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

TLDR

Solving PDEs with random inputs typically requires many uncoupled deterministic solves, and full tensor grids become infeasible as dimensionality grows, making sparse grids a promising alternative to Monte Carlo. The study proposes a Smolyak‑type sparse grid stochastic collocation method for estimating statistics of PDE solutions with random coefficients and seeks to determine when it outperforms Monte Carlo. The method evaluates finite‑element solutions at a deterministic sparse grid of random inputs, derives L^q error bounds for the fully discrete solution, and assesses computational efficiency. The authors show algebraic convergence in the number of collocation points, quantify dimensional effects, and demonstrate through theory and numerical experiments that the sparse grid method outperforms full tensor grids and Monte Carlo for many problems.

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.