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QMC Galerkin Discretization of Parametric Operator Equations

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Monte Carlo and Quasi-Monte Carlo Methods 2012

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 65))

Abstract

We extend recent results from Kuo et al. (SIAM J Numer Anal 50:3351–3374, 2012) of QMC quadrature and Finite Element discretization for parametric, scalar second order elliptic partial differential equations to general QMC-Galerkin discretizations of parametric operator equations, which depend on possibly countably many parameters. Such problems typically arise in the numerical solution of differential and integral equations with random field inputs. The present setting covers general second order elliptic equations which are possibly indefinite (Helmholtz equation), or which are given in saddle point variational form (such as mixed formulations). They also cover nonsymmetric variational formulations which appear in space-time Galerkin discretizations of parabolic problems or countably parametric nonlinear initial value problems (Hansen and Schwab, Vietnam J. Math 2013, to appear).

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References

  1. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  2. Charrier, J.: Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 216–246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen, A., DeVore, R., Schwab, Ch.: Convergence rates of best N-term Galerkin approximation for a class of elliptic sPDEs., Found. Comput. Math. 10, 615–646 (2010).

    Google Scholar 

  4. Cools, R., Kuo, F.Y., Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput. 28, 2162–2188 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dick, J.: On the convergence rate of the component-by-component construction of good lattice rules. J. Complexity 20, 493–522 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge, UK (2010)

    MATH  Google Scholar 

  7. Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Liberating the weights. J. Complexity 20, 593–623 (2004)

    Article  MATH  Google Scholar 

  8. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, Heidelberg (2004)

    Google Scholar 

  9. 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 log-normal random coefficients. Report 2013–14, Seminar for Applied Mathematics, ETH Zurich, http://www.sam.math.ethz.ch/sam_reports/index.php?id=2013-14 (in review)

  10. Hansen, M., Schwab, Ch.: Analytic regularity and best N-term approximation of high dimensional parametric initial value problems. Vietnam J. Math. 41/2, 181–215 (2013)

    Google Scholar 

  11. Hansen, M., Schwab, Ch.: Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs. Math. Nachr. (2013, to appear)

    Google Scholar 

  12. Hoang, V.H., Schwab, Ch.: Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs: analytic regularity and gpc-approximation. Report 2010-11, Seminar for Applied Mathematics, ETH Zürich. SIAM J. Math. Anal. 45/5, 3050–3083 (2013)

    Google Scholar 

  13. Hoang, V.H., Schwab, Ch.: Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs. Report 2010-19, Seminar for Applied Mathematics, ETH Zürich. Anal. Appl. (Singapore) 10(3) (2012)

    Google Scholar 

  14. Hoang, V.H., Schwab, Ch.: Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs. Anal. Appl. (Singapore) (2013, to appear)

    Google Scholar 

  15. Kunoth, A., Schwab, Ch.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo methods for very high dimensional integration: the standard weighted-space setting and beyond. ANZIAM J. 53, 1–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kuo, F.Y., Schwab, Ch., Sloan, I.H.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Corrigendum for Quasi-Monte Carlo methods for very high dimensional integration: the standard weighted-space setting and beyond. ANZIAM J. (2013, to appear)

    Google Scholar 

  19. Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Multi-level Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient. Report 2012-25, Seminar für Angewandte Mathematik, ETH Zürich (in review)

    Google Scholar 

  20. Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Waterhouse, B.J.: Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands. J. Complexity 26, 135–160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Woźniakowski, H.: Liberating the dimension. J. Complexity 26, 422–454 (2010)

    Article  MATH  Google Scholar 

  22. Kuo, F.Y., Wasilkowski, G.W., Waterhouse, B.J.: Randomly shifted lattice rules for unbounded integrals. J. Complexity 22, 630–651 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Larcher, G., Leobacher, G., Scheicher, K.: On the tractability of the Brownian bridge algorithm. J. Complexity 19, 511–528 (2004)

    Article  MathSciNet  Google Scholar 

  24. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  25. Nistor, V., Schwab, Ch.: High order Galerkin approximations for parametric second order elliptic partial differential equations. Report 2012-21 Seminar for Applied Mathematics, ETH Zürich. Math. Method Model Appl. Sci. (2013, to appear)

    Google Scholar 

  26. Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp. 75, 903–920 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schwab, Ch., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011). Cambridge University Press

    Google Scholar 

  28. Schwab, Ch., Todor, R.A.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217, 100–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)

    MATH  Google Scholar 

  30. Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40, 1650–1665 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sloan, I.H., Wang, X., Woźniakowski, H.: Finite-order weights imply tractability of multivariate integration. J. Complexity 20, 46–74 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sloan, I.H., Woźniakowski, H.: When are Quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14, 1–33 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, X.: Strong tractability of multivariate integration using Quasi-Monte Carlo algorithms. Math. Comp. 72, 823–838 (2002)

    Article  Google Scholar 

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Acknowledgements

The author is supported by ERC under Grant AdG 247277.

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Authors and Affiliations

  1. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

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  1. Christoph Schwab
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Correspondence to Christoph Schwab .

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Editors and Affiliations

  1. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Josef Dick

  2. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Frances Y. Kuo

  3. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Gareth W. Peters

  4. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Ian H. Sloan

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Schwab, C. (2013). QMC Galerkin Discretization of Parametric Operator Equations. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_32

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