Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling
Importance sampling is a variance reduction technique that is used to improve the efficiency of Monte Carlo estimation. Importance sampling uses the trick of sampling from a distribution, which is located around the zone of interest of the primary distribution thereby reducing the number of realizat...
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MDPI AG
2022-11-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/27/6/99 |
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| author | Kiran Pannerselvam Deepanshu Yadav Palaniappan Ramu |
| author_facet | Kiran Pannerselvam Deepanshu Yadav Palaniappan Ramu |
| author_sort | Kiran Pannerselvam |
| collection | DOAJ |
| description | Importance sampling is a variance reduction technique that is used to improve the efficiency of Monte Carlo estimation. Importance sampling uses the trick of sampling from a distribution, which is located around the zone of interest of the primary distribution thereby reducing the number of realizations required for an estimate. In the context of reliability-based structural design, the limit state is usually separable and is of the form Capacity (<i>C</i>)–Response (<i>R</i>). The zone of interest for importance sampling is observed to be the region where these distributions overlap each other. However, often the distribution information of <i>C</i> and <i>R</i> themselves are not known, and one has only scarce realizations of them. In this work, we propose approximating the probability density function and the cumulative distribution function using kernel functions and employ these approximations to find the parameters of the importance sampling density (ISD) to eventually estimate the reliability. In the proposed approach, in addition to ISD parameters, the approximations also played a critical role in affecting the accuracy of the probability estimates. We assume an ISD which follows a normal distribution whose mean is defined by the most probable point (MPP) of failure, and the standard deviation is empirically chosen such that most of the importance sample realizations lie within the means of <i>R</i> and <i>C</i>. Since the probability estimate depends on the approximation, which in turn depends on the underlying samples, we use bootstrap to quantify the variation associated with the low failure probability estimate. The method is investigated with different tailed distributions of <i>R</i> and <i>C</i>. Based on the observations, a modified Hill estimator is utilized to address scenarios with heavy-tailed distributions where the distribution approximations perform poorly. The proposed approach is tested on benchmark reliability examples and along with surrogate modeling techniques is implemented on four reliability-based design optimization examples of which one is a multi-objective optimization problem. |
| first_indexed | 2024-03-09T16:08:38Z |
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| language | English |
| last_indexed | 2024-03-09T16:08:38Z |
| publishDate | 2022-11-01 |
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| series | Mathematical and Computational Applications |
| spelling | doaj.art-fb0e9374c5e545abb9635049853b41152023-11-24T16:30:51ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472022-11-012769910.3390/mca27060099Scarce Sample-Based Reliability Estimation and Optimization Using Importance SamplingKiran Pannerselvam0Deepanshu Yadav1Palaniappan Ramu2Advanced Design, Optimization and Probabilistic Techniques (ADOPT) Laboratory, Department of Engineering Design, Indian Institute of Technology Madras, Chennai 600036, IndiaAdvanced Design, Optimization and Probabilistic Techniques (ADOPT) Laboratory, Department of Engineering Design, Indian Institute of Technology Madras, Chennai 600036, IndiaAdvanced Design, Optimization and Probabilistic Techniques (ADOPT) Laboratory, Department of Engineering Design, Indian Institute of Technology Madras, Chennai 600036, IndiaImportance sampling is a variance reduction technique that is used to improve the efficiency of Monte Carlo estimation. Importance sampling uses the trick of sampling from a distribution, which is located around the zone of interest of the primary distribution thereby reducing the number of realizations required for an estimate. In the context of reliability-based structural design, the limit state is usually separable and is of the form Capacity (<i>C</i>)–Response (<i>R</i>). The zone of interest for importance sampling is observed to be the region where these distributions overlap each other. However, often the distribution information of <i>C</i> and <i>R</i> themselves are not known, and one has only scarce realizations of them. In this work, we propose approximating the probability density function and the cumulative distribution function using kernel functions and employ these approximations to find the parameters of the importance sampling density (ISD) to eventually estimate the reliability. In the proposed approach, in addition to ISD parameters, the approximations also played a critical role in affecting the accuracy of the probability estimates. We assume an ISD which follows a normal distribution whose mean is defined by the most probable point (MPP) of failure, and the standard deviation is empirically chosen such that most of the importance sample realizations lie within the means of <i>R</i> and <i>C</i>. Since the probability estimate depends on the approximation, which in turn depends on the underlying samples, we use bootstrap to quantify the variation associated with the low failure probability estimate. The method is investigated with different tailed distributions of <i>R</i> and <i>C</i>. Based on the observations, a modified Hill estimator is utilized to address scenarios with heavy-tailed distributions where the distribution approximations perform poorly. The proposed approach is tested on benchmark reliability examples and along with surrogate modeling techniques is implemented on four reliability-based design optimization examples of which one is a multi-objective optimization problem.https://www.mdpi.com/2297-8747/27/6/99reliabilityimportance samplingscarce datasurrogateRBDOMOO |
| spellingShingle | Kiran Pannerselvam Deepanshu Yadav Palaniappan Ramu Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling Mathematical and Computational Applications reliability importance sampling scarce data surrogate RBDO MOO |
| title | Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling |
| title_full | Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling |
| title_fullStr | Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling |
| title_full_unstemmed | Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling |
| title_short | Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling |
| title_sort | scarce sample based reliability estimation and optimization using importance sampling |
| topic | reliability importance sampling scarce data surrogate RBDO MOO |
| url | https://www.mdpi.com/2297-8747/27/6/99 |
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