
[62]Serre, D., Systèmes Hyperboliques de Lois de Conservation, partie I, Diderot, 1996, Paris.Google Scholar
[63]Serre, D., Systèmes Hyperboliques de Lois de Conservation, partie II, Diderot, 1996, Paris.Google Scholar
[64]Simon, F., Guillen, P., Sagaut, P. and Lucor, D., A gPC-based approach to uncertain transonic aerodynamics, Comput. Methods Appl. Mech. Eng., 199(17-20) (2010), 1091–1099.Google Scholar
[65]Tarantola, A., Inverse Problem Theory-Methods for Data Fitting and Model Parameter Estimation, Elsevier, 1987.Google Scholar
[66]Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM: Philadelphia, 2005.Google Scholar
[67]Tatang, M., Pan, W., Prinn, R. and McRae, G., An efficient method for parametric uncertainty analysis of numerical geophysical models, J. Geophys. Res., 102 (1997), 21925–21932.Google Scholar
[68]Toro, E. F., Riemann Solver and Numerical Methods for Fluid Dynamics, Springer-Verlag, 1997.Google Scholar
[69]Tryoen, J., Adaptive anisotropic stochastic discretization schemes for uncertain conservation laws, in ASME-FEDSM Proceedings, Montreal, Canada, 2010.Google Scholar
[70]Vetter and Sturtevant, , Experiments on the Richtmyer-Meshkov instability of an air/SF6 interface, Shock Waves, 4 (1995), 247–252.Google Scholar
[71]Vos, P., Time Dependent Polynomial Chaos, Master of science thesis, Delft University of Technology, Faculty of Aerospace Engineering, 2006.Google Scholar
[72]Wan, X. and Karniadakis, G. E., An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209 (2005), 617–642.Google Scholar
[73]Wan, X. and Karniadakis, G. E., An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209(2) (2005), 617–642.Google Scholar
[74]Wan, X. and Karniadakis, G. E., Multi-element generalized Polynomial Chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28(3) (2006), 901–928.Google Scholar
[75]Wan, X. and Karniadakis, G. E., Stochastic heat transfer enhancement in a grooved channel, J. Fluid Mech., 565 (2006), 255–278.Google Scholar
[76]Wan, Xiaoliang and Em Karniadakis, George, Error control in multi-element generalized Polynomial Chaos method for elliptic problems with random coefficients, Commun. Com- put. Phys., 5(2-4) (2009), 793–820.Google Scholar
[77]Wang, Jingbo and Zabaras, Nicholas, Hierarchical bayesian models for inverse problems in heat conduction, Inverse Problems, 21 (2005), 183–206.Google Scholar
[78]Wiener, N., The homogeneous chaos, Amer. J. Math., 60(4) (1938), 897–936.Google Scholar
[79]Wiener, N., The homogeneous chaos, Amer. J. Math., 60 (1938), 897–936.Google Scholar
[80]Witteveen, J. and Iaccarino, G., Subcell resolution in simplex stochastic collocation for spatial discontinuities, J. Comput. Phys., 251 (2013), 17–52.Google Scholar
[81]Xiu, D. and Karniadakis, G. E., The Wiener-Askey Polynomial Chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619–644.CrossRefGoogle Scholar
[82]Xiu, D., Lucor, D., Su, C.-H. and Karniadakis, G. E., Stochastic modeling of flow-structure interactions using generalized Polynomial Chaos, J. Fluid Eng., 124(1) (2002), 51–59.Google Scholar
[83]Xiu, Dongbin, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton Univ. Press, 2010.Google Scholar
[84]Yang, R. and Berger, J. O., A Catalog of Noninformative Priors, Institute of Statistics and Decision Sciences, Duke University, 1996.Google Scholar