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Multilevel QMC with Product Weights for Affine-Parametric, Elliptic PDEs

  • Robert N. Gantner
  • Lukas Herrmann
  • Christoph Schwab
Chapter

Abstract

We present an error analysis of higher order Quasi-Monte Carlo (QMC) integration and of randomly shifted QMC lattice rules for parametric operator equations with uncertain input data taking values in Banach spaces. Parametric expansions of these input data in locally supported bases such as splines or wavelets was shown in Gantner et al. (SIAM J Numer Anal 56(1):111–135, 2018) to allow for dimension independent convergence rates of combined QMC-Galerkin approximations. In the present work, we review and refine the results in that reference to the multilevel setting, along the lines of Kuo et al. (Found Comput Math 15(2):441–449, 2015) where randomly shifted lattice rules and globally supported representations were considered, and also the results of Dick et al. (SIAM J Numer Anal 54(4):2541–2568, 2016) in the particular situation of locally supported bases in the parametrization of uncertain input data. In particular, we show that locally supported basis functions allow for multilevel QMC quadrature with product weights, and prove new error vs. work estimates superior to those in these references (albeit at stronger, mixed regularity assumptions on the parametric integrand functions than what was required in the single-level QMC error analysis in the first reference above). Numerical experiments on a model affine-parametric elliptic problem confirm the analysis.

Notes

Acknowledgements

This work was supported in part by the Swiss National Science Foundation (SNSF) under grant SNF 159940 and SNF 149819. The authors thank Fabian Keller for his contribution to the implementation in two space dimensions and Fabian Müller for his help with generating the bisection-tree meshes.

References

  1. 1.
    Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Pelteret, J.P., Turcksin, B., Wells, D.: The deal.II library, version 8.5. J. Numer. Math. (2017).  https://doi.org/10.1515/jnma-2017-0058
  2. 2.
    Babuška, I., Kellogg, R.B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33(4), 447–471 (1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bachmayr, M., Cohen, A., Migliorati, G.: Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. ESAIM Math. Model. Numer. Anal. 51(1), 321–339 (2017)Google Scholar
  4. 4.
    Chen, P., Schwab, Ch.: Model order reduction methods in computational uncertainty quantification. In: Handbook of Uncertainty Quantification, pp. 1–53. Springer International Publishing, Cham (2016)Google Scholar
  5. 5.
    Dashti, M., Stuart, A.: The Bayesian approach to inverse problems. In: Handbook of Uncertainty Quantification, pp. 1–118. Springer International Publishing, Cham (2016)Google Scholar
  6. 6.
    Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dick, J., Kuo, F.Y., Le Gia, Q.T., Nuyens, D., Schwab, Ch.: Higher order QMC Petrov-Galerkin discretization for affine parametric operator equations with random field inputs. SIAM J. Numer. Anal. 52(6), 2676–2702 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dick, J., Kuo, F.Y., Le Gia, Q.T., Schwab, Ch.: Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations. SIAM J. Numer. Anal. 54(4), 2541–2568 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gantner, R.N.: A generic C++ library for multilevel quasi-Monte Carlo. In: Proceedings of the Platform for Advanced Scientific Computing Conference, PASC’16, pp. 11:1–11:12. ACM, New York, NY (2016)Google Scholar
  10. 10.
    Gantner, R.N., Schwab, Ch.: Computational higher order quasi-Monte Carlo integration. In: Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, April 2014, vol. 163, pp. 271–288. Springer, Cham (2016)Google Scholar
  11. 11.
    Gantner, R.N., Herrmann, L., Schwab, Ch.: Quasi-Monte Carlo integration for affine-parametric, elliptic PDEs: local supports and product weights. SIAM J. Numer. Anal. 56(1), 111–135 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gaspoz, F.D., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917–936 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Giles, M.B.: Multilevel Monte Carlo methods. Acta Numer. 24, 259–328 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Graham, I.G., Kuo, F.Y., Nichols, J.A., Scheichl, R., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131(2), 329–368 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Herrmann, L., Schwab, Ch.: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights. Technical Report 2016-39 (revised), Seminar for Applied Mathematics, ETH Zürich (2016)Google Scholar
  16. 16.
    Herrmann, L., Schwab, Ch.: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients. Technical Report 2017-19, Seminar for Applied Mathematics, ETH Zürich, Zürich (2017)Google Scholar
  17. 17.
    Herrmann, L., Schwab, Ch.: QMC algorithms with product weights for lognormal-parametric, elliptic PDEs. Technical Report 2017-04 (revised), Seminar for Applied Mathematics, ETH Zürich, Zürich (2017)Google Scholar
  18. 18.
    Hilber, N., Reichmann, O., Schwab, Ch., Winter, Ch.: Computational methods for quantitative finance. In: Finite Element Methods for Derivative Pricing. Springer Finance. Springer, Heidelberg (2013)Google Scholar
  19. 19.
    Kuo, F.Y., Nuyens, D.: Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation. Found. Comput. Math. 16(6), 1631–1696 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond. ANZIAM J. 53(1), 1–37 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50(6), 3351–3374 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kuo, F.Y., Schwab, Ch., Sloan, I.H.: 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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kuo, F., Scheichl, R., Schwab, Ch., Sloan, I., Ullmann, E.: Multilevel quasi-Monte Carlo methods for lognormal diffusion problems. Math. Comput. 86(308), 2827–2860 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75(254), 903–920 (electronic) (2006)Google Scholar
  25. 25.
    Nuyens, D., Cools, R.: Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complex. 22(1), 4–28 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schwab, Ch., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford Science Publications. The Clarendon Press/Oxford University Press, Oxford/New York (1994)zbMATHGoogle Scholar
  28. 28.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complex. 14(1), 1–33 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40(5), 1650–1665 (2002)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sloan, I.H., Kuo, F.Y., Joe, S.: On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput. 71(240), 1609–1640 (2002)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Robert N. Gantner
    • 1
  • Lukas Herrmann
    • 1
  • Christoph Schwab
    • 1
  1. 1.Seminar for Applied MathematicsETH ZürichZurichSwitzerland

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