Free Access
Volume 54, Number 2, March-April 2020
Page(s) 649 - 677
Published online 03 March 2020
  1. B. Arras, M. Bachmayr and A. Cohen, Sequential sampling for optimal weighted least squares approximations in hierarchical spaces. Preprint arXiv:1805.10801 (2018). [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. [Google Scholar]
  3. T. Bagby, L. Bos and N. Levenberg, Multivariate simultaneous approximation. Constr. Approx. 18 (2002) 569. [Google Scholar]
  4. A. Chkifa, A. Cohen, G. Migliorati, F. Nobile and R. Tempone, Discrete least squares polynomial approximation with random evaluations – application to parametric and stochastic elliptic PDEs. ESAIM: M2AN 49 (2015) 815–837. [CrossRef] [EDP Sciences] [Google Scholar]
  5. 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. [Google Scholar]
  6. A. Cohen and G. Migliorati, Optimal weighted least-squares methods. Preprint arXiv:1608.00512 (2016). [Google Scholar]
  7. A. Cohen, R. Devore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. App. 9 (2011) 11–47. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Cohen, M.A. Davenport and D. Leviatan, On the stability and accuracy of least squares approximations. Found. Comput. Math. 13 (2013) 819–834. [CrossRef] [MathSciNet] [Google Scholar]
  9. M.K. Deb, I.M. Babuška and J. Tinsley Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190 (2001) 6359–6372. [Google Scholar]
  10. R.A. DeVore, Nonlinear approximation. Acta Numer. 7 (1998) 51–150. [CrossRef] [Google Scholar]
  11. D. Dũng, V.N. Temlyakov and T. Ullrich, Hyperbolic cross approximation. Preprint arXiv:1601.03978 (2016). [Google Scholar]
  12. J.E. Gentle, Random number generation and Monte Carlo methods, 2nd edition. In: Statistics and Computing. Springer, New York (2003). [Google Scholar]
  13. T. Gerstner and M. Griebel, Dimension–adaptive tensor–product quadrature. Computing 71 (2003) 65–87. [CrossRef] [MathSciNet] [Google Scholar]
  14. M.B. Giles, Multilevel monte carlo path simulation. Oper. Res. 56 (2008) 607–617. [Google Scholar]
  15. M. Griebel and C. Rieger, Reproducing kernel Hilbert spaces for parametric partial differential equations. SIAM/ASA J. Uncertainty Quant. 5 (2017) 111–137. [CrossRef] [Google Scholar]
  16. A.-L. Haji-Ali, F. Nobile, L. Tamellini and R. Tempone, Multi-index stochastic collocation convergence rates for random PDEs with parametric regularity. Found. Comput. Math. 16 (2016) 1555–1605. [CrossRef] [Google Scholar]
  17. 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]
  18. J. Hampton and A. Doostan, Coherence motivated sampling and convergence analysis of least squares polynomial chaos regression. Comput. Methods Appl. Mech. Eng. 290 (2015) 73–97. [Google Scholar]
  19. H. Harbrecht, M. Peters and M. Siebenmorgen, Multilevel accelerated quadrature for PDEs with log-normally distributed diffusion coefficient. SIAM/ASA J. Uncertainty Quant. 4 (2016) 520–551. [CrossRef] [Google Scholar]
  20. M. Hegland, Adaptive sparse grids. ANZIAM J. 44 (2003) 335–353. [CrossRef] [Google Scholar]
  21. S. Heinrich, Multilevel Monte Carlo methods. In: International Conference on Large-Scale Scientific Computing. Springer (2001) 58–67. [Google Scholar]
  22. F. Kuo, R. Scheichl, C. Schwab, I. Sloan and E. Ullmann, Multilevel quasi-Monte Carlo methods for lognormal diffusion problems. Math. Comput. 86 (2017) 2827–2860. [Google Scholar]
  23. O. Le Matre and O. Knio, Spectral Methods for Uncertainty Quantification. Springer (2010). [CrossRef] [Google Scholar]
  24. E. Levin and D.S. Lubinsky, Christoffel functions, orthogonal polynomials, and Nevai’s conjecture for Freud weights. Constr. Approx. 8 (1992) 463–535. [Google Scholar]
  25. J.S. Liu, Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Stat. Comput. 6 (1996) 113–119. [Google Scholar]
  26. J.S. Liu, Monte Carlo Strategies in Scientific Computing. Springer Science & Business Media (2008). [Google Scholar]
  27. G. Mastroianni and V. Totik, Weighted polynomial inequalities with doubling and a weights. Constr. Approx. 16 (2000) 37–71. [Google Scholar]
  28. G. Migliorati, F. Nobile and R. Tempone, Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points. J. Multivariate Anal. 142 (2015) 167–182. [CrossRef] [Google Scholar]
  29. A. Narayan, J. Jakeman and T. Zhou, A Christoffel function weighted least squares algorithm for collocation approximations. Math. Comput. 86 (2017) 1913–1947. [Google Scholar]
  30. P. Nevai, T. Erdélyi and A.P. Magnus, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM J. Math. Anal. 25 (1994) 602–614. [CrossRef] [Google Scholar]
  31. F. Nobile, R. Tempone and S. Wolfers, Sparse approximation of multilinear problems with applications to kernel-based methods in UQ. Numer. Math. 139 (2018) 247–280. [Google Scholar]
  32. A. Quarteroni, Some results of Bernstein and Jackson type for polynomial approximation in Lp-spaces. Jpn J. Appl. Math. 1 (1984) 173–181. [CrossRef] [MathSciNet] [Google Scholar]
  33. G. Szegö, Orthogonal polynomials, 4th edition. In: Vol. XXIII of American Mathematical Society, Colloquium Publications. American Mathematical Society, Providence, RI (1975). [Google Scholar]
  34. J.A. Tropp, User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12 (2012) 389–434. [CrossRef] [Google Scholar]

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