The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data.

This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.

[1]  P. Erdös,et al.  Interpolation , 1953, An Introduction to Scientific, Symbolic, and Graphical Computation.

[2]  N. Wiener The Homogeneous Chaos , 1938 .

[3]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

[4]  M. Loève,et al.  Elementary Probability Theory , 1977 .

[5]  V. K. Dzjadyk,et al.  On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points , 1983 .

[6]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[7]  Multivariate Polynomial Interpolation , 1990 .

[8]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[9]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[10]  Henryk Wozniakowski,et al.  Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems , 1995, J. Complex..

[11]  Menner A Tatang,et al.  Direct incorporation of uncertainty in chemical and environmental engineering systems , 1995 .

[12]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[13]  Alberto Guadagnini,et al.  Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains: 2. Computational examples , 1999 .

[14]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[15]  Ivo Babuška,et al.  On solving elliptic stochastic partial differential equations , 2002 .

[16]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[17]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[18]  Daniel M. Tartakovsky,et al.  Groundwater flow in heterogeneous composite aquifers , 2002 .

[19]  M. Grigoriu Stochastic Calculus: Applications in Science and Engineering , 2002 .

[20]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[21]  Ivo Babuška,et al.  SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE EXPERIMENTAL DATA , 2003 .

[22]  Thomas Gerstner,et al.  Dimension–Adaptive Tensor–Product Quadrature , 2003, Computing.

[23]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[24]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[25]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[26]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[27]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[28]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[29]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[30]  I. Babuska,et al.  Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation , 2005 .

[31]  Radu Alexandru Todor,et al.  Sparse perturbation algorithms for elliptic PDE's with stochastic data , 2005 .

[32]  Marcus Sarkis,et al.  Stochastic Galerkin Method for Elliptic Spdes: A White Noise Approach , 2006 .

[33]  J. Burkardt,et al.  REDUCED ORDER MODELING OF SOME NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[34]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[35]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[36]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[37]  Thomas A. Zang,et al.  Stochastic approaches to uncertainty quantification in CFD simulations , 2005, Numerical Algorithms.