Abstract
In this chapter we review methods for formulating partial differential equations based on the random field representations outlined in Chap. 2 These include the stochastic Galerkin method, which is the predominant choice in this book, as well as other methods that frequently occur in the literature, e.g., stochastic collocation methods and spectral projection. We also briefly discuss methods that are not polynomial chaos methods themselves but are viable alternatives.
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References
Abgrall R (2008) A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems. Rapport de recherche. http://hal.inria.fr/inria-00325315/en/
Abgrall R, Congedo PM, Corre C, Galera S (2010) A simple semi-intrusive method for uncertainty quantification of shocked flows, comparison with a non-intrusive polynomial chaos method. In: ECCOMAS CFD, Lisbon
Babuška IM, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005–1034
Bäck J, Nobile F, Tamellini L, Tempone R (2011) Implementation of optimal Galerkin and collocation approximations of PDEs with random coefficients. ESAIM Proc 33:10–21. doi:10.1051/proc/201133002, http://dx.doi.org/10.1051/proc/201133002
Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element: a non intrusive approach by regression. Eur J Comput Mech 15:81–92
Blatman G, Sudret B (2008) Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus Mécanique 336(6):518–523
Clenshaw CW, Curtis AR (1960) A method for numerical integration on an automatic computer. Numerische Mathematik 2:197
Doostan A, Owhadi H (2011) A non-adapted sparse approximation of PDEs with stochastic inputs. J Comput Phys 230(8):3015–3034. doi:10.1016/j.jcp.2011.01.002, http://dx.doi.org/10.1016/j.jcp.2011.01.002
Ganapathysubramanian B, Zabaras N (2007) Sparse grid collocation schemes for stochastic natural convection problems. J Comput Phys 225(1):652–685. doi:10.1016/j.jcp.2006.12.014, http://dx.doi.org/10.1016/j.jcp.2006.12.014
Gautschi W (1982) On generating orthogonal polynomials. SIAM J Sci Stat Comput 3:289–317. doi:10.1137/0903018
Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York
Golub GH, Welsch JH (1967) Calculation of Gauss quadrature rules. Tech rep, Stanford
Hosder S, Walters R, Balch M (2007) Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: AIAA-2007-1939, 9th AIAA non-deterministic approaches conference, Honolulu
Keese A, Matthies H (2003) Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations. Tech rep, Institute of Scientific Computing TU Braunschweig, Brunswick
Mathelin L, Hussaini MY (2003) A stochastic collocation algorithm for uncertainty analysis. Tech Rep 2003-212153, NASA Langley Research Center
Mathelin L, Hussaini MY, Zang TA, Bataille F (2003) Uncertainty propagation for turbulent, compressible flow in a quasi-1D nozzle using stochastic methods. In: AIAA-2003-4240, 16TH AIAA CFD conference, Orlando, pp 23–26
Migliorati G, Nobile F, von Schwerin E, Tempone R (2013) Approximation of quantities of interest in stochastic PDEs by the random discrete L 2 projection on polynomial spaces. SIAM J Sci Comput 35(3):A1440–A1460
Reagan MT, Najm HN, Ghanem RG, Knio OM (2003) Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combust Flame 132(3):545–555
Tuminaro RS, Phipps ET, Miller CW, Elman HC (2011) Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int J Uncertain Quantif 1(1):19–33
Wan X, Karniadakis GE (2005) An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J Comput Phys 209:617–642. doi:http://dx.doi.org/10.1016/j.jcp.2005.03.023, http://dx.doi.org/10.1016/j.jcp.2005.03.023
Wan X, Karniadakis GE (2006) Long-term behavior of polynomial chaos in stochastic flow simulations. Comput Methods Appl Math Eng 195:5582–5596
Wan X, Karniadakis GE (2006) Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J Sci Comput 28(3):901–928. doi:10.1137/050627630, http://dx.doi.org/10.1137/050627630
Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2(2):293–309
Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, Princeton. http://www.worldcat.org/isbn/9780691142128
Xiu D, Hesthaven JS (2005) High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput 27:1118–1139. doi:10.1137/040615201, http://dl.acm.org/citation.cfm?id=1103418.1108714
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Pettersson, M.P., Iaccarino, G., Nordström, J. (2015). Polynomial Chaos Methods. In: Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-10714-1_3
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