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


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.


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) 



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.


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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

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