Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

IMA Journal of Numerical Analysis

Volumn 27, Issue 2, 2007, Pages 232-261

  Todor, Radu Alexandru ^a   Schwab, Christoph ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

Karh nen Lo ve expansion; Partial differential equations with stochastic coefficients; Polynomial chaos; Sparse tensor product approximation

Indexed keywords

BOUNDARY VALUE PROBLEMS; CHAOS THEORY; DIFFUSION; POLYNOMIAL APPROXIMATION; STOCHASTIC MODELS;

DETERMINISTIC PROBLEMS; ELLIPTIC BOUNDARY VALUE PROBLEM; INFINITE DIMENSIONAL; POLYNOMIAL CHAOS; PRODUCT PROBABILITY SPACE; REGULARITY ASSUMPTION; SPARSE TENSOR PRODUCTS; STOCHASTIC COEFFICIENTS;

STOCHASTIC SYSTEMS;

EID: 34047123580     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drl025     Document Type: Article

References (19)

1. BABUŠKA, I., TEMPONE, R. & ZOURARIS, G. E. (2004) Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42, 800-825 (electronic).
2. DAVIS, P. J. (1963) Interpolation and Approximation. New York: Blaisdell Publishing Co.
3. DETTINGER, M. D. & WILSON, J. L. (1981) First order analysis of uncertainty in numerical models of ground-water flow, 1, mathematical development. Water Resour. Res., 17, 149-161.
4. FRAUENFELDER, P., SCHWAB, C. & TODOR, R. A. (2005) Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng., 194, 205-228.
5. GHANEM, R. G. & SPANOS, P. D. (1997) Spectral techniques for stochastic finite elements. Arch. Comput. Methods Eng., 4, 63-100.
6. KELLER, J. B. (1964) Stochastic equations and wave propagation in random media. Proc. Sympos. Appl. Math., vol. XVI. (R. Bellmann ed.). Providence, RI: American Mathematical Society, pp. 145-170.
7. KLEIBER, M. & HIEN, T. D. (1992) The Stochastic Finite Element Method. Chichester, UK: John Wiley.
8. KÖNIG, H. (1986) Eigenvalue Distribution of Compact Operators. Operator Theory: Advances and Applications, vol. 16. Basel, Switzerland: Birkhäuser.
9. LOÈVE, M. (1977) Probability Theory. I. 4th edn. Graduate Texts in Mathematics, vol. 45. New York: Springer.
10. LOÈVE, M. (1978) Probability Theory. 11. 4th edn. Graduate Texts in Mathematics, vol. 46. New York: Springer.
11. MATTHIES, H. G. & KEESE, A. (2005) Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng., 194, 1295-1331.
12. PIETSCH, A. (1987) Eigenvalues and s-Numbers. Cambridge Studies in Advanced Mathematics, vol. 13. Cambridge: Cambridge University Press.
13. PINKUS, A. (1985) n-Widths in Approximation Theory. New York: Springer.
14. SCHOUTENS, W. (2000) Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, vol. 146. New York: Springer.
15. SCHWAB, C. & TODOR, R. A. (2006) Karhúnen-Loève approximation of random fields in domains by generalized fast multipole methods. Journ. Comp. Phys., 217, 100-122.
16. TODOR, R. A. (2006) Robust eigenvalue computation for smoothing operators. SIAM J. Numer. Anal. (to appear).
17. TODOR, R. A. (2005) Sparse perturbation algorithms for elliptic PDE's with stochastic data. Ph.D. Thesis No. 16192, ETH Zurich, Switzerland.
18. WIENER, N. (1938) The homogeneous chaos. Am. J. Math., 60, 897-936.
19. XIU, D. & KARNIADAKIS, G. E. (2002) The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24, 619-644 (electronic).