TY - JOUR

T1 - Data-driven and active learning of variance-based sensitivity indices with Bayesian probabilistic integration

AU - Song, Jingwen

AU - Wei, Pengfei

AU - Valdebenito, Marcos A.

AU - Faes, Matthias

AU - Beer, Michael

N1 - Funding Information: This work is supported by the National Natural Science Foundation of China under Grant No. 51905430, the Sino-German Mobility Program under Grant No. M-0175, the ANID (Agency for Research and Development, Chile) under its program FONDECYT, Grant No. 1180271, and the Research Foundation Flanders (FWO) under Grant No. 12P3519N. The first author is supported by the program of China Scholarships Council (CSC). The second to forth authors are all supported by the Alexander von Humboldt Foundation of Germany.

PY - 2022/1/15

Y1 - 2022/1/15

N2 - Variance-based sensitivity indices play an important role in scientific computation and data mining, thus the significance of developing numerical methods for efficient and reliable estimation of these sensitivity indices based on (expensive) computer simulators and/or data cannot be emphasized too much. In this article, the estimation of these sensitivity indices is treated as a statistical inference problem. Two principle lemmas are first proposed as rules of thumb for making the inference. After that, the posterior features for all the (partial) variance terms involved in the main and total effect indices are analytically derived (not in closed form) based on Bayesian Probabilistic Integration (BPI). This forms a data-driven method for estimating the sensitivity indices as well as the involved discretization errors. Further, to improve the efficiency of the developed method for expensive simulators, an acquisition function, named Posterior Variance Contribution (PVC), is utilized for realizing optimal designs of experiments, based on which an adaptive BPI method is established. The application of this framework is illustrated for the calculation of the main and total effect indices, but the proposed two principle lemmas also apply to the calculation of interaction effect indices. The performance of the development is demonstrated by an illustrative numerical example and three engineering benchmarks with finite element models.

AB - Variance-based sensitivity indices play an important role in scientific computation and data mining, thus the significance of developing numerical methods for efficient and reliable estimation of these sensitivity indices based on (expensive) computer simulators and/or data cannot be emphasized too much. In this article, the estimation of these sensitivity indices is treated as a statistical inference problem. Two principle lemmas are first proposed as rules of thumb for making the inference. After that, the posterior features for all the (partial) variance terms involved in the main and total effect indices are analytically derived (not in closed form) based on Bayesian Probabilistic Integration (BPI). This forms a data-driven method for estimating the sensitivity indices as well as the involved discretization errors. Further, to improve the efficiency of the developed method for expensive simulators, an acquisition function, named Posterior Variance Contribution (PVC), is utilized for realizing optimal designs of experiments, based on which an adaptive BPI method is established. The application of this framework is illustrated for the calculation of the main and total effect indices, but the proposed two principle lemmas also apply to the calculation of interaction effect indices. The performance of the development is demonstrated by an illustrative numerical example and three engineering benchmarks with finite element models.

KW - Adaptive experiment design

KW - Bayesian probabilistic integration

KW - Data-driven

KW - Gaussian process regression

KW - Posterior variance contribution

KW - Variance-based sensitivity

UR - http://www.scopus.com/inward/record.url?scp=85108119561&partnerID=8YFLogxK

U2 - 10.1016/j.ymssp.2021.108106

DO - 10.1016/j.ymssp.2021.108106

M3 - Article

AN - SCOPUS:85108119561

VL - 163

JO - Mechanical Systems and Signal Processing

JF - Mechanical Systems and Signal Processing

SN - 0888-3270

M1 - 108106

ER -