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

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.Comment: In section 4, numerical examples 3-5, replaced the mean of the error with the quantiles and mean of the error. Added section 4.6 to compare different method

A General Framework for Enhancing Sparsity of Generalized Polynomial Chaos Expansions

Compressive sensing has become a powerful addition to uncertainty quantification when only limited data is available. In this paper we provide a general framework to enhance the sparsity of the representation of uncertainty in the form of generalized polynomial chaos expansion. We use alternating direction method to identify new sets of random variables through iterative rotations such that the new representation of the uncertainty is sparser. Consequently, we increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. We demonstrate that the previously developed iterative method to enhance the sparsity of Hermite polynomial expansion is a special case of this general framework. Moreover, we use Legendre and Chebyshev polynomials expansions to demonstrate the effectiveness of this method with applications in solving stochastic partial differential equations and high-dimensional (O(100)) problems.Comment: Corrected the lemmas in the previous version using perturbation theory of singular value decomposition. arXiv admin note: text overlap with arXiv:1506.0434

The l1-l2 minimization with rotation for sparse approximation in uncertainty quantification

This paper proposes a combination of rotational compressive sensing with the l1-l2 minimization to estimate coefficients of generalized polynomial chaos (gPC) used in uncertainty quantification. In particular, we aim to identify a rotation matrix such that the gPC of a set of random variables after the rotation has a sparser representation. However, this rotational approach alters the underlying linear system to be solved, which makes finding the sparse coefficients much more difficult than the case without rotation. We further adopt the l1-l2 minimization that is more suited for such ill-posed problems in compressive sensing (CS) than the classic l1 approach. We conduct extensive experiments on standard gPC problem settings, showing superior performance of the proposed combination of rotation and l1-l2 minimization over the ones without rotation and with rotation but using the l1 minimization

Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

Sparse polynomial chaos expansions are a popular surrogate modelling method that takes advantage of the properties of polynomial chaos expansions (PCE), the sparsity-of-effects principle, and powerful sparse regression solvers to approximate computer models with many input parameters, relying on only few model evaluations. Within the last decade, a large number of algorithms for the computation of sparse PCE have been published in the applied math and engineering literature. We present an extensive review of the existing methods and develop a framework to classify the algorithms. Furthermore, we conduct a unique benchmark on a selection of methods to identify which approaches work best in practical applications. Comparing their accuracy on several benchmark models of varying dimensionality and complexity, we find that the choice of sparse regression solver and sampling scheme for the computation of a sparse PCE surrogate can make a significant difference, of up to several orders of magnitude in the resulting mean-square error. Different methods seem to be superior in different regimes of model dimensionality and experimental design size