Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation

TL;DR

This paper introduces an adaptive Leja sparse grid method for high-dimensional stochastic collocation, demonstrating its advantages over traditional methods in terms of flexibility and convergence properties for approximating parameterized functions.

Contribution

It develops a novel adaptive sparse grid approach using Leja sequences, with proven convergence to the limiting distribution and improved performance in high-dimensional approximation.

Findings

Leja sequences can be arbitrarily refined for adaptive sparse grid construction.

The weighted Leja sequences converge to the distribution of Gauss quadrature nodes.

Leja-based sparse grids outperform standard methods like Clenshaw-Curtis in several high-dimensional problems.

Abstract

We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja-sparse grid approach to several high-dimensional and problems and demonstrate that Leja sequences are often superior to more standard sparse grid constructions (e.g. Clenshaw-Curtis), at least for interpolatory metrics.