A multilevel adaptive sparse grid stochastic collocation approach to the non-smooth forward propagation of uncertainty in discretized problems

TL;DR

This paper introduces the MLASGC method, combining adaptive sparse grid stochastic collocation with multilevel discretizations to efficiently propagate uncertainty in non-smooth problems, outperforming existing methods.

Contribution

The paper presents the novel MLASGC approach that integrates multilevel discretizations with adaptive sparse grid collocation for uncertainty propagation in non-smooth problems.

Findings

MLASGC achieves an error decay rate of approximately t^{-0.95}.

MLASGC significantly outperforms ALSGC and MLMC in computational efficiency.

Preliminary results on a low-dimensional problem show promising accuracy and efficiency.

Abstract

This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic collocation method (ALSGC) with a hierarchy of successively finer spatial discretizations (e.g. finite elements) of the underlying deterministic problem. To achieve this, we build strongly upon ideas from the Multilevel Monte Carlo method (MLMC), which represents a well-established technique for the reduction of computational complexity in problems affected by both deterministic and stochastic error contributions. The resulting approach is termed the Multilevel Adaptive Sparse Grid Collocation (MLASGC) method. Preliminary results for a low-dimensional, non-smooth parametric ODE problem are promising: the proposed MLASGC method exhibits an error/cost-relation of $\varepsilon \sim t^{-0.95}$ and therefore significantly outperforms the single-level ALSGC ($\varepsilon \sim t^{-0.65}$) and MLMC methods ($\varepsilon \lesssim t^{-0.5}$).