Sparse polynomial chaos expansion based on D-MORPH regression

Abstract

Polynomial chaos expansion (PCE) is widely used by engineers and modelers in various engineering fields for uncertainty analysis. The computational cost of full PCE is unaffordable for the “curse of dimensionality” of the expansion coefficients. In this paper, a new method for developing sparse PCE is proposed based on the diffeomorphic modulation under observable response preserving homotopy (D-MORPH) algorithm. D-MORPH is a regression technique, it can construct the full PCE models with model evaluations much less than the unknown coefficients. This technique determines the unknown coefficients by minimizing the least-squared error and an objective function. For the purpose of developing sparse PCE, an iterative reweighted algorithm is proposed to construct the objective function. As a result, the objective in D-MORPH regression is converted to minimize the ℓ1 norm of PCE coefficients, and the sparse PCE is established after the proposed algorithm converges to the optimal value. To validate the performance of the developed methodology, several benchmark examples are investigated. The accuracy and efficiency are compared to the well-established least angle regression (LAR) sparse PCE, and results show that the developed method is superior to the LAR-based sparse PCE in terms of efficiency and accuracy.