[Paper Review] Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure

Ling Li, Julien Bect+1 more|arXiv (Cornell University)|Jul 9, 2012

Advanced Multi-Objective Optimization Algorithms36 references20 citations

TL;DR

This paper proposes Bayesian Subset Simulation (BSS), a kriging-based algorithm that combines Subset Simulation with sequential Bayesian experimental design to estimate small failure probabilities with significantly fewer expensive computer model evaluations. By using Gaussian process models to guide adaptive sampling, BSS reduces the number of function evaluations from 4,600 in standard Subset Simulation to an average of 104, while maintaining low bias and coefficient of variation.

ABSTRACT

The estimation of small probabilities of failure from computer simulations is a classical problem in engineering, and the Subset Simulation algorithm proposed by Au & Beck (Prob. Eng. Mech., 2001) has become one of the most popular method to solve it. Subset simulation has been shown to provide significant savings in the number of simulations to achieve a given accuracy of estimation, with respect to many other Monte Carlo approaches. The number of simulations remains still quite high however, and this method can be impractical for applications where an expensive-to-evaluate computer model is involved. We propose a new algorithm, called Bayesian Subset Simulation, that takes the best from the Subset Simulation algorithm and from sequential Bayesian methods based on kriging (also known as Gaussian process modeling). The performance of this new algorithm is illustrated using a test case from the literature. We are able to report promising results. In addition, we provide a numerical study of the statistical properties of the estimator.

Motivation & Objective

• To address the challenge of estimating small probabilities of failure in systems where performance functions are expensive-to-evaluate computer models.
• To reduce the number of required function evaluations compared to standard Subset Simulation and Monte Carlo methods.
• To improve computational efficiency by integrating kriging-based sequential design into the Subset Simulation framework.
• To maintain low bias and variance in the failure probability estimator despite reduced sampling.
• To provide a statistically sound, adaptive sampling strategy that focuses evaluations on critical regions of the input space.

Proposed method

• The algorithm decomposes the small failure probability into a product of conditional probabilities using intermediate thresholds, following the Subset Simulation framework.
• A Gaussian process prior with a Matérn covariance function is used to model the performance function, enabling uncertainty quantification over the failure region.
• An adaptive sampling strategy (SUR) selects new evaluation points to maximize information gain, particularly focusing on regions near the failure threshold.
• The intermediate thresholds are chosen adaptively based on a criterion that ensures each conditional probability is approximately 0.1.
• Gibbs sampling with a Gaussian proposal distribution is used to generate samples within the current failure set, and kriging predictions guide the selection of new points.
• The algorithm uses REML to re-estimate covariance parameters after each new evaluation, ensuring model adaptivity.

Experimental results

Research questions

• RQ1 Can kriging-based adaptive sampling significantly reduce the number of expensive function evaluations required to estimate small failure probabilities?
• RQ2 How does the performance of Bayesian Subset Simulation compare to standard Subset Simulation and Monte Carlo in terms of bias, variance, and computational cost?
• RQ3 Does the integration of sequential design with Subset Simulation maintain statistical reliability while reducing the number of simulations?
• RQ4 What is the impact of adaptive sampling on the convergence and accuracy of the failure probability estimator?
• RQ5 Can the proposed method achieve reliable estimation with fewer than 100 function evaluations in high-dimensional or complex failure regions?

Key findings

• The Bayesian Subset Simulation algorithm reduced the average number of function evaluations from 4,600 in standard Subset Simulation to 104, with a range of 94 to 109 across 50 runs.
• The estimated failure probability using BSS was 3.7020 × 10⁻⁵, compared to the reference Monte Carlo value of 3.8500 × 10⁻⁵, resulting in a relative bias of 4.4%.
• The coefficient of variation for BSS was 16.7%, significantly lower than the 63.2% for standard Subset Simulation, indicating better precision with fewer samples.
• The algorithm achieved a high level of accuracy with minimal computational cost, demonstrating that kriging-based adaptive sampling effectively targets informative regions of the input space.
• The design of experiments showed that evaluations were concentrated near the failure threshold, confirming the algorithm's ability to focus sampling where it is most needed.
• The use of REML for online covariance parameter re-estimation improved model fidelity and contributed to stable performance across runs.