Multi-Index Stochastic Collocation for random PDEs

TL;DR

This paper introduces the Multi-Index Stochastic Collocation (MISC) method for efficiently computing statistics of PDE solutions with random data, optimizing mixed differences for improved convergence.

Contribution

The paper proposes a novel MISC method with an optimization procedure for selecting mixed differences, enhancing efficiency over existing multi-level collocation methods.

Findings

MISC achieves convergence rates dictated by deterministic solver accuracy.

Optimization of mixed differences improves computational efficiency.

MISC outperforms several existing stochastic collocation methods in tests.

Abstract

In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.