A sequential sparse polynomial chaos expansion using Bayesian regression for geotechnical reliability estimations |
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| Authors: | Qiujing Pan Xingru Qu Leilei Liu Daniel Dias |
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| Institution: | 1. School of Civil Engineering, Central South University, Changsha, Hunan, 410075 China;2. Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring, Ministry of Education, School of Geosciences and Info-Physics, Central South University, 932 South Lushan Road, Changsha, 410083 China;3. Laboratory 3SR, Grenoble Alpes University, CNRS UMR, 5521 Grenoble, France |
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| Abstract: | Polynomial chaos expansions (PCEs) have been widely employed to estimate failure probabilities in geotechnical engineering. However, PCEs suffer from two deficiencies: (a) PCE coefficients are solved by the least-square minimization method which easily causes overfitting issues; (b) building a high order PCE is often computationally expensive. In order to overcome the aforementioned drawbacks, the Bayesian regression technique is employed to evaluate PCE coefficients, which not only provides a sparse solution but also avoids overfitting. With the aid of the predictive means and variances given by Bayesian analysis, a learning function is proposed to sequentially select the most informative samples that are critical to build a PCE. This sequential learning scheme can highly enhance the computational efficiency of PCEs. Besides, importance sampling (IS) is incorporated into the sequential learning (SL)-PCEs to deal with geotechnical problems with small failure probabilities. The proposed method of SL-PCE-IS is applied to three illustrative examples, which shows that the improved PCE method is more effective and efficient than the common PCEs method, leading to accurate estimations of small failure probabilities using fewer training samples. |
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| Keywords: | Bayesian regression failure probability importance sampling polynomial chaos expansions reliability analysis |
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