An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

Journal of Computational Physics

Volumn 228, Issue 8, 2009, Pages 3084-3113

  Ma, Xiang ^a   Zabaras, Nicholas ^a  

^a School of Civil and Environmental Engineering   (United States)

Author keywords

Adaptive sparse grid; Collocation; Discontinuities; Hierarchical multiscale method; Smolyak algorithm; Sparse grid; Stochastic partial differential equations

Indexed keywords

DOMAIN DECOMPOSITION METHODS; FINITE ELEMENT METHOD; ORDINARY DIFFERENTIAL EQUATIONS; POLYNOMIALS; STOCHASTIC SYSTEMS;

ADAPTIVE SPARSE GRID; COLLOCATION; DISCONTINUITY; HIERARCHICAL MULTISCALE METHOD; MULTISCALE METHOD; SMOLYAK ALGORITHMS; SPARSE GRID; STOCHASTIC PARTIAL DIFFERENTIAL EQUATION; STOCHASTICS;

MONTE CARLO METHODS;

EID: 61349191533     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.01.006     Document Type: Article

References (43)

1. Iman R.L., and Conover W.J. Small sample sensitivity analysis techniques for computer models, with an application to risk assessment. Commun. Statist. A9 (1980) 1749-1842
2. Ghanem R., and Spanos P.D. Stochastic Finite Elements: A Spectral Approach (1991), Springer-Verlag, New York
3. Wiener N. The homogeneous chaos. Am. J. Math. 60 (1938) 897-936
4. Ghanem R. Probabilistic characterization of transport in heterogeneous media. Comput. Meth. Appl. Mech. Eng. 158 (1998) 199-220
5. Ghanem R., and Sarkar A. Mid-frequency structural dynamics with parameter uncertainty. Comput. Meth. Appl. Mech. Eng. 191 (2002) 5499-5513
6. Ghanem R. Higher order sensitivity of heat conduction problems to random data using the spectral stochastic finite element method. ASME J. Heat Transfer 121 (1999) 290-299
7. Xiu D., and Karniadakis G.E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002) 619-644
8. Xiu D., and Karniadakis G.E. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Meth. Appl. Mech. Eng. 191 (2002) 4927-4948
9. Xiu D., and Karniadakis G.E. Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003) 137-167
10. Xiu D., and Karniadakis G.E. A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Transfer 46 (2003) 4681-4693
11. Babuska I., Tempone R., and Zouraris G.E. Galerkin finite elements approximation of stochastic finite elements. SIAM J. Numer. Anal. 42 (2004) 800-825
12. Deb M.K., Babuska I.K., and Oden J.T. Solution of stochastic partial differential equations using the Galerkin finite element techniques. Comput. Meth. Appl. Mech. Eng. 190 (2001) 6359-6372
13. Babuska I., Tempone R., and Zouraris G.E. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Meth. Appl. Mech. Eng. 194 (2005) 1251-1294
14. Velamur Asokan B., and Zabaras N. Using stochastic analysis to capture unstable equilibrium in natural convection. J. Comput. Phys. 208 (2005) 134-153
15. Le Mai{dotless}̂tre O.P., Najm H.N., Ghanem R.G., and Knio O.M. Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197 (2004) 28-57
16. Le Mai{dotless}̂tre O.P., Najm H.N., Ghanem R.G., and Knio O.M. Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197 (2004) 502-531
17. Wan X., and Karniadakis G.E. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209 (2005) 617-642
18. Wan X., and Karniadakis G.E. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (2006) 901-928
19. Wan X., and Karniadakis G.E. Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Meth. Appl. Math. Eng. 195 (2006) 41-43
20. Babuska I., Nobile F., and Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. ISIAM J. Numer. Anal. 45 (2007) 1005-1034
21. Smolyak S. Quadrature and interpolation formulas for tensor product of certain classes of functions. Soviet Math. Dokl. 4 (1963) 240-243
22. Gerstner T., and Griebel M. Numerical integration using sparse grids. Numer. Algorithms 18 (1998) 209-232
23. Barthelmann V., Novak E., and Ritter K. High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12 (2000) 273-288
24. Klimke A., and Wohlmuth B. Algorithm 847: spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Trans. Math. Software 31 (2005)
25. A. Klimke, Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids, Ph.D. Thesis, Universität Stuttgart, Shaker Verlag, Aachen, 2006.
26. Griebel M. Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing 61 (1998) 151-179
27. Bungartz H.J., and Griebel M. Sparse grids. Acta Numerica 13 (2004) 147-269
28. Xiu D., and Hesthaven J.S. High order collocation methods for the differential equation with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118-1139
29. Xiu D. Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2 (2007) 293-309
30. Nobile F., Tempone R., and Webster C. A sparse grid collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2008) 2309-2345
31. Foo J., Wan X., and Karniadakis G.E. The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys. 227 (2008) 9572-9595
32. D.L. Donoho, Interpolating wavelet transforms, preprint, Stanford University, 1992.
33. Bertoluzza S., and Naldi G. A wavelet collocation method for the numerical solution of partial differential equations. Appl. Comput. Harmon. Anal. 3 (1996) 1-9
34. Vasilyev O.V., and Paolucci S. A dynamically adaptive multilevel wavelet collocation method for solving differential equations in a finite domain. J. Comput. Phys. 125 (1996) 498-512
35. Vasilyev O.V., and Paolucci S. A fast adaptive wavelet collocation algorithm for multidimensional PDEs. J. Comput. Phys. 138 (1997) 16-56
36. Grestner T., and Griebel M. Dimension adaptive tensor product quadrature. Computing 71 (2003) 65-87
37. Ganapathysubramanian B., and Zabaras N. Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225 (2007) 652-685
38. Nobile F., Tempone R., and Webster C. An anisotropic sparse grid collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2411-2442
39. M. Griebel, Sparse grids and related approximation schemes for high dimensional problems, in: Proceedings of the Conference on Foundations of Computational Mathematics, Santander, Spain, 2005.
40. Oksendal B. Stochastic Differential Equations: An introduction with applications (1998), Springer-Verlag, New York
41. Bungartz H.-J., and Dirnstorfer S. Multivariate quadrature on adaptive sparse grids. Computing 71 (2003) 89-114
42. A. Klimke, Sparse Grid Interpolation Toolbox - User's Guide, IANS report 2006/001, University of Stuttgart, 2006.
43. Ma X., and Zabaras N. A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media. J. Comput. Phys. 227 (2008) 8448-8471