Efficient estimation of failure probability

Jing Li, Purdue University

Abstract

Computation of failure probability is a fundamental problem in structural reliability analysis, which requires a multi-dimensional integral over the failure domain with respect to the probability measure. Such problems seldom have closed-from analytical solutions. To accurately estimate the failure probability of a given system requires sampling of the system response and can be computationally expensive. We demonstrate that the straightforward sampling of a surrogate model can lead to erroneous results, no matter how accurate the surrogate model is. Therefore, the purpose of this work is to construct an efficient approach that takes advantage of simulations on the surrogate model and maintains the accuracy of the failure probability estimate. A hybrid approach by sampling both the surrogate model in a "large" portion of the probability space and the original system in a "small" portion is proposed. The resulting algorithm is significantly more efficient than the traditional sampling method, and is more accurate and robust than the straightforward surrogate model approach. Combined with cross entropy(CE) method and importance sampling(IS) method, computing rare failure probability is a main application of this hybrid surrogate approach. Both direct algorithm and iterative algorithm are discussed in detail in the cross entropy optimization procedure and the final failure integral procedure with importance sampling. This surrogate-based CE-IS approach is highly efficient for rare failure probability computation—it incurs much reduced simulation efforts compared to the original CE-IS methods. In many cases, the new method is capable of capturing failure probability as small as 10–6 with only several hundred samples. Computation of the failure probability naturally requires the knowledge of the probability distribution of the underlying random inputs. However, for many complex systems it is often impossible to have complete information of the probability distributions. In such cases the uncertainty is usually referred to as epistemic uncertainty and straightforward computation of the failure probability is not available. We develop a method to estimate both the upper bound and lower bound of the failure probability subject to epistemic uncertainty. The bounds are rigorously derived using the variational formulas of relative entropy. We thoroughly examine the properties of the bounds which shed light on the construction of numerical algorithms to efficiently compute the bounds. This methodology is not only feasible for computing the failure probability but also applicable when estimating the bounds of the statistical mean of an integrable function given a subtle change in the inequality involving cross entropy and exponential integral.

Degree

Ph.D.

Advisors

Xiu, Purdue University.

Subject Area

Mathematics

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