A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
Top Cited Papers
- 1 January 2008
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Numerical Analysis
- Vol. 46 (5), 2309-2345
- https://doi.org/10.1137/060663660
Abstract
This work proposes and analyzes 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 as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using Lq norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approachesKeywords
This publication has 24 references indexed in Scilit:
- A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input DataSIAM Journal on Numerical Analysis, 2007
- Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulationComputer Methods in Applied Mechanics and Engineering, 2005
- Sparse gridsActa Numerica, 2004
- Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential EquationsSIAM Journal on Numerical Analysis, 2004
- SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE EXPERIMENTAL DATAMathematical Models and Methods in Applied Sciences, 2003
- On solving elliptic stochastic partial differential equationsComputer Methods in Applied Mechanics and Engineering, 2002
- High dimensional polynomial interpolation on sparse gridsAdvances in Computational Mathematics, 2000
- On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal pointsAnalysis Mathematica, 1983
- A method for numerical integration on an automatic computerNumerische Mathematik, 1960
- On Interpolation IAnnals of Mathematics, 1937