Free Access
Volume 54, Number 4, July-August 2020
Page(s) 1259 - 1307
Published online 18 June 2020
  1. 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] [Google Scholar]
  2. V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids. Multivariate polynomial interpolation. Adv. Comput. Math. 12 (2000) 273–288. [Google Scholar]
  3. J. Beck, R. Tempone, F. Nobile and L. Tamellini, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Models Methods Appl. Sci. 22 (2012) 1250023, 33. [Google Scholar]
  4. H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numer. 13 (2004) 147–269. [CrossRef] [MathSciNet] [Google Scholar]
  5. J.-P. Calvi and P.V. Manh, Lagrange interpolation at real projections of Leja sequences for the unit disk. Proc. Amer. Math. Soc. 140 (2012) 4271–4284. [CrossRef] [Google Scholar]
  6. J.-P. Calvi and M. Van Phung, On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation. J. Approx. Theory 163 (2011) 608–622. [Google Scholar]
  7. S.B. Chae, Holomorphy and Calculus in Normed Spaces. With an appendix by Angus E. Taylor. In: Vol. 92 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (1985). [Google Scholar]
  8. M.A. Chkifa, On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection. J. Approx. Theory 166 (2013) 176–200. [Google Scholar]
  9. A. Chkifa, A. Cohen, R. DeVore, C. Schwab, Adaptive algorithms for sparse polynomial approximation of parametric and stochastic elliptic pdes. ESAIM: M2AN 47 (2013) 253–280. [CrossRef] [EDP Sciences] [Google Scholar]
  10. A. Cohen, R. 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]
  11. 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]
  12. A. Cohen, A. Chkifa and C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric pdes. J. Math. Pures Appl. 103 (2015) 400–428. [Google Scholar]
  13. A. Cohen, C. Schwab and J. Zech, Shape holomorphy of the stationary Navier-Stokes equations. SIAM J. Math. Anal. 50 (2018) 1720–1752. [CrossRef] [Google Scholar]
  14. J. Dick, F.Y. Kuo, Q.T.L. Gia, D. Nuyens and C. Schwab, Higher order QMC Petrov-Galerkin discretization for affine parametric operator equations with random field inputs. SIAM J. Numer. Anal. 52 (2014) 2676–2702. [Google Scholar]
  15. J. Dick, F.Y. Kuo and I.H. Sloan, High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22 (2013) 133–288. [CrossRef] [Google Scholar]
  16. R.N. Gantner and C. Schwab, Computational higher order quasi-Monte Carlo integration, In: Vol. 163 Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014 (2016) 271–288. [Google Scholar]
  17. T. Gerstner and M. Griebel, Numerical integration using sparse grids. Numer. Algorithms 18 (1998) 209–232. [Google Scholar]
  18. T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature. Computing 71 (2003) 65–87. [CrossRef] [MathSciNet] [Google Scholar]
  19. I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, C. Schwab and I.H. Sloan, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131 (2015) 329–368. [Google Scholar]
  20. M. Griebel and J. Oettershagen, On tensor product approximation of analytic functions. J. Approx. Theory 207 (2016) 348–379. [Google Scholar]
  21. A.-L. Haji-Ali, F. Nobile, L. Tamellini and R. Tempone, Multi-index stochastic collocation for random PDEs. Comput. Methods Appl. Mech. Eng. 306 (2016) 95–122. [Google Scholar]
  22. A.-L. Haji-Ali, H. Harbrecht, M. Peters and M. Siebenmorgen, Novel results for the anisotropic sparse grid quadrature. J. Complexity 47 (2018) 62–85. [CrossRef] [Google Scholar]
  23. H. Harbrecht, M. Peters and M. Siebenmorgen, Analysis of the domain mapping method for elliptic diffusion problems on random domains. Numer. Math. 134 (2016) 823–856. [Google Scholar]
  24. M. Hervé, Analyticity in Infinite-dimensional Spaces. In: Vol. 10 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (1989). [Google Scholar]
  25. R. Hiptmair, L. Scarabosio, C. Schillings and C. Schwab, Large deformation shape uncertainty quantification in acoustic scattering. Adv. Comput. Math. 44 (2018) 1475–1518. [Google Scholar]
  26. V.H. Hoang and C. Schwab, N-term Wiener chaos approximation rate for elliptic PDEs with lognormal Gaussian random inputs. Math. Models Methods Appl. Sci. 24 (2014) 797–826. [Google Scholar]
  27. C. Jerez-Hanckes, C. Schwab and J. Zech, Electromagnetic wave scattering by random surfaces: shape holomorphy. Math. Mod. Meth. Appl. Sci. 27 (2017) 2229–2259. [CrossRef] [Google Scholar]
  28. F.Y. Kuo and D. Nuyens, Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation. Found. Comput. Math. 16 (2016) 1631–1696. [CrossRef] [Google Scholar]
  29. F. Kuo, R. Scheichl, C. Schwab, I. Sloan and E. Ullmann, Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems. Technical Report 2015–22, Seminar for Applied Mathematics, ETH Zürich (2016). [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. [Google Scholar]
  31. H. Robbins, A remark on Stirling’s formula. Amer. Math. Monthly 62 (1955) 26–29. [Google Scholar]
  32. C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems. Inverse Prob. 29 (2013) 065011. [CrossRef] [Google Scholar]
  33. C. Schwab and C. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic pdes. Acta Numer. 20 (2011) 291–467. [CrossRef] [MathSciNet] [Google Scholar]
  34. C. Schwab and A.M. Stuart, Sparse deterministic approximation of Bayesian inverse problems. Inverse Prob. 28 (2012) 045003. [CrossRef] [Google Scholar]
  35. S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4 (1963) 240–243. [Google Scholar]
  36. R.A. Todor and C. Schwab, Convergence rates of sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 44 (2007) 232–261. [CrossRef] [MathSciNet] [Google Scholar]
  37. J. Zech, Sparse-grid approximation of high-dimensional parametric PDEs. Ph.D. thesis. Dissertation 25683, ETH Zürich (2018). doi: 10.3929/ethz-b-000340651. [Google Scholar]
  38. J. Zech, D. Dung and C. Schwab, Multilevel approximation of parametric and stochastic PDEs. Math. Models Methods Appl. Sci. 29 (2019) 1753–1817. [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