Sliced-Inverse-Regression-Aided Rotated Compressive Sensing Method for Uncertainty Quantification

TL;DR

This paper introduces a novel approach combining sliced inverse regression with compressed sensing to improve uncertainty quantification efficiency and accuracy, especially with limited data, by enhancing sparsity and enabling dimension reduction.

Contribution

It proposes two new algorithms that leverage SIR for initial guesses and dimension reduction, significantly improving uncertainty quantification with minimal data and no prior sparsity knowledge.

Findings

Enhanced sparsity leads to more efficient uncertainty quantification.

Algorithms perform well with high-dimensional problems up to 500 dimensions.

Methods require no prior knowledge of system sparsity.

Abstract

Compressive-sensing-based uncertainty quantification methods have become a pow- erful tool for problems with limited data. In this work, we use the sliced inverse regression (SIR) method to provide an initial guess for the alternating direction method, which is used to en- hance sparsity of the Hermite polynomial expansion of stochastic quantity of interest. The sparsity improvement increases both the efficiency and accuracy of the compressive-sensing- based uncertainty quantification method. We demonstrate that the initial guess from SIR is more suitable for cases when the available data are limited (Algorithm 4). We also propose another algorithm (Algorithm 5) that performs dimension reduction first with SIR. Then it constructs a Hermite polynomial expansion of the reduced model. This method affords the ability to approximate the statistics accurately with even less available data. Both methods are non-intrusive and require no a priori information of the sparsity of the system. The effec- tiveness of these two methods (Algorithms 4 and 5) are demonstrated using problems with up to 500 random dimensions.