A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

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 L q 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 approaches.

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

“…The choice of {y k } Q k=1 and {P k } Q k=1 governs the accuracy and efficiency of this algorithm [45,47,65,3]. We will use sparse-grid knots {y k } Q k=1 and sparse-grid polynomials {P k } Q k=1 .…”

“…The material in this section is based on previous work by many authors, including [26,28,6,3,46,48,66].…”

“…The number of points in N I can further be reduced if the 1D quadrature nodes are nested, that is, N i k ⊂ N i+1 k [48]. Convergence results for sparse-grid stochastic collocation methods for PDEs with random input data can be found, for example, in [3,45,46,47]. These papers generate sequences of multi-index sets in a predetermined manner.…”

A multiscale preconditioner for stochastic mortar mixed finite elements

“…For example, if k collocation points are used in each stochastic dimension, e M ¼ k N . To cope with this problem, more advanced collocation techniques are possible such as the so called probabilistic collocation method (see e.g., [26]) and the Smolyak sparse grids (see e.g., [44,32,31]). These advanced collocation techniques will not be considered in this paper, but it is straightforward to extend the multiscale preconditioning strategy defined in Section 6 to these methods.…”

“…However, the stochastic collocation method shares the approximation properties of the stochastic finite element method [6,41,15], making it more efficient than MCS. Choices of collocation points include tensor product of zeros of orthogonal polynomials [5,45], sparse grid approximations [17,32,40,45], and probabilistic collocation [26].…”