Free Access
Issue
ESAIM: M2AN
Volume 46, Number 6, November-December 2012
Page(s) 1485 - 1508
DOI https://doi.org/10.1051/m2an/2012013
Published online 13 June 2012
  1. I. Babuška and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191 (2002) 4093–4122. [CrossRef] [MathSciNet] [Google Scholar]
  2. I.M. 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 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  3. I.M. 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 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  4. A. Barinka, Fast Evaluation Tools for Adaptive Wavelet Schemes. Ph.D. thesis, RWTH Aachen (2005). [Google Scholar]
  5. M. Bieri and C. Schwab, Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 198 (2009) 1149–1170. [CrossRef] [MathSciNet] [Google Scholar]
  6. M. Bieri, R. Andreev and C. Schwab, Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput. 31 (2009/2010) 4281–4304. [CrossRef] [MathSciNet] [Google Scholar]
  7. P. Binev, W. Dahmen and R.A. DeVore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Chkifa, A. Cohen, R. DeVore and C. Schwab, Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. Technical Report 44, SAM, ETHZ (2011). [Google Scholar]
  9. A. Cohen, W. Dahmen and R.A. DeVore, Adaptive wavelet methods for elliptic operator equations : convergence rates. Math. Comput. 70 (2001) 27–75 (electronic). [Google Scholar]
  10. A. Cohen, W. Dahmen and R.A. DeVore, Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math. 2 (2002) 203–245. [CrossRef] [MathSciNet] [Google Scholar]
  11. A. Cohen, R.A. DeVore and C. Schwab, Convergence rates of best -term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10 (2010) 615–646. [CrossRef] [MathSciNet] [Google Scholar]
  12. A. Cohen, R. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Anal. Appl. (Singap.) 9 (2011) 11–47. [CrossRef] [MathSciNet] [Google Scholar]
  13. S. Dahlke, M. Fornasier and T. Raasch, Adaptive frame methods for elliptic operator equations. Adv. Comput. Math. 27 (2007) 27–63. [CrossRef] [MathSciNet] [Google Scholar]
  14. S. Dahlke, T. Raasch, M. Werner, M. Fornasier and R. Stevenson, Adaptive frame methods for elliptic operator equations : the steepest descent approach. IMA J. Numer. Anal. 27 (2007) 717–740. [CrossRef] [MathSciNet] [Google Scholar]
  15. M.K. Deb, I.M. Babuška and J.T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190 (2001) 6359–6372. [CrossRef] [MathSciNet] [Google Scholar]
  16. T.J. Dijkema, C. Schwab and R. Stevenson, An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constr. Approx. 30 (2009) 423–455. [CrossRef] [MathSciNet] [Google Scholar]
  17. W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. [CrossRef] [MathSciNet] [Google Scholar]
  18. P. Frauenfelder, C. Schwab and R.A. Todor, Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194 (2005) 205–228. [CrossRef] [MathSciNet] [Google Scholar]
  19. T. Gantumur, H. Harbrecht and R. Stevenson, An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput. 76 (2007) 615–629 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  20. W. Gautschi, Orthogonal polynomials : computation and approximation, in Numer. Math. Sci. Comput. Oxford University Press, Oxford Science Publications, New York (2004). [Google Scholar]
  21. R.G. Ghanem and P.D. Spanos, Stochastic finite elements : a spectral approach. Springer-Verlag, New York (1991). [Google Scholar]
  22. C.J. Gittelson, Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations. Ph.D. thesis, ETH Dissertation No. 19533. ETH Zürich (2011). [Google Scholar]
  23. C.J. Gittelson, An adaptive stochastic Galerkin method for random elliptic operators. Math. Comput. (2011). To appear. [Google Scholar]
  24. C.J. Gittelson, Convergence Rates of Multilevel and Sparse Tensor Approximations for a Random Elliptic PDE (2012). Submitted. [Google Scholar]
  25. I.G. Graham, F.Y. Kuo, D. Nuyens, R. Scheichl and I.H. Sloan, Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230 (2011) 3668–3694. [CrossRef] [Google Scholar]
  26. R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras I, Elementary theory, Reprint of the 1983 original, in Graduate Studies in Mathematics. Amer. Math. Soc. 15 (1997). [Google Scholar]
  27. H.G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194 (2005) 1295–1331. [CrossRef] [MathSciNet] [Google Scholar]
  28. A. Metselaar, Handling Wavelet Expansions in Numerical Methods. Ph.D. thesis, University of Twente (2002). [Google Scholar]
  29. P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466–488 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  30. F. Nobile, R. Tempone and C.G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2411–2442. [CrossRef] [MathSciNet] [Google Scholar]
  31. W. Rudin, Functional analysis, 2nd edition. International Series in Pure Appl. Math. McGraw-Hill Inc., New York (1991). [Google Scholar]
  32. C. Schwab and C.J. Gittelson, Sparse tensor discretization of high-dimensional parametric and stochastic PDEs. Acta Numer. 20 (2011) 291–467. [CrossRef] [MathSciNet] [Google Scholar]
  33. R. Stevenson, Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal. 41 (2003) 1074–1100 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  34. M.H. Stone, The generalized Weierstrass approximation theorem. Math. Mag. 21 (1948) 237–254. [CrossRef] [Google Scholar]
  35. G. Szegő, Orthogonal polynomials, 4th edition, in Colloq. Publ. XXIII. Amer. Math. Soc. (1975). [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. X. Wan and G.E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209 (2005) 617–642. [CrossRef] [Google Scholar]
  38. X. Wan and G.E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (2006) 901–928 (electronic). [CrossRef] [Google Scholar]
  39. X. Wan and G.E. Karniadakis, Solving elliptic problems with non-Gaussian spatially-dependent random coefficients. Comput. Methods Appl. Mech. Eng. 198 (2009) 1985–1995. [CrossRef] [Google Scholar]
  40. D. Xiu, Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2 (2007) 293–309. [Google Scholar]
  41. D. Xiu, Numerical methods for stochastic computations : A spectral method approach. Princeton University Press, Princeton, NJ (2010). [Google Scholar]
  42. D. Xiu and J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118–1139 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  43. D. Xiu and G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002) 619–644 (electronic). [CrossRef] [MathSciNet] [Google Scholar]

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