This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on the decomposition of the solution into its mean and a random field using a generic Karhunen-Loève expansion. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spatial dimensions that are spectrally represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos bases for the stochastic dimension and eigenfunction bases for the spatial dimension. Dynamic orthogonality is used to derive closed-form equations for the time evolution of mean, spatial and the stochastic fields. The resultant system of equations consists of a partial differential equation (PDE) that defines the dynamic evolution of the mean, a set of PDEs to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian inference for efficient exploration of the posterior distribution. The efficacy of the proposed method is investigated for calibration of a 2D transient diffusion simulator with an uncertain source location and diffusivity. The computational efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.
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BayarriM. Berger J. Paulo R. Sacks J. Cafeo J. Cavendish J. Lin C. and Tu J. (2007). A framework for validation of computer models Technometrics 49(2): 138-153.
Besag J. Green P. Higdon D. and Mengersen K. (1995). Bayesian computation and stochastic systems Statistical Science 10(1): 3-41.
Bieri M. and Schwab C. (2009). Sparse high order FEM for elliptic sPDEs Computer Methods in Applied Mechanics and Engineering 198(13-14): 1149-1170.
Bortz A. Kalos M. and Lebowitz J. (1975). A new algorithm for Monte Carlo simulation in Ising spin systems Journal of Computational Physics 17(1): 10-18.
Cai B. Meyer R. and Perron F. (2008). Metropolis-Hastings algorithms with adaptive proposals Statistics and Computing 18(4): 421-433.
Cameron R. and Martin W. (1947). The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals The Annals of Mathematics 48(2): 385-392.
Cheng M. Hou T. and Zhang Z. (2013a). A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations. I: Derivation and algorithms Journal of Computational Physics 242(1): 843-868.
Cheng M. Hou T. and Zhang Z. (2013b). A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations. II: Adaptivity and generalizations Journal of Computational Physics 242(1): 753-776.
Cotter S. Dashti M. Robinson J. and Stuart A. (2012). Variational data assimilation using targeted random walks International Journal for Numerical Methods in Fluids 68(4).
Cotter S. Roberts G. Stuart A. and White D. (2013). MCMC methods for functions: Modifying old algorithms to make them faster Statistical Science 28(3): 424-446.
Dumbser M. and Munz C.-D. (2007). On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes International Journal of Applied Mathematics and Computer Science 17(3): 297-310 DOI: 10.2478/v10006-007-0024-1.
Gamerman D. and Lopes H. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference Chapman and Hall/CRC Boca Raton FL.
Ghanem R. and Red-Horse J. (1999). Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach Physica D 133(1-4): 137-144.
Ghanem R. and Spanos P. (1991). Spectral stochastic finite-element formulation for reliability analysis Journal of Engineering Mechanics 117(10): 2351-2372.
Ghanem R. and Spanos P. (2003). Stochastic Finite Elements: A Spectral Approach Dover Publications New York NY.
Gilks W. Roberts G. and Sahu S. (1998). Adaptive Markov chain Monte Carlo through regeneration Journal of American Statistical Association 93(443): 1045-1054.
Goldstein M. and Rougier J. (2005). Probabilistic formulations for transferring inferences from mathematical models to physical systems SIAM Journal of Scientific Computing 26(2): 467-487.
Hastings W. (1970). Monte Carlo sampling methods using Markov chains and their applications Biometrika 57(1): 97-109.
Higdon D. Kennedy M. Cavendish J. Cafeo J. and Ryne R. (2005). Combining field data and computer simulations for calibration and prediction SIAM Journal of Scientific Computing 26(2): 448-446.
Hoang V. Schwab C. and Stuart A. (2013). Complexity analysis of accelerated MCMC methods for Bayesian inversion Inverse Problems 29(8): 085010
Janiszowski K. and Wnuk P. (2016). Identification of parameteric models with a priori knowledge of process properties International Journal of Applied Mathematics and Computer Science 26(4): 767-776 DOI: 10.1515/amcs-2016-0054.
Kamiński M. (2015). Symbolic computing in probabilistic and stochastic analysis International Journal of Applied Mathematics and Computer Science 25(4): 961-973 DOI: 10.1515/amcs-2015-0069.
Karczewska A. Pozmej P. Szczeci´nski M. and Boguniewicz B. (2016). A finite element method for extended KdV equations International Journal of Applied Mathematics and Computer Science 26(3): 555-567 DOI: 10.1515/amcs-2016-0039.
Kelly D. and Smith C. (2009). Bayesian inference in probabilistic risk assessment-the current state of the art Reliability Engineering and System Safety 94(2): 628-643.
Kennedy M. and O’Hagan A. (2001). Bayesian calibration of computer models Journal of the Royal Statistical Society B: Statistical Methodology 63(3): 425-464.
Knio O. and Maitre O. (2006). Uncertainty propagation in CFD using polynomial chaos decomposition Fluid Dynamics Research 38(9): 616-640.
Lucor D. Xiu D. Su C. and Karniadakis G. (2003). Predictability and uncertainty in CFD International Journal for Numerical Methods in Fluids 43(5): 483-505.
Marzouk Y. and Najm H. (2007). Stochastic spectral methods for efficient Bayesian solution of inverse problems Journal of Computational Physics 224(2): 560-586.
Marzouk Y. and Najm H. (2009). Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems Journal of Computational Physics 228(6): 18621902.
Mathelin L. Hussaini M. Zang T. and Bataille F. (2004). Uncertainty propagation for a turbulent compressible nozzle flow using stochastic methods AIAA Journal 42(8): 1669-1676.
Mehta U. (1991). Some aspects of uncertainty in computational fluid dynamics results Journal of Fluid Engineering 113(4): 538-543.
Mehta U. (1996). Guide to credible computer simulations of fluid flows Journal of Propulsion and Power 12(5): 940-948.
Metropolis N. Rosenbluth A. Rosenbluth M. Teller A. and Teller E. (1953). Equation of state calculations by fast computing machines The Journal of Chemical Physics 21(6): 1087-1092.
Narayanan V. and Zabaras N. (2004). Stochastic inverse heat conduction using spectral approach International Journal for Numerical Methods in Engineering 60(9): 1569-1593.
Oberkampf W. DeLand S. Rutherford B. Diegert K. and Alvin K. (2002). Error and uncertainty in modeling and simulation Reliability Engineering and System Safety 75(3): 335-357.
O’Hagan A. (2006). Bayesian analysis of computer code outputs: A tutorial Reliability Engineering and System Safety 91(10-11): 1290-1300.
Oreskes N. Shrader-Frechett K. and Belitz K. (1994). Verification validation and confirmation of numerical models in earth sciences Science 263(5147): 641-647.
Paulo R. (2005). Default priors for Gaussian processes The Annals of Statistics 33(2): 556-582.
Poette G. Despres B. and Lucor D. (2009). Uncertainty quantification for systems of conservation laws Journal of Computational Physics 228(7): 2443-2467.
Sapsis T. and Lermusiaux P. (2009). Dynamically orthogonal field equations for continuous stochastic dynamical systems Physica D 238: 2347-2360.
Sapsis T. and Lermusiaux P. (2012). Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty Physica D 241(1): 60-76.
Schwab C. and Gittelson C. (2011). Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs Acta Numerica 20: 291-467.
Schwab C. and Stuart A. (2012). Sparse deterministic approximation of Bayesian inverse problems Inverse Problems 28(4): 1-32.
Tagade P. and Choi H.-L. (2012). An efficient Bayesian calibration approach using dynamically biorthogonal field equations ASME International Design Engineering Technical Conference and Computers and Information in Engineering Conference Chicago IL USA pp. 873-882.
Tagade P. and Choi H.-L. (2013). A generalized polynomial chaos-based method for efficient Bayesian calibration of uncertain computational models Inverse Problems in Science and Engineering 22(4): 602-624.
Tagade P. and Sudhakar K. (2011). Inferencing component maps of gas turbine engine using Bayesian framework Journal of Propulsion and Power 27(1): 94-104.
Tagade P. Sudhakar K. and Sane S. (2009). Bayesian framework for calibration of gas turbine simulator Journal of Propulsion and Power 25(4): 987-992.
Trucano T. Swiler L. Igusa T. Oberkampf W. and M. P. (2006). Calibration validation and sensitivity analysis: What’s what Reliability Engineering and System Safety 91(10-11): 1331-1357.
Venturi D. (2011). A fully symmetric nonlinear biorthogonal decomposition theory for random fields Physica D 240(4-5): 415-425.
Wiener N. (1938). The homogeneous chaos American Journal of Mathematics 60(4): 897-936.
Wiener N. (1958). Nonlinear Problems in Random Theory John Wiley&Sons New York NY.
Xiu D. and Karniadakis E. (2002). The Weiner-Askey polynomial chaos for stochastic differential equations SIAM Journal of Scientific Computing 24(2): 619-644.
Xiu D. and Karniadakis G. (2003). Modeling uncertainty in flow simulations via generalized polynomial chaos Journal of Computational Physics 187(1): 137-167.
Zaidi A. Ould Bouamama B. and Tagina M. (2012). Bayesian reliability models of Weibull systems: State of the art International Journal of Applied Mathematics and Computer Science 22(3): 585-600 DOI: 10.2478/v10006-012-0045-2.