A two-scale nonperturbative approach to uncertainty analysis of diffusion in random composites

Multiscale Modeling and Simulation

Volumn 2, Issue 4, 2004, Pages 662-674

  Xiu, Dongbin ^a   Tartakovsky, Daniel M ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Moment equations; Polynomial chaos; Random domain decomposition; Random fields

Indexed keywords

DECOMPOSITION; DOMAIN DECOMPOSITION METHODS; POROUS MATERIALS;

MOMENT EQUATIONS; PHYSICAL SYSTEMS; POLYNOMIAL CHAOS; POLYNOMIAL CHAOS EXPANSION; RANDOM COMPOSITES; RANDOM DOMAINS; RANDOM FIELDS; SPATIAL ARRANGEMENTS;

UNCERTAINTY ANALYSIS;

EID: 84986296677     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/03060268X     Document Type: Article

References (30)

1. R. Askey and J. Wilson, Some Basic Hypergeometric Polynomials that Generalize Jacobi Polynomials, Mem. Amer. Math. Soc. 319, AMS, Providence, RI, 1995.
2. I. Babuška, R. Tempone, and G. Zouraris, Galerkin Finite Element Approximations of Stochastic Elliptic Differential Equations, Tech. report 02-38, TICAM, Austin, TX, 2002.
3. R. Cameron and W. Martin, The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals, Ann. of Math. (2), 48 (1947), pp. 385-392.
4. J. H. Cushman, The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles, Kluwer Academic Publishers, New York, 1997.
5. G. Dagan, Flow and Transport in Porous Formations, Springer-Verlag, New York, 1989.
6. G. Dagan and S. P. Neuman, eds., Subsurface Flow and Transport: A Stochastic Approach, Cambridge University Press, New York, 1997.
7. M. Deb, I. Babuška, and J. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6359-6372.
8. U. Frisch, Wave propagation in random media, in Probabilistic Methods in Applied Mathematics, Vol. 1, A. T. Bharucha-Reid, ed., Academic Press, New York, 1968, pp. 75-198.
9. L. W. Gelhar, Stochastic Subsurface Hydrology, Prentice-Hall, Englewood Cliffs, NJ, 1993.
10. R. Ghanem, Scales of fluctuation and the propagation of uncertainty in random porous media, Water Resour. Res., 34 (1998), p. 2123.
11. R. Ghanem, Stochastic finite elements for heterogeneous media with multiple random non- Gaussian properties, ASCE J. Eng. Mech., 125 (1999), pp. 26-40.
12. R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
13. A. Guadagnini, L. Guadagnini, D. M. Tartakovsky, and C. L. Winter, Random domain decomposition for flow in heterogeneous stratified aquifers, Stochastic Environ. Res. and Risk Assessment (SERRA), 17 (2003), pp. 394-407.
14. L. Guadagnini, A. Guadagnini, and D. M. Tartakovsky, Probabilistic reconstruction of geologic facies, J. Hydrol., 294 (2004), pp. 57-67.
15. K. D. Jarman and T. F. Russell, Eulerian moment equations for stochastic immiscible flow, Multiscale Model. Simul., 1 (2003), pp. 598-608.
16. R. Koekoek and R. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue, Tech. report 98-17, Department of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1998.
17. H. Ogura, Orthogonal functionals of the Poisson process, IEEE Trans. Information Theory, 18 (1972), pp. 473-481.
18. Y. Rubin, Applied Stochastic Hydrogeology, Oxford University Press, New York, 2003.
19. W. Schoutens, Stochastic Processes in the Askey Scheme, Ph.D. thesis, K. U. Leuven, Leuven, The Netherlands, 1999.
20. D. M. Tartakovsky, A. Guadagnini, and M. Riva, Stochastic averaging of nonlinear flows in heterogeneous porous media, J. Fluid Mech., 492 (2003), pp. 47-62.
21. D. M. Tartakovsky and S. P. Neuman, Extension of "Transient flow in bounded randomly heterogeneous domains 1. Exact conditional moment equations and recursive approximations," Water Resour. Res., 35 (1999), pp. 1921-1925.
22. N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897-936.
23. C. L. Winter and D. M. Tartakovsky, Mean flow in composite porous media, Geophys. Res. Lett., 27 (2000), pp. 1759-1762.
24. C. L. Winter and D. M. Tartakovsky, Groundwater flow in heterogeneous composite aquifers, Water Resour. Res., 38 (2002), doi:10.1029/2001WR000450.
25. C. L. Winter, D. M. Tartakovsky, and A. Guadagnini, Numerical solutions of moment equations for flow in heterogeneous composite aquifers, Water Resour. Res., 38 (2002), doi:10.1029/2001WR000222.
26. D. Xiu and G. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Math. Engrg., 191 (2002), pp. 4927-4948.
27. D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619-644.
28. D. Xiu and G. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187 (2003), pp. 137-167.
29. D. Xiu and G. Karniadakis, A new stochastic approach to transient heat conduction modeling with uncertainty, Inter. J. Heat Mass Trans., 46 (2003), pp. 4681-4693.
30. A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. I. Basic Results, Springer-Verlag, New York, 1987.