A robust and efficient stepwise regression method for building sparse polynomial chaos expansions

• Published in: Journal of computational physics Vol. 332; pp. 461 - 474
• Main Authors: Abraham, Simon, Raisee, Mehrdad, Ghorbaniasl, Ghader, Contino, Francesco, Lacor, Chris
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.03.2017; Elsevier Science Ltd
• Subjects: ALGORITHMS; BENCHMARKS; CAPTURE; CHAOS THEORY; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPARATIVE EVALUATIONS; Computational efficiency; Computational mathematics; Computational physics; Computer simulation; COMPUTERIZED SIMULATION; Computing time; EFFICIENCY; EXPANSION; Least angle regression; Mathematical analysis; MATHEMATICAL SOLUTIONS; Parameter uncertainty; Personal computers; POLYNOMIALS; PROBABILISTIC ESTIMATION; Probabilistic methods; Probability; Regression analysis; Regression-based polynomial chaos; Robustness (mathematics); Sparse polynomial chaos expansion; Statistical analysis; Stepwise regression; STOCHASTIC PROCESSES; Studies; UNCERTAINTY PRINCIPLE; Uncertainty quantification; Uncertainty quantification; Stepwise regression; Regression-based polynomial chaos; Sparse polynomial chaos expansion; Least angle regression
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2016.12.015

Polynomial Chaos (PC) expansions are widely used in various engineering fields for quantifying uncertainties arising from uncertain parameters. The computational cost of classical PC solution schemes is unaffordable as the number of deterministic simulations to be calculated grows dramatically with the number of stochastic dimension. This considerably restricts the practical use of PC at the industrial level. A common approach to address such problems is to make use of sparse PC expansions. This paper presents a non-intrusive regression-based method for building sparse PC expansions. The most important PC contributions are detected sequentially through an automatic search procedure. The variable selection criterion is based on efficient tools relevant to probabilistic method. Two benchmark analytical functions are used to validate the proposed algorithm. The computational efficiency of the method is then illustrated by a more realistic CFD application, consisting of the non-deterministic flow around a transonic airfoil subject to geometrical uncertainties. To assess the performance of the developed methodology, a detailed comparison is made with the well established LAR-based selection technique. The results show that the developed sparse regression technique is able to identify the most significant PC contributions describing the problem. Moreover, the most important stochastic features are captured at a reduced computational cost compared to the LAR method. The results also demonstrate the superior robustness of the method by repeating the analyses using random experimental designs.