Improved cross entropy-based importance sampling with a flexible mixture model

Papaioannou, Iason; Geyer, Sebastian; Straub, Daniel

doi:10.1016/j.ress.2019.106564

Cited by 67 publications

(48 citation statements)

References 29 publications

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“…where p(•) is the pdf of the fitted distribution. In particular, we consider the GM [17, Section 1] and the vMFNM distribution model [41]. Indeed, the vMFNM performs well in high-dimension while the performance of the GM deteriorates with increase of the dimension.…”

Section: Estimation Of the Probability Of Failurementioning

confidence: 99%

“…Often, the evaluation of the LSF requires the evaluation of a computational expensive model, a partial differential equation, which makes crude Monte Carlo sampling [16,45] prohibitive. Variance reduction techniques like Subset Simulation (SuS) [3,4], Sequential Importance Sampling (SIS) [42,54] or the cross-entropy based Importance Sampling (IS) method [29,41,56] have been developed to reduce computational costs while preserving an accurate estimate. In line sampling [8,28,43], sampling is performed on a hyperplane perpendicular to an important direction.…”

Section: Introductionmentioning

confidence: 99%

“…This procedure is similar to [29,Algorithm 3.1]. In particular, we apply the Gaussian mixture (GM) and the von Mises-Fisher-Nakagami mixture (vMFNM) distribution model, which have also been used in the context of the cross-entropy method [41]. Due to the properties of IS, we achieve an unbiased estimator for the probability of failure if the density of the fitted distribution admits positive values within the whole failure domain.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The ensemble Kalman filter for rare event estimation

Wagner¹,

Papaioannou²,

Ullmann³

2021

Preprint

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We present a novel sampling-based method for estimating probabilities of rare or failure events. Our approach is founded on the Ensemble Kalman filter (EnKF) for inverse problems. Therefore, we reformulate the rare event problem as an inverse problem and apply the EnKF to generate failure samples. To estimate the probability of failure, we use the final EnKF samples to fit a distribution model and apply Importance Sampling with respect to the fitted distribution. This leads to an unbiased estimator if the density of the fitted distribution admits positive values within the whole failure domain. To handle multi-modal failure domains, we localise the covariance matrices in the EnKF update step around each particle and fit a mixture distribution model in the Importance Sampling step. For affine linear limit-state functions, we investigate the continuous-time limit and large time properties of the EnKF update. We prove that the mean of the particles converges to a convex combination of the most likely failure point and the mean of the optimal Importance Sampling density if the EnKF is applied without noise. We provide numerical experiments to compare the performance of the EnKF with Sequential Importance Sampling.

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Section: Estimation Of the Probability Of Failurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The ensemble Kalman filter for rare event estimation

Wagner¹,

Papaioannou²,

Ullmann³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Although the method is applied to high-dimensional problems, it requires the generation of a large number of samples; moreover, its performance degrades in low-dimensional problems. Hence, the more flexible von Mises-Fisher-Nakagami mixture distribution is proposed in the improved CE method [28] to extend the applicability to low and moderate dimensions. [28] also proposes a smooth approximation of the optimal biasing distribution that allows information from all the samples to be used in fitting the parametric distribution.…”

mentioning

confidence: 99%

“…Hence, the more flexible von Mises-Fisher-Nakagami mixture distribution is proposed in the improved CE method [28] to extend the applicability to low and moderate dimensions. [28] also proposes a smooth approximation of the optimal biasing distribution that allows information from all the samples to be used in fitting the parametric distribution.…”

mentioning

confidence: 99%

Cross-entropy-based importance sampling with failure-informed dimension reduction for rare event simulation

Uribe¹,

Papaioannou²,

Marzouk³

et al. 2020

Preprint

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The estimation of rare event or failure probabilities in high dimensions is of interest in many areas of science and technology. We consider problems where the rare event is expressed in terms of a computationally costly numerical model. Importance sampling with the cross-entropy method offers an efficient way to address such problems provided that a suitable parametric family of biasing densities is employed. Although some existing parametric distribution families are designed to perform efficiently in high dimensions, their applicability within the cross-entropy method is limited to problems with dimension of O(10 2 ). In this work, rather than directly building sampling densities in high dimensions, we focus on identifying the intrinsic low-dimensional structure of the rare event simulation problem. To this end, we exploit a connection between rare event simulation and Bayesian inverse problems. This allows us to adapt dimension reduction techniques from Bayesian inference to construct new, effectively low-dimensional, biasing distributions within the cross-entropy method. In particular, we employ the approach in [47], as it enables control of the error in the approximation of the optimal biasing distribution. We illustrate our method using two standard high-dimensional reliability benchmark problems and one structural mechanics application involving random fields.

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Copula‐based importance sampling method for efficient slope reliability analysis

Zhou

Xiao

Jiang

et al. 2023

Num Anal Meth Geomechanics

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Efficient slope reliability analysis that accounts for the interdependencies of random variables is challenging. In this study, an efficient reliability method is proposed based on importance sampling (IS) and copulas. First, the IS method based on the design point is introduced to construct the marginal IS density. Copulas are subsequently adopted to capture the interdependencies and establish both the initial joint distributions and the joint IS densities of random variable pairs. The IS algorithm is modified to consider the dependencies through not only the estimation of the probability distribution but also random sampling. Finally, a framework of the copula‐based IS method is constructed. The applicability, accuracy, efficiency, and robustness of the proposed method were validated by a classical slope case. The results showed that with the proposed method, slope reliability could be efficiently assessed with a relatively high accuracy. The dependencies of the variables significantly affected the IS and final slope reliability, whereas the IS density had a significantly lower effect. The proposed method combines the advantages of IS and copulas well. Interdependencies of the variables can be properly characterized during sampling. Moreover, the proposed method was shown to be robust regardless of the system failure probability level. It can be further applied to improve the random finite element method in the future.

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Improved cross entropy-based importance sampling with a flexible mixture model

Cited by 67 publications

References 29 publications

The ensemble Kalman filter for rare event estimation

The ensemble Kalman filter for rare event estimation

Cross-entropy-based importance sampling with failure-informed dimension reduction for rare event simulation

Copula‐based importance sampling method for efficient slope reliability analysis

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