Advertisement

Foundations of Computational Mathematics

, Volume 16, Issue 6, pp 1555–1605 | Cite as

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

  • Abdul-Lateef Haji-Ali
  • Fabio Nobile
  • Lorenzo Tamellini
  • Raúl TemponeEmail author
Article

Abstract

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.

Keywords

Multi-level Multi-index Stochastic Collocation Infinite dimensional integration Elliptic partial differential equations with random coefficients Finite element method Uncertainty quantification Random partial differential equations Multivariate approximation Sparse grids Stochastic Collocation methods Multi-level methods Combination technique 

Mathematics Subject Classification

41A10 (approx by polynomials) 65C20 (models, numerical methods) 65N30 (Finite elements) 65N05 (Finite differences) 

Notes

Acknowledgments

F. Nobile and L. Tamellini received support from the Center for ADvanced MOdeling Science (CADMOS) and partial support by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media”. L. Tamellini also received support from the Gruppo Nazionale Calcolo Scientifico - Istituto Nazionale di Alta Matematica “Francesco Severi” (GNCS-INDAM). R. Tempone is a member of the KAUST Strategic Research Initiative, Center for Uncertainty Quantification in Computational Sciences and Engineering. R. Tempone received support from the KAUST CRG3 Award Ref: 2281.

References

  1. 1.
    I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Review, 52 (2010), 317–355.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Barth, C. Schwab, and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numerische Mathematik, 119 (2011), 123–161.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Beck, F. Nobile, L. Tamellini, and R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250023.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Bieri, A sparse composite collocation finite element method for elliptic SPDEs., SIAM Journal on Numerical Analysis, 49 (2011), 2277–2301.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. I. Bogachev, Measure Theory, Vol. 1, Springer Berlin Heidelberg, 2007.CrossRefzbMATHGoogle Scholar
  6. 6.
    H. J. Bungartz and M. Griebel, Sparse grids, Acta Numerica, 13 (2004), 147–269.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. J. Bungartz, M. Griebel, D. Röschke, and C. Zenger, Pointwise convergence of the combination technique for the Laplace equation, East-West Journal of Numerical Mathematics, 2 (1994), 21–45.MathSciNetzbMATHGoogle Scholar
  8. 8.
    J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients, SIAM Journal on Numerical Analysis, 50 (2012), 216–246.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. Charrier, R. Scheichl, and A. Teckentrup, Finite element error analysis of elliptic pdes with random coefficients and its application to multilevel Monte Carlo methods, SIAM Journal on Numerical Analysis, 51 (2013), 322–352.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. Chkifa, On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection, Journal of Approximation Theory, 166 (2013), 176–200.Google Scholar
  11. 11.
    A. Cohen, R. Devore, and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’S, Analysis and Applications, 9 (2011), 11–47.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    N. Collier, A.-L. Haji-Ali, F. Nobile, E. von Schwerin, and R. Tempone, A continuation multilevel Monte Carlo algorithm, BIT Numerical Mathematics, 55 (2015), 399–432.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Transactions of the American Mathematical Society, 348 (1996), 503–520.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. Dũng and M. Griebel, Hyperbolic cross approximation in infinite dimensions, Journal of Complexity, 33 (2016), 55–88.Google Scholar
  15. 15.
    B. Ganapathysubramanian and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, jcp, 225 (2007), 652–685.MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607–617.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    W. J. Gordon and C. A. Hall, Construction of curvilinear co-ordinate systems and applications to mesh generation, International Journal for Numerical Methods in Engineering, 7 (1973), 461–477.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    I. G. Graham, R. Scheichl, and E. Ullmann, Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods, Stochastic Partial Differential Equations: Analysis and Computations, (2015), 1–35.Google Scholar
  19. 19.
    M. Griebel and H. Harbrecht, On the convergence of the combination technique, in Sparse Grids and Applications - Munich 2012, J. Garcke and D. Pflüger, eds., vol. 97 of Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2014, 55–74.Google Scholar
  20. 20.
    M. Griebel and S. Knapek, Optimized general sparse grid approximation spaces for operator equations, Mathematics of Computation, 78 (2009), 2223–2257.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and R. Beauwens, eds., IMACS, Elsevier, North Holland, 1992, pp. 263–281.Google Scholar
  22. 22.
    A.-L. Haji-Ali, F. Nobile, L. Tamellini, and R. Tempone, Multi-index stochastic collocation for random PDEs, Computer Methods in Applied Mechanics and Engineering, 306 (2016), 95–122.Google Scholar
  23. 23.
    A.-L. Haji-Ali, F. Nobile, and R. Tempone, Multi-index Monte Carlo: when sparsity meets sampling, Numerische Mathematik, 132 (2015), 767–806.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    A.-L. Haji-Ali, F. Nobile, E. von Schwerin, and R. Tempone, Optimization of mesh hierarchies in multilevel Monte Carlo samplers, Stochastic Partial Differential Equations: Analysis and Computations, 4 (2015), 76–112.MathSciNetCrossRefGoogle Scholar
  25. 25.
    H. Harbrecht, M. Peters, and M. Siebenmorgen, On multilevel quadrature for elliptic stochastic partial differential equations, in Sparse Grids and Applications, vol. 88 of Lecture Notes in Computational Science and Engineering, Springer, 2013, 161–179.Google Scholar
  26. 26.
    M. Hegland, J. Garcke, and V. Challis, The combination technique and some generalisations, Linear Algebra and its Applications, 420 (2007), 249–275.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Heinrich, Multilevel Monte Carlo methods, in Large-Scale Scientific Computing, vol. 2179 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2001, 58–67.Google Scholar
  28. 28.
    T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, nurbs, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4135–4195.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    F. Y. Kuo, C. Schwab, and I. Sloan, Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients, Foundations of Computational Mathematics, 15 (2015), 411–449.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    S. Martello and P. Toth, Knapsack problems: algorithms and computer implementations, Wiley-Interscience series in discrete mathematics and optimization, J. Wiley & Sons, 1990.Google Scholar
  31. 31.
    A. Narayan and J. D. Jakeman, Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation, SIAM Journal on Scientific Computing, 36 (2014), A2952–A2983.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    F. Nobile, L. Tamellini, and R. Tempone, Comparison of Clenshaw-Curtis and Leja Quasi-Optimal Sparse Grids for the Approximation of Random PDEs, in Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM ’14, R. M. Kirby, M. Berzins, and J. S. Hesthaven, eds., vol. 106 of Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2015, 475–482.Google Scholar
  33. 33.
    F. Nobile, L. Tamellini, and R. Tempone, Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs, Numerische Mathematik, 134(2) (2016), 343–388.Google Scholar
  34. 34.
    F. Nobile, R. Tempone, and C. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM Journal on Numerical Analysis, 46 (2008), 2411–2442.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    F. Nobile, R. Tempone, and C. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM Journal on Numerical Analysis, 46 (2008), 2309–2345.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Problems, 29 (2013), 065011.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    A. Teckentrup, P. Jantsch, C. G. Webster, and M. Gunzburger, A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), 1046–1074.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    H. W. van Wyk, Multilevel sparse grid methods for elliptic partial differential equations with random coefficients, arXiv preprint arXiv:1404.0963, 2014.
  39. 39.
    G. W. Wasilkowski and H. Wozniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, Journal of Complexity, 11 (1995), 1–56.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    D. Xiu and J. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing, 27 (2005), 1118–1139.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations, W. Hackbusch, ed., vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg, 1991, pp. 241–251.Google Scholar

Copyright information

© SFoCM 2016

Authors and Affiliations

  • Abdul-Lateef Haji-Ali
    • 1
  • Fabio Nobile
    • 2
  • Lorenzo Tamellini
    • 3
    • 4
  • Raúl Tempone
    • 1
    Email author
  1. 1.CEMSEKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia
  2. 2.MATHICSE-CSQIEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly
  4. 4.CNR-IMATIPaviaItaly

Personalised recommendations