Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces

SIAM Journal on Scientific Computing

Volumn 34, Issue 2, 2012, Pages

  Witteveen, Jeroen A S ^a   Iaccarino, Gianluca ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Essentially extremum diminishing; Extrapolation; Nonhypercube; Random sampling; Stochastic collocation

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; PROBABILITY; SAMPLING; STOCHASTIC SYSTEMS;

COMPUTATIONAL PROBLEM; ESSENTIALLY EXTREMUM DIMINISHING; HYPER-CUBES; NONHYPERCUBE; PROBABILITY SPACES; RANDOM SAMPLING; STOCHASTIC COLLOCATION; STOCHASTIC COLLOCATION METHOD; STRUCTURED GRID; UNCERTAINTY QUANTIFICATIONS;

EXTRAPOLATION;

EID: 84861394267     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/100817504     Document Type: Article

References (37)

1. N. Agarwal and N. R. Aluru, A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties, J. Comput. Phys., 228 (2009), pp. 7662-7688.
2. I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), pp. 1005- 1034.
3. I. Babuška, R. Tempone, and G. E. Zouraris, Galerkin finite element approximation of stochastic finite elements, SIAM J. Numer. Anal., 42 (2004), pp. 800-825. (Pubitemid 40645215)
4. P. G. Constantine, D. F. Gleich, and G. Iaccarino, Spectral methods for parameterized matrix equations, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2681-2699.
5. M. Deb, I. M. Babuška, and J. T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6359-6372. (Pubitemid 32984202)
6. M. De Berg, O. Cheong, M. Van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag, Berlin, 2008.
7. J. Foo, X. Wan, and G. E. Karniadakis, The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications, J. Comput. Phys., 227 (2008), pp. 9572-9595.
8. J. Foo and G. E. Karniadakis, Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229 (2010), pp. 1536-1557.
9. B. Ganapathysubramanian and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comput. Phys., 225 (2007), pp. 652-685.
10. R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer- Verlag, New York, 1991.
11. J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, Methuen & Co. Ltd., London, 1964.
12. S. Hosder, R. W. Walters, and R. Perez, A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations, in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2006, AIAA-2006-891.
13. O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, A stochastic projection method for fluid flow, I. Basic formulation, J. Comput. Phys., 173 (2001), pp. 481-511. (Pubitemid 33082573)
14. O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, Uncertainty propagation using Wiener-Haar expansions, J. Comput. Phys., 197 (2004), pp. 28-57. (Pubitemid 38739642)
15. O. P. Le Maître, H. N. Najm, R. G. Ghanem, and O. M. Knio, Multi-resolution analysis of Wiener-type uncertainty propagation schemes, J. Comput. Phys., 197 (2004), pp. 502-531. (Pubitemid 38955945)
16. O. P. Le Maître, M. T. Reagan, H. N. Najm, R. G. Ghanem, and O. M. Knio, A stochastic projection method for fluid flow. II. Random process, J. Comput. Phys., 181 (2002), pp. 9-44.
17. X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228 (2009), pp. 3084-3113.
18. L. Mathelin and O. P. Le Maître, Dual-based a posteriori error estimate for stochastic finite element methods, Commun. Appl. Math. Comput. Sci., 2 (2007), pp. 83-115.
19. H. N. Najm, Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, Annu. Rev. Fluid Mech., 41 (2009), pp. 35-52.
20. F. Nobile, R. Tempone, and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), pp. 2309-2345.
21. F. Nobile, R. Tempone, and C. G. Webster, An anisotropic sparse grid collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), pp. 2411-2442.
22. M. T. Reagan, H. N. Najm, R. G. Ghanem, and O. M. Knio, Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection, Combust. Flame, 132 (2003), pp. 545-555. (Pubitemid 36344128)
23. S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963), pp. 240-243.
24. 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 (1997), pp. 21925-21932.
25. X. Wan and G. E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209 (2005), pp. 617-642. (Pubitemid 40919643)
26. X. Wan and G. E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28 (2006), pp. 901-928. (Pubitemid 46785408)
27. N. Wiener, Nonlinear Problems in Random Theory, MIT Technology Press and John Wiley & Sons, New York, 1958.
28. J. A. S. Witteveen and H. Bijl, A monomial chaos approach for efficient uncertainty quantification in nonlinear problems, SIAM J. Sci. Comput., 30 (2008), pp. 1296-1317.
29. J. A. S. Witteveen and H. Bijl, Effect of randomness on multi-frequency aeroelastic responses resolved by unsteady adaptive stochastic finite elements, J. Comput. Phys., 228 (2009), pp. 7025-7045.
30. J. A. S. Witteveen and H. Bijl, A TVD uncertainty quantification method with bounded error applied to transonic airfoil flutter, Commun. Comput. Phys., 6 (2009), pp. 406-432.
31. J. A. S. Witteveen and H. Bijl, Transonic velocity fluctuations simulated using extremum diminishing uncertainty quantification based on inverse distance weighting, Theor. Comp. Fluid Dyn., in press.
32. J. A. S. Witteveen and G. Iaccarino, Refinement criteria for simplex stochastic collocation with local extremum diminishing robustness, SIAM J. Sci. Comput., in press.
33. J. A. S. Witteveen, A. Loeven, and H. Bijl, An adaptive stochastic finite elements approach based on Newton-Cotes quadrature in simplex elements, Comput. & Fluids, 38 (2009), pp. 1270-1288.
34. D. Xiu, Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), pp. 293-309.
35. D. Xiu, Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys., 5 (2009), pp. 242-272.
36. D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619-644.
37. D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118-1139.