References
- Bayarri,M., 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.