Free Access
Issue
ESAIM: M2AN
Volume 51, Number 1, January-February 2017
Page(s) 341 - 363
DOI https://doi.org/10.1051/m2an/2016051
Published online 23 December 2016
  1. 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]
  2. M. Bachmayr, A. Cohen and G. Migliorati, Sparse polynomial approximation of parametric elliptic PDEs. Part I: Affine coefficients, ESAIM: M2AN 51 (2017) 321–339. [CrossRef] [EDP Sciences]
  3. J. Beck, F. Nobile, L. Tamellini and R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Models Methods Appl. Sci. 22 (2012) 1–33. [CrossRef]
  4. J. Beck, F. Nobile, L. Tamellini and R. Tempone, Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients. Comput. Math. Appl. 67 (2014) 732–751. [CrossRef]
  5. J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012) 216–246. [CrossRef]
  6. A. Chkifa, A. Cohen and C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103 (2015) 400–428. [CrossRef]
  7. A. Cohen, Numerical analysis of wavelet methods, Studies in Mathematics and its Applications. Elsevier, Amsterdam (2003).
  8. A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs. Acta Numer. 24 (2015) 1–159. [CrossRef] [MathSciNet]
  9. A. Cohen, R. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic PDEs. Anal. Appl. 9 (2011) 11–47. [CrossRef] [MathSciNet]
  10. M. Dashti and A.M. Stuart, The Bayesian Approach to Inverse Problems. Handbook of Uncertainty Quantification, edited by R. Ghanem, D. Higdon and H. Owhadi. Springer (2015).
  11. R. DeVore, Nonlinear Approximation, Acta Numer. 7 (1998) 51–150. [CrossRef]
  12. O. Ernst and B. Sprungk, Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case, in Sparse Grids and Applications – Munich (2012), edited by J. Garcke and D. Pflüger. Vol. 97 of Lect. Notes Comput. Sci. Eng. Springer International Publishing Switzerland (2014).
  13. 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]
  14. R. Ghanem and P. Spanos, Spectral techniques for stochastic finite elements. Arch. Comput. Methods Engrg. 4 (1997) 63–100. [CrossRef]
  15. R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, 2nd edition, Dover (2007).
  16. C. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2010) 237–263. [CrossRef] [MathSciNet]
  17. I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, Ch. Schwab and I. H. Sloan, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131 (2015) 329–368. [CrossRef] [MathSciNet]
  18. M. Hairer, An Introduction to Stochastic PDEs. Lecture notes. Available at http://www.hairer.org (2009).
  19. V.H. Hoang and C. Schwab, N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. M3AS 24 (2014) 797–826.
  20. O. Knio and O. Le Maitre, Spectral Methods for Uncertainty Quantication: With Applications to Computational Fluid Dynamics. Springer (2010).
  21. F.Y. Kuo, R. Scheichl, Ch. Schwab, I.H. Sloan and E. Ullmann, Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems, arXiv:1507.01090, to appear in Math. of Comp. (2015).
  22. A. Mugler and H.-J. Starkloff, On the convergence of the stochastic Galerkin methods for random elliptic partial differential equations. ESAIM: M2AN 47 (2013) 1237–1263. [CrossRef] [EDP Sciences]
  23. D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press (2010).

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