Multi-element generalized polynomial chaos for bitrary probability measures

SIAM Journal on Scientific Computing

Volumn 28, Issue 3, 2006, Pages 901-928

  Wan, Xiaoliang ^a   Karniadakis, George Em ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Polynomial chaos; Random inputs; Stochastic differential equations; Uncertainty

Indexed keywords

CHAOS THEORY; DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; PROBABILITY DENSITY FUNCTION; PROBLEM SOLVING; RANDOM PROCESSES;

POLYNOMIAL CHAOS; RANDOM INPUTS; STOCHASTIC DIFFERENTIAL EQUATIONS;

POLYNOMIAL APPROXIMATION;

EID: 33749326978     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/050627630     Document Type: Article

References (34)

1. I. BABUŠKA, R. TEMPONE, AND G. E. ZOURARIS, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), pp. 800-825.
2. D. BOLEY AND G. H. GOLUB, A survey of matrix inverse eigenvalue problems, Inverse Problems, 3 (1987), pp. 595-622.
3. R. CAMERON AND W. MARTIN, The orthogonal development of non-linear functionals in series of Fourier-Hermite functional, Ann. of Math. (2), 48 (1947), pp. 385-392.
4. M. K. DEB, I. 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.
5. H. J. FISCHER, On the condition of orthogonal polynomials via modified moments, Z. Anal. Anwendungen, 15 (1996), pp. 1-18.
6. H. J. FISCHER, On generating orthogonal polynomials for discrete measures, Z. Anal. Anwendungen, 17 (1998), pp. 183-205.
7. P. FRAUENFELDER, C. SCHWAB, AND R. A. TODOR, Finite elements for elliptic problems with stochastic coefficients, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 205-228.
8. W. GAUTSCHI, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput., 3 (1982), pp. 289-317.
9. W. GAUTSCHI, Algorithm 726: ORTHPOL - A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software, 20 (1994), pp. 21-62.
10. N. K. GHADDAR, G. E. KARNIADAKIS, AND A. T. PATERA, A conservative isoparametric spectral element method for forced convection: Application to fully developed flow in periodic geometries, Num. Heat Transfer, 9 (1986), pp. 277-300.
11. N. K. GHADDAR, M. MAGEN, B. B. MIKIC, AND A. T. PATERA, Numerical investigation of incompressible flow in grooved channels. II. Resonance and oscillatory heat-transfer enhancement, J. Fluid Mech., 168 (1986), pp. 541-567.
12. R. G. GHANEM, Ingredients for a general purpose stochastic finite element formulation, Com put. Methods Appl. Mech. Engrg., 168 (1999), pp. 19-34.
13. R. G. GHANEM, Stochastic finite elements for heterogeneous media with multiple random non-Gaussian properties, ASCE J. Engrg. Mech., 125 (1999), pp. 26-40.
14. R. GHANEM AND P. SPANOS, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
15. R. D. HENDERSON, Dynamic refinement algorithms for spectral element methods, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 395-411.
16. C. HERMAN AND E. KANG, Heat transfer enhancement in a grooved channel with curved vanes, Int. J. Heat Mass Transfer, 45 (2002), pp. 3741-3757.
17. G. KARNIADAKIS AND S. SHERWIN, Spectral/hp Element Methods for CFD, Oxford University Press, New York, 1999.
18. O. P. LE MAITRE, H. N. NJAM, R. G. GHANEM, AND O. M. KNIO, Uncertainty propagation using Wiener-Haar expansions, J. Comput. Phys., 197 (2004), pp. 28-57.
19. O. P. LE MAITRE, H. N. NJAM, R. G. GHANEM, AND O. M. KNIO, Multi-resolution analysis of Wiener-type uncertainty propagation schemes, J. Comput. Phys., 197 (2004), pp. 502-531.
20. H. G. MATTHIES AND C. G. BUCKER, Finite elements for stochastic media problems, Comput. Methods Appl. Mech. Engrg., 168 (1999), pp. 3-17.
21. H. G. MATTHIES AND A. KEESE, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 1295-1331.
22. C. MAVRIPLIS, Adaptive mesh strategies for the spectral element method, Comput. Methods Appl. Mech. Engrg., 116 (1994), pp. 77-86.
23. S. E. NOTARIS, Interpolatory quadrature formulae with Chebyshev abscissae, J. Comput. Appl. Math., 133 (2001), pp. 507-517.
24. S. A. ORSZAG AND L. R. BISSONNETTE, Dynamical properties of truncated Wiener-H ermite expansions, Physics Fluids, 10 (1967), pp. 2603-2613.
25. C. SCHWAB, p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1998.
26. G. SZEGÖ, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1959.
27. X. WAN, Multi-Element Generalized Polynomial Chaos for Differential Equations with Random Imputs: Algorithms and Applications, Ph.D. thesis, Division of Applied Mathematics, Brown University, Providence, RI, in preparation.
28. 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.
29. X. WAN AND G. E. KARNIADAKIS, Long-term behavior of polynomial chaos in stochastic flow simulations, Comput. Methods Appl. Mech. Engrg., in press.
30. X. WAN, D. XIU, AND G. E. KARNIADAKIS, Stochastic solutions for the two-dimensional advection-diffusion equation, SIAM J. Sci. Comput., 26 (2004), pp. 578-590.
31. N. WIENER, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897-936.
32. D. XIU AND G. E. KARNIADAKIS, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619-644.
33. D. XIU AND G. E. KARNIADAKIS, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187 (2003), pp. 137-167.
34. D. XIU AND G. E. KARNIADAKIS, A new stochastic approach to transient heat conduction modeling with uncertainty, Int. J. Heat Mass Transfer, 46 (2003), pp. 4681-4693.