Efficient polynomial chaos approximations: Active, local and basis-adapted
Authors: Ghauch Ziad Georges 
Published: June 2021

Metamodels provide an efficient means for the approximation of response surfaces of systems, particularly for resource-intensive experiment designs. It is oftentimes the case that interest is focused on a specific region of the parameter space. We propose an efficient recipe for the local approximation of response surfaces using Polynomial Chaos techniques. For systems embedded in high-dimensional settings, a basis-adapted spectral representation is exploited locally for dimension reduction. The proposed approach comprises an initial heuristic global solution for parameter space exploration using an approximate global Polynomial Chaos metamodel, followed by a local design being refined through an active learning scheme. The problem of turbulent flow around a symmetric airfoil is considered. Statistical estimators based on the local, active, basis-adapted approach show less bias and faster convergence as compared to the estimators from a global solution.
References

B. J. Bichon, M. S. Eldred, L. P. Swiler, S. Mahadevan, and J. M. McFarland. Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J. 46(10):2459–2468, 2008. doi: 10.2514/1.34321.
G. E. P. Box and N. R. Draper. Empirical Model-Building and Response Surfaces. Wiley, 1987.
V. Dubourg, B. Sudret, and F. Deheeger. Metamodel-based importance sampling for structural reliability analysis. Prob. Eng. Mech. 33:47–57, 2013. doi: 10.1016/j.probengmech.2013.02.002.
R. G. Ghanem and P. D. Spanos. Stochastic finite element: A spectral approach. Dover, 1991. doi: 10.1007/978-1-4612-3094-6.
Z. G. Ghauch. Leveraging adapted polynomial chaos metamodels for real-time Bayesian updating. J. Verif. Valid. Uncert. 4(4):041003, 2020. doi: 10.1115/1.4045693.
Z. G. Ghauch, V. Aitharaju, W. R. Rodgers, P. Pasupuleti, A. Dereims, and R. G. Ghanem. Integrated stochastic analysis of fiber composites manufacturing using adapted polynomial chaos expansions. Compos. Part A: Appl. Sci. 118:179–193, 2019. doi: 10.1016/j.compositesa.2018.12.029.
M. E. Johnson, L. M. Moore, and D. Ylvisaker. Minimax and maximin distance designs. J. Stat. Plan. Infer. 26(2):131–148, 1990. doi: 10.1016/0378-3758(90)90122-B.
A. Notin, N. Gayton, J. L. Dulong, M. Lemaire, P. Villon, and H. Jaffal. RPCM: A strategy to perform reliability analysis using polynomial chaos and resampling. Euro. J. Comput. Mech. 19(8):795–830, 2010. doi: 10.3166/ejcm.19.795-830.
OpenCFD. OpenFOAM User’s Guide. 2019. https://www.openfoam.com/documentation/user-guide.
V. Picheny, D. Ginsbourger, O. Roustant, R. T Haftka, and N.-H. Kim. Adaptive designs of experiments for accurate approximation of target regions. J. Mech. Design. 132(7):071008, 2010. doi: 10.1115/1.4001873.
C. Thimmisetty, P. Tsilifis, and R. Ghanem. Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem. AI EDAM 31(3):265–276, 2017. doi: 10.1017/S0890060417000166.
R. Tipireddy and R. Ghanem. Basis adaptation in homogeneous chaos spaces. J. Comput. Phys. 259:304–317, 2014. doi: 10.1016/j.jcp.2013.12.009.
P. Tsilifis and R. G. Ghanem. Reduced Wiener chaos representation of random fields via basis adaptation and projection. J. Comput. Phys. 341:102–120, 2017. doi: 10.1016/j.jcp.2017.04.009.
Turbulence Modeling Resource. NASA Langley Research Center. Washington, DC, 2018. http://turbmodels.larc.nasa.gov/.