JOURNAL OF COMPUTATIONAL PHYSICS | 卷:381 |
Data-driven polynomial chaos expansions: A weighted least-square approximation | |
Article | |
Guo, Ling1  Liu, Yongle2  Zhou, Tao3  | |
[1] Shanghai Normal Univ, Dept Math, Shanghai, Peoples R China | |
[2] Southern Univ Sci & Technol, Dept Math, Shenzhen, Peoples R China | |
[3] Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, LSEC, Beijing, Peoples R China | |
关键词: Uncertainty quantification; Data-driven polynomial chaos expansions; Weighted least-squares; Equilibrium measure; | |
DOI : 10.1016/j.jcp.2018.12.020 | |
来源: Elsevier | |
【 摘 要 】
In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of data-driven polynomial chaos is to use statistical moments of the input random variables to develop an arbitrary polynomial chaos expansion, and then use such data-driven bases to perform UQ computations. Here we adopt the bases construction procedure by following (Ahlfeld et al. (2016), [1]), where the bases are computed by using matrix operations on the Hankel matrix of moments. Different from previous works, in the postprocessing part, we propose a weighted least-squares approach to solve UQ problems. This approach includes a sampling strategy and a least-squares solver. The main features of our approach are two folds: On one hand, our sampling strategy is independent of the random input. More precisely, we propose to sampling with the equilibrium measure, and this measure is also independent of the data-driven bases. Thus, this procedure can be done in prior (or in a off-line manner). On the other hand, we propose to solve a Christoffel function weighted least-square problem, and this strategy is quasi-linearly stable - the required number of PDE solvers depends linearly (up to a logarithmic factor) on the number of (data-driven) bases. This new approach is thus promising in dealing with a class of problems with epistemic uncertainties. A number of numerical tests are presented to show the effectiveness of our approach. (C) 2019 Elsevier Inc. All rights reserved.
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