EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 47 / No 5 (September-October 2013) ESAIM: M2AN, 47 5 (2013) 1237-1263 References
Free Access
Issue
ESAIM: M2AN
Volume 47, Number 5, September-October 2013
Page(s) 1237 - 1263
DOI https://doi.org/10.1051/m2an/2013066
Published online 09 July 2013
  1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. Dover Publications, Inc, New York (1965). [Google Scholar]
  2. I. Babuška, F. Nobile and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data. SIAM J. Numer. Anal. 45 (2007) 1005–1034. [CrossRef] [MathSciNet] [Google Scholar]
  3. 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]
  4. 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]
  5. M. Bieri, R. Andreev and C. Schwab, Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput. 31 (2009) 4281–4304. [CrossRef] [MathSciNet] [Google Scholar]
  6. M. Bieri and C. Schwab, Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1149–1170. [CrossRef] [MathSciNet] [Google Scholar]
  7. A. Bobrowski, Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge UK (2005). [Google Scholar]
  8. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed. Texts in Appl. Math., vol. 15. Springer-Verlag, New York (2002). [Google Scholar]
  9. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg (2006). [Google Scholar]
  10. A. Cohen, R. DeVore and C. Schwab, Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs. Foundations Comput. Math. 10 (2010) 615–646. [Google Scholar]
  11. 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) 6359–6372. [CrossRef] [MathSciNet] [Google Scholar]
  12. M. Eiermann, O.G. Ernst and E. Ullmann, Computational aspects of the stochastic finite element method. Comput. Visualiz. Sci. 10 (2007) 3–15. [Google Scholar]
  13. O. Ernst, A. Mugler, E. Ullmann and H.J. Starkloff, On the convergence of generalized polynomial chaos. ESAIM: M2AN 46 (2012) 317–339. [Google Scholar]
  14. R.V. Field and M. Grigoriu, Convergence Properties of Polynomial Chaos Approximations for L2-Random Variables, Sandia Report SAND2007-1262 (2007). [Google Scholar]
  15. P. Frauenfelder, C. Schwab and R.A. Todor, Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194 (2005) 205–228. [CrossRef] [MathSciNet] [Google Scholar]
  16. R.A. Freeze, A Stochastical-Conceptual Analysis of One-Dimensional Groundwater Flow in Nonuniform Homogeneous Media. Water Resources Research (1975) 725–741. [Google Scholar]
  17. J. Galvis and M. Sarkis, Approximating infinity–dimensional stochastic Darcy‘s Equations without uniform ellipticity. SIAM J. Numer. Anal. 47 (2009) 3624–3651. [CrossRef] [MathSciNet] [Google Scholar]
  18. J. Galvis and M. Sarkis, Regularity results for the ordinary product stochastic pressure equation, to appear in SIAM J. Math. Anal. (preprint 2011) 1–31. [Google Scholar]
  19. R. Ghanem, Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Engrg. 168 (1999) 19–34. [CrossRef] [MathSciNet] [Google Scholar]
  20. R. Ghanem, Stochastic Finite Elements with Multiple Random Non-Gaussian Properties. J. Engrg. Mech. 125 (1999) 26–40. [CrossRef] [Google Scholar]
  21. R. Ghanem and S. Dham, Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media. Transport in Porous Media 32 (1998) 239–262. [CrossRef] [MathSciNet] [Google Scholar]
  22. R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). [Google Scholar]
  23. C.J. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2010) 237–263. [Google Scholar]
  24. E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case. J. Computat. Appl. Math. 84 (1997) 257–275. [CrossRef] [Google Scholar]
  25. E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications. Amer. Math. Soc. 31 (1957). [Google Scholar]
  26. O. Kallenberg, Foundations of modern probability. Springer-Verlag, Berlin (2001). [Google Scholar]
  27. O.P. Le Maître and O.M. Knio, Spectral Methods for Uncertainty Quantification. Scientific Computation: With Applications to Computational Fluid Dynamics. Springer-Verlag (2010). [Google Scholar]
  28. M. Loève, Probability Theory II. 4th Edition. Springer-Verlag, New York, Heidelberg, Berlin (1978). [Google Scholar]
  29. D. Lucor, C. H Su and G.E. Karniadakis, Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Engrg. 60 (2004) 571–596. [CrossRef] [Google Scholar]
  30. P. Maroni and Z. Rocha, Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algor. 47 (2008) 291–314. [CrossRef] [Google Scholar]
  31. H.G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1295–1331. [Google Scholar]
  32. A. Mugler and H.J. Starkloff, On elliptic partial differential equations with random coefficients. Stud. Univ. Babes-Bolyai Math. 56 (2011) 473–487. [MathSciNet] [Google Scholar]
  33. A. Narayan and J.S. Hesthaven, Computation of connection coefficients and measure modifications for orthogonal polynomials. BIT Numer. Math. (2011). [Google Scholar]
  34. W. Nowak, Geostatistical Methods for the Identification of Flow and Transport Parameters in the Subsurface, Ph.D. Thesis. Universität Stuttgart (2005). [Google Scholar]
  35. C. Schwab and C.J. Gittelson, Sparse tensor discretizations of high–dimensional parametric and stochastic PDEs. Acta Numer. 20 (2011) 291–467. [CrossRef] [MathSciNet] [Google Scholar]
  36. 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]
  37. H. Umegaki and A.T. Bharucha-Reid, Banach Space-Valued Random Variables and Tensor Products of Banach Spaces. J. Math. Anal. Appl. 31 (1970) 49–67. [CrossRef] [MathSciNet] [Google Scholar]
  38. X. Wan and G.E. Karniadakis, Beyond Wiener Askey Expansions: Handling Arbitrary PDFs. J. Scient. Comput. 27 (2006) 455–464. [Google Scholar]
  39. N. Wiener, Homogeneous Chaos. Amer. J. Math. 60 (1938) 897–936. [Google Scholar]
  40. D. Xiu, Numerical methods for stochastic computations: A spectral method approach. Princeton Univ. Press, Princeton and NJ (2010). [Google Scholar]
  41. 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]
  42. 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]
  43. D. Xiu and G.E. Karniadakis, A new stochastic approach to transient heat conduction modeling with uncertainty. Inter. J. Heat and Mass Transfer 46 (2003) 4681–4693. [CrossRef] [Google Scholar]
  44. D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003) 137–167. [Google Scholar]
  45. D. Xiu, D. Lucor, C. H Su and G.E. Karniadakis, Performance Evaluation of Generalized Polynomial Chaos, Computational Science – ICCS 2003, edited by P.M.A. Sloot, D. Abramson, A.V. Bogdanov, J.J. Dongarra, Albert Y. Zomaya and Y.E. Gorbachev, Lect. Notes Comput. Sci., vol. 2660. Springer Verlag (2003). [Google Scholar]
  46. D. Zhang, Stochastic Methods for Flow in Porous Media. Coping with Uncertainties. Academic Press, San Diego, CA (2002). [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