A comparative study of stochastic collocation methods for flow in spatially correlated random fields

Communications in Computational Physics

Volumn 6, Issue 3, 2009, Pages 509-535

  Chang, Haibin ^a   Zhang, Dongxiao ^b  

^a PEKING UNIVERSITY   (China)

^b University of Southern California   (United States)

Author keywords

Probabilistic collocation method; Smolyak sparse grid method; Stochastic collocation method; Stochastic partial differential equations

Indexed keywords

EID: 67650935289     PISSN: 18152406     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (26)

1. I. Babuska, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. Technical report, MOX, Dipartimento di Matematica, 2005.
2. I. Babuska, R. Tempone and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Number. Anal. 42 (2004), 800-825.
3. I. Babuska, R. Tempone and G. E. Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods Appl. Mech. Engrg., 194(12-16) (2005), 1251-1294.
4. E Ballio and A. Guadagnini, Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology, Water Resour. Res., 40 (2004), W04603, doi: 10.1029/2003WR002876.
5. C. Canuto, M. Y. Hussaini, A. Quateroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.
6. M. K. Deb, I. Babuska and J. T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg., 190 (2001), 6359-6372.
7. Y. Ding, T. Li, D. Zhang and P Zhang, Adaptive Stroud stochastic collocation method for flow in random porous media via Karhunen-Loeve expansion. Commun. Comput. Phys., 4(2008), 102-123.
8. R. Ghanem, Scales of fluctuation and the propagation of uncertainty in random porous media, Water Resour. Res., 34(9) (1998), 2123-2136.
9. R. Ghanem and R Spanos, Stochastic Finite Element: A Spectral Approach, Springer, New York, 1991.
10. A. Keese, A review of recent developments in the numerical solution of stochastic partial differential equations (stochastic finite elements), Institute fur Wissenschaftliches Rechnen, 2003.
11. Y. S. Kumar, J. Li, C. Talarico and J. Wang, A probabilistic collocation method based statistical gate delay model considering process variations and multiple input switching, Proceedings of Design, Automation and Test in Europe, 7-11 March (2005), 770-775.
12. H. Li and D. Zhang, Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resources Research, 43 (2007), W09409, doi:10.1029/2006WR005673.
13. L. Mathelin, M. Y. Hussaini and T. A. Zang, Stochastic approaches to uncertainty quantification in CFD simulations, Numerical Algorithms, 38 (2005), 209-236.
14. I. P. Mysovskih, Proof of the minimality of the number of nodes in the cubature formula for a hypersphere, USSR Comput. Math. Math. Phys., 6 (1966), 15-27.
15. F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Sci. Comput., 46 (2008), 2309-2345.
16. S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl, 4 (1963), 240-243.
17. A. H. Stroud, Remarks on the disposition of points in numerical integration formulas, Mathematical Tables and Other Aids to Computation, 11 (1957), 257-261.
18. M. A. Tatang, W. Pan, R. G. Prinn and G. J. McRae, An efficient method for parametric uncertainty analysis of numerical geophysical models, J. Geophys. Res., 102(D18) (1997), 21,925-21, 931.
19. G. W. Wasilkowski and H. Wozniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complexity, 11 (1995), 1-56.
20. M. Webster, M. A. Tatang and G. J. Mcrae, Application of the probabilistic collocation method for an uncertainty analysis of a simple ocean model, MIT Joint Program on the Science and Policy of Global Change Report Series No. 4, Massachusetts Institute of Technology, 1996.
21. N. Wiener, The homogeneous chaos, Am. J. Math., 60 (1938), 897-936.
22. D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput., 27(3) (2005), 1118-1139.
23. D. Xiu and G. E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Eng., 191 (2002), 4927-4948.
24. D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644.
25. D. Zhang, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, Academic Press, San Diego, CA, 2002.
26. D. Zhang and Z. Lu, An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Loeve and polynomial expansions, J. Comput. Phys., 194 (2004), 773-794.