  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.
