Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations

TL;DR

This paper introduces an iterative rotation method to enhance the sparsity of Hermite polynomial expansions, improving the efficiency and accuracy of uncertainty quantification in high-dimensional stochastic problems.

Contribution

It proposes a novel iterative rotation technique to create sparser Hermite polynomial bases, advancing compressive sensing methods for uncertainty quantification.

Findings

Improved sparsity leads to more efficient uncertainty quantification.

Effective in high-dimensional stochastic PDEs with around 100 dimensions.

Demonstrates enhanced accuracy over traditional methods.

Abstract

Compressive sensing has become a powerful addition to uncertainty quantification in recent years. This paper identifies new bases for random variables through linear mappings such that the representation of the quantity of interest is more sparse with new basis functions associated with the new random variables. This sparsity increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. Specifically, we consider rotation-based linear mappings which are determined iteratively for Hermite polynomial expansions. We demonstrate the effectiveness of the new method with applications in solving stochastic partial differential equations and high-dimensional ($\mathcal{O}(100)$) problems.