Free Access
Volume 46, Number 2, November-December 2012
Page(s) 317 - 339
Published online 12 October 2011
  1. M. Arnst, R. Ghanem and C. Soize, Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys. 229 (2010) 3134–3154. [CrossRef] [MathSciNet] [Google Scholar]
  2. I. Babuška, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800–825. [CrossRef] [MathSciNet] [Google Scholar]
  3. I. Babuška, R. Tempone and G.E. Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1251–1294. [CrossRef] [MathSciNet] [Google Scholar]
  4. C. Berg, Moment problems and polynomial approximation. Ann. Fac. Sci. Toulouse Math. (Numéro spécial Stieltjes) 6 (1996) 9–32. [CrossRef] [Google Scholar]
  5. C. Berg and J.P.R. Christensen, Density questions in the classical theory of moments. Ann. Inst. Fourier 31 (1981) 99–114. [CrossRef] [MathSciNet] [Google Scholar]
  6. A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005). [Google Scholar]
  7. R.H. Cameron and W.T. Martin, The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals. Ann. Math. 48 (1947) 385–392. [CrossRef] [Google Scholar]
  8. T.S. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978). [Google Scholar]
  9. J.H. Curtiss, A note on the theory of moment generating functions. Ann. Stat. 13 (1942) 430–433. [CrossRef] [Google Scholar]
  10. B.J. Debusschere, H.N. Najm, Ph.P. Pébay, O.M. Knio, R.G. Ghanem and O.P. le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698–719. [CrossRef] [MathSciNet] [Google Scholar]
  11. R.V. Field Jr. and M. Grigoriu, On the accuracy of the polynomial chaos expansion. Probab. Engrg. Mech. 19 (2004) 65–80. [CrossRef] [Google Scholar]
  12. G. Freud, Orthogonal Polynomials. Akademiai, Budapest (1971). [Google Scholar]
  13. W. Gautschi, Orthogonal Polynomials: Computation and Approximation. Oxford University Press (2004). [Google Scholar]
  14. R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). [Google Scholar]
  15. A. Gut, On the moment problem. Bernoulli 8 (2002) 407–421. [MathSciNet] [Google Scholar]
  16. T. Hida, Brownian Motion. Springer, New York (1980). [Google Scholar]
  17. K. Itô, Multiple Wiener integral. J. Math. Soc. Jpn 3 (1951) 157–169. [Google Scholar]
  18. S. Janson, Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997). [Google Scholar]
  19. O. Kallenberg, Foundations of Modern Probability, 2nd edition. Springer-Verlag, New York (2002). [Google Scholar]
  20. G. Kallianpur, Stochastic Filtering Theory. Springer, New York (1980). [Google Scholar]
  21. G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edition. Oxford University Press (2005). [Google Scholar]
  22. G.E. Karniadakis, C.-H. Shu, D. Xiu, D. Lucor, C. Schwab and R.-A. Todor, Generalized polynomial chaos solution for differential equations with random inputs. Technical Report 2005-1, Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland (2005). [Google Scholar]
  23. A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933). [Google Scholar]
  24. G.D. Lin, On the moment problems. Stat. Probab. Lett. 35 (1997) 85–90 [Google Scholar]
  25. . Correction: G.D. Lin, On the moment problems. Stat. Probab. Lett. 50 (2000) 205. [Google Scholar]
  26. P. Masani, Wiener’s contributions to generalized harmonic analysis, prediction theory and filter theory. Bull. Amer. Math. Soc. 72 (1966) 73–125. [CrossRef] [MathSciNet] [Google Scholar]
  27. P.R. Masani, Norbert Wiener, 1894–1964. Number 5 in Vita mathematica, Birkhäuser (1990). [Google Scholar]
  28. H.G. Matthies and C. Bucher, Finite elements for stochastic media problems. Comput. Methods Appl. Mech. Engrg. 168 (1999) 3–17. [CrossRef] [MathSciNet] [Google Scholar]
  29. A. Mugler and H.-J. Starkloff, On elliptic partial differential equations with random coefficients, Stud. Univ. Babes-Bolyai Math. 56 (2011) 473–487. [Google Scholar]
  30. A.T. Patera, A spectral element method for fluid dynamics – laminar flow in a channel expansion. J. Comput. Phys. 54 (1984) 468–488. [CrossRef] [Google Scholar]
  31. R.E.A.C. Payley and N. Wiener, Fourier Transforms in the Complex Domain. Number XIX in Colloquium Publications. Amer. Math. Soc. (1934). [Google Scholar]
  32. L.C. Petersen, On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand. 51 (1982) 361–366. [MathSciNet] [Google Scholar]
  33. M. Reed and B. Simon, Methods of modern mathematical physics, Functional analysis 1. Academic press, New York (1972). [Google Scholar]
  34. M. Riesz, Sur le problème des moments et le théorème de Parseval correspondant. Acta Litt. Ac. Scient. Univ. Hung. 1 (1923) 209–225. [Google Scholar]
  35. R.A. Roybal, A reproducing kernel condition for indeterminacy in the multidimensional moment problem. Proc. Amer. Math. Soc. 135 (2007) 3967–3975. [CrossRef] [MathSciNet] [Google Scholar]
  36. I.E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956) 106–134. [CrossRef] [MathSciNet] [Google Scholar]
  37. A.N. Shiryaev, Probability. Springer-Verlag, New York (1996). [Google Scholar]
  38. I.C. Simpson, Numerical integration over a semi-infinite interval using the lognormal distribution. Numer. Math. 31 (1978) 71–76. [CrossRef] [MathSciNet] [Google Scholar]
  39. C. Soize and R. Ghanem, Physical systems with random uncertainties: Chaos representations with arbitrary probability measures. SIAM J. Sci. Comput. 26 (2004) 395–410. [Google Scholar]
  40. H.-J. Starkloff, On the number of independent basic random variables for the approximate solution of random equations, in Celebration of Prof. Dr. Wilfried Grecksch’s 60th Birthday, edited by C. Tammer and F. Heyde. Shaker Verlag, Aachen (2008) 195–211. [Google Scholar]
  41. J.M. Stoyanov, Counterexamples in Probability, 2nd edition. John Wiley & Sons Ltd., Chichester, UK (1997). [Google Scholar]
  42. G. Szegö, Orthogonal Polynomials. American Mathematical Society, Providence, Rhode Island (1939). [Google Scholar]
  43. R.-A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232–261. [CrossRef] [MathSciNet] [Google Scholar]
  44. N. Wiener, Differential space. J. Math. Phys. 2 (1923) 131–174. [Google Scholar]
  45. N. Wiener, Generalized harmonic analysis. Acta Math. 55 (1930) 117–258. [CrossRef] [MathSciNet] [Google Scholar]
  46. N. Wiener, The homogeneous chaos. Amer. J. Math. 60 (1938) 897–936. [Google Scholar]
  47. D. Xiu and J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118–1139. [CrossRef] [MathSciNet] [Google Scholar]
  48. D. Xiu and G.E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927–4948. [Google Scholar]
  49. D. Xiu and G.E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002) 619–644. [CrossRef] [MathSciNet] [Google Scholar]
  50. D. Xiu and G.E. Karniadakis, A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Trans. 46 (2003) 4681–4693. [Google Scholar]
  51. D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phy. 187 (2003) 137–167. [Google Scholar]
  52. D. Xiu, D. Lucor, C.-H. Su and G.E. Karniadakis, Stochastic modeling of flow-structure interactions using generalized polynomial chaos. J. Fluids Eng. 124 (2002) 51–59. [CrossRef] [Google Scholar]
  53. D. Xiu, D. Lucor, C.-H. Su and G.E. Karniadakis, Performance evaluation of generalized polynomial chaos, in Computational Science – ICCS 2003, Lecture Notes in Computer Science 2660, edited by P.M.A. Sloot, D. Abramson, A.V. Bogdanov, J.J. Dongarra, A.Y. Zomaya and Y.E. Gorbachev. Springer-Verlag (2003). [Google Scholar]
  54. Y. Xu, On orthogonal polynomials in several variables, in Special functions, q-series, and related topics, edited by M. Ismail, D.R. Masson and M. Rahman. Fields Institute Communications 14 (1997) 247–270. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you