Abstract
We review recent results on dimension-robust higher order convergence rates of Quasi-Monte Carlo Petrov-Galerkin approximations for response functionals of infinite-dimensional, parametric operator equations which arise in computational uncertainty quantification.
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References
A. Chkifa, A. Cohen, C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103, 400–428 (2015)
A. Cohen, R. DeVore, C. Schwab, Convergence rates of best N-term Galerkin approximation for a class of elliptic sPDEs. Found. Comput. Math. 10, 615–646 (2010)
A. Cohen, R. DeVore, C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. Appl. 9, 1–37 (2011)
J. Dick, Walsh spaces containing smooth functions and Quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008)
J. Dick, F. Pillichshammer, Digital Nets and Sequences (Cambridge University Press, Cambridge, 2010)
J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens, C. Schwab, Higher order QMC Galerkin discretization for parametric operator equations. SIAM J. Numer. Anal. 52(6), 2676–2702 (2014)
J. Dick, F.Y. Kuo, Q.T. Le Gia, C. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations. Research Report 2014–14, SAM, ETH Zürich, 2014. Available at arXiv:1406.4432
R. Gantner, C. Schwab, Computational higher order Quasi-Monte Carlo integration. Report 2014–25, Seminar for Applied Mathematics, ETH Zürich, 2014 (to appear in Proc. MCQMC2014, Springer Publ., 2015)
T. Goda, J. Dick, Construction of interlaced scrambled polynomial lattice rules of arbitrary high order. Found. Comput. Math. (2015). doi:10.1007/s10208-014-9226-8
M. Hansen, C. Schwab, Analytic regularity and best N-term approximation of high dimensional, parametric initial value problems. Vietnam J. Math. 41(2), 181–215 (2013)
A. Kunoth, C. Schwab, Analytic regularity and GPC approximation for stochastic control problems constrained by linear parametric elliptic and parabolic PDEs. SIAM J. Control Optim. 51, 2442–2471 (2013)
F.Y. Kuo, C. Schwab, I.H. Sloan, Quasi-Monte Carlo methods for very high dimensional integration: the standard weighted-space setting and beyond. ANZIAM J. 53, 1–37 (2011)
F.Y. Kuo, C. Schwab, I.H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient. SIAM J. Numer. Anal. 50, 3351–3374 (2012)
F.Y. Kuo, C. Schwab, I.H. Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic PDEs with random coefficients. Found. Comput. Math. 15(2), 411–449 (2015)
S. Mishra, C. Schwab, J. Sukys, Multi-Level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws. Lecture Notes in Computational Science and Engineering, vol. 92. SAM Report 2012-08 (2013), pp. 225–294
D. Nuyens, R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75, 903–920 (2006)
C. Schwab, QMC Galerkin discretizations of parametric operator equations, in Monte Carlo and Quasi-Monte Carlo Methods 2012, ed. by J. Dick, F.Y. Kuo, G. W. Peters, I.H. Sloan (Springer, Berlin, 2013), pp. 613–630
Acknowledgements
Josef Dick is the recipient of an Australian Research Council Queen Elizabeth II Fellowship (project number DP1097023). Quoc T. Le Gia was supported partially by the ARC Discovery Grant DP120101816. The work of Christoph Schwab was supported in part by European Research Council AdG grant STAHDPDE 247277, and the Swiss National Science Foundation under Grant No. 200021-149819.
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Dick, J., Le Gia, Q.T., Schwab, C. (2015). Higher Order Quasi Monte-Carlo Integration in Uncertainty Quantification. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_41
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DOI: https://doi.org/10.1007/978-3-319-19800-2_41
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