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

Compressive-sensing-based uncertainty quantification methods have become a powerful 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 enhance sparsity of the Hermite polynomial expansion of a 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 suitable for cases when the available data are limited (Algorithm 3.2). We also propose another algorithm (Algorithm 3.3) 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 nonintrusive and require no a priori information of the sparsity of the system. The effectiveness of these two methods (Algorithms 3.2 and 3.3) is demonstrated using problems with up to 500 random dimensions.