Confidence-based design optimization using multivariate kernel density estimation under insufficient input data非完备输入数据下基于多元核密度估计的置信度设计优化
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Jung YS, Kim MJ, Cho HY, Hu WF, Lee IJ, 2024. Confidence-based design optimization using multivariate kernel density estimation under insufficient input data. Probabilistic Engineering Mechanics, 78: 103702.DOI: 10.1016/j.probengmech.2024.103702
为进行精确可靠性分析，广泛研究了可靠性设计优化 (reliability-based design optimization, RBDO) 中输入统计模型的不确定性量化，可通过其特征、累积经验和可用数据进行估计。然而，现有可靠性设计优化研究中随机变量的不确定性量化采用 Bayes 定理量化不确定性参数分布。此外，由于缺乏信息和难以描述高维相关性，经常低估随机变量的相关性。因此，合理量化输入统计模型及其不确定性一直是一项挑战。估计，采用多元核密度估计 (kernel density estimation, KDE) 进行数据驱动下置信度设计优化 (confidence-based design optimization, CBDO)，以有效量化输入模型不确定性。由于输入分布仅通过给定的输入数据建立，因此不需要对输入分布进行任何假设。此外，通过使用交叉验证误差的 Bayes 定理，使用引导和最佳自适应带宽矩阵量化由数据不足导致的输入模型不确定性。因此，对于给定输入数据，所提置信度设计优化可通过多元核密度估计找到可靠性设计优化的保守最优值，既能处理随机变量的偶然不确定性，又能处理有限数量输入数据引起的认知不确定性。关键词: 可靠性设计优化, 置信度设计优化, 认知不确定性, 输入模型不确定性, 核密度估计The uncertainty quantification of the input statistical model in reliability-based design optimization (RBDO) has been widely investigated for accurate reliability analysis, and it could be estimated through its characteristics, cumulative experiences, and available data. However, uncertainty quantification of random variables in existing RBDO studies has exploited parametric distributions quantifying the uncertainty through the Bayes' theorem. In addition, a correlation between random variables is often underestimated due to a lack of knowledge and difficulty to describe the high-dimensional correlation. Hence, it has been a challenge to properly quantify input statistical model and its uncertainty. Therefore, a multivariate kernel density estimation (KDE) is employed to perform data-driven confidence-based design optimization (CBDO) for effective quantification of input model uncertainty. Any assumption on input distribution is not necessary since it is established only with the given input data. Moreover, the input model uncertainty due to insufficient data is quantified using bootstrapping and optimal adaptive bandwidth matrices through the Bayes’ theorem using cross-validation error. Consequently, the proposed CBDO with given input data is capable of finding a conservative optimum of RBDO accounting for both aleatory uncertainty of random variables and epistemic uncertainty induced by a limited number of input data through the multivariate KDE.Keywords: Reliability-based Design Optimization (RBDO); Confidence-based Design Optimization (CBDO); Epistemic Uncertainty; Input model Uncertainty; Kernel Density Estimation (KDE).Fig. 1. Illustration of the proposed process to obtain the critical multivariate KDE
Fig. 2. Flowchart of the proposed CBDO
Fig. 3. Contours of G_1, G_2 and G_3 in 2-D mathematical example
图 4: 各约束函数最优下可靠度直方图及各约束函数收敛下临界输入数据与带宽矩阵描述的真实与临界概率密度函数Fig. 4. Histogram of reliability for (a) G_1 and (b) G_2 at optimum and true and critical PDFs described by critical input data and bandwidth matrices for (c) G_1 and (d) G_2 at convergence.
图 5: 不同目标置信度下案例 2 最优成本与可靠度Fig. 5. Cost and reliabilities at optimum of Case 2 for various target confidences
图 6: 案例 1 不同约束函数的成本与可靠度箱线图Fig. 6. Boxplots of cost and reliabilities of G_1 and G_2 for Case 1
图 7: 100 次重复试验下不同约束函数的成本函数值与可靠度箱线图Fig. 7. Boxplots of cost function values and reliabilities of G_1 and G_2 for 100 repeated tests
图 8: 100 次重复试验下不同约束函数的成本函数值与可靠度箱线图Fig. 8. Boxplots of cost function values and reliabilities of G_1, G_2 and G_3 for 100 repeated tests
作者信息 | Authors
韩国科学技术院 (Korea Advanced Institute of Science & Technology) 机械工程系
韩国科学技术院 (Korea Advanced Institute of Science & Technology) 机械工程系
韩国木浦大学 (Mokpo National University) 机械工程系
浙江大学 (Zhejiang University) 流体动力与机电系统国家重点实验室
Ik-Jin Lee, 通讯作者 (Corresp.)韩国科学技术院 (Korea Advanced Institute of Science & Technology) 机械工程系Email: ikjin.lee@kaist.ac.kr
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
