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

Abstract

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.

Improvement of the cross-entropy method in high dimension for failure probability estimation through a one-dimensional projection without gradient estimation

“…The idea of CE-m * , and other projection algorithms such as [19], is to only estimate variance parameters in relevant directions. In [19] the choice of the directions is suggested by a theoretical result which provides an upper bound on the KL divergence.…”

“…The idea of CE-m * , and other projection algorithms such as [19], is to only estimate variance parameters in relevant directions. In [19] the choice of the directions is suggested by a theoretical result which provides an upper bound on the KL divergence. However, the computation of these directions requires to compute the gradient of the failure function which presents two drawbacks: 1) the failure function needs to be differentiable and 2) computing its gradient is often expensive (and sometimes even out of reach).…”

“…Closer to the spirit of our paper, the recent paper [19] proposes to construct a subspace, called the Failure-Informed Subspace (FIS), which identifies a low-dimensional structure of the problem and then applies CE on the FIS by projecting the samples: the resulting algorithm is a variation of iCE called iCEred. The FIS is chosen based on a theoretical bound which suggests that it induces a small KL divergence: however its computation requires the evaluation of the gradient of the limit-state function.…”

“…CE-m * and iCE-m * . In the present paper, we propose two new algorithms, which we call CE-m * and iCE-m * , which pursue the line of research initiated in [19], namely circumventing the curse of dimensionality faced by CE (and iCE) by projecting on a suitable low-dimensional space. They are extremely simple and in particular do not require any additional model evaluation compared to CE or iCE.…”

Output-Weighted Optimal Sampling for Bayesian Experimental Design and Uncertainty Quantification

“…For example, importance sampling utilizes a biasing distribution to concentrate sampling on the regions of the input space that generate extreme outcomes [25,45]. In practice, constructing the optimal biasing distribution is quite challenging, and approximations based on large-deviation theory [12], the cross-entropy method [49], or other techniques [52] are often inevitable. Another example is the subsetsimulation approach [2], where the probability of a rare event is expressed as a product of larger conditional probabilities computed by Markov chain Monte Carlo simulation.…”