The evaluation of failure probabilities is a fundamental problem in reliability analysis and risk management of systems with uncertain inputs. We consider systems described by PDEs with random coefficients together with efficient approximation schemes. This includes stochastic finite elements, collocation, reduced basis, and advanced Monte Carlo methods. Efficient evaluation and updating of small failure probabilities and rare events remains a significant computational challenge. This mini-symposium brings together tools from applied probability, numerical analysis, and computational science and engineering. We showcase advances in analysis and computational treatment of rare events and failure probabilities, including variance reduction, advanced meta-models, and multilevel Monte Carlo.
08:30
On the Asymptotic Normality of Adaptive Multilevel Splitting
Arnaud Guyader | Sorbonne Université | France
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Arnaud Guyader | Sorbonne Université | France
Adaptive Multilevel Splitting (AMS), also known as Subset Simulation, is a generic Monte Carlo method for Markov processes that simulates rare events and estimates associated probabilities. Despite its practical efficiency, there are almost no theoretical results on the convergence of this algorithm. The purpose of this talk is to prove both consistency and asymptotic normality results in a general setting. This is done by associating to the original Markov process a level-indexed process, also called a stochastic wave, and by showing that AMS can then be seen as a Fleming-Viot type particle system.
09:00
Error analysis of probabilities of rare events with approximate models
Fabian Wagner | Technical University of Munich | Germany
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Fabian Wagner | Technical University of Munich | Germany
Jonas Latz | University of Cambridge | United Kingdom
Iason Papaioannou | Technical University of Munich | Germany
Elisabeth Ullmann | Technical University of Munich | Germany
The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit state function, which depends on the solution of a partial differential equation (PDE). However, in many applications the PDE cannot be solved analytically. We can only evaluate an approximation of the exact PDE solution. Therefore, the probability of rare events is estimated with respect to an approximation of the limit state function. This leads to an approximation error in the estimate of the probability of rare events. Indeed, we are interested in the relative approximation error with respect to the discretization level. We prove an upper bound of the relative error which scales like the discretization accuracy of the PDE under certain assumptions. Hence, we derive a relationship between the required accuracy of the probability of rare events estimate and the PDE discretization level.
09:30
A Weight-bounded Importance Sampling Method for Variance Reduction
Tengchao Yu | Shanghai Jiao Tong University | China
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Tengchao Yu | Shanghai Jiao Tong University | China
Jinglai Li | University of Liverpool | United Kingdom
Importance sampling (IS) is an important technique to reduce the estimation variance in Monte Carlo simulations. In many practical problems, however, the use of the IS method may result in unbounded variance, and thus fail to provide reliable estimates. To address the issue, we propose a method which can prevent the risk of unbounded variance; the proposed method performs the standard IS for the integral of interest in a region only in which the IS weight is bounded and we use the result as an approximation to the original integral. It can be verified that the resulting estimator has a finite variance. Moreover, we also provide a normality test based method to identify the region with bounded IS weight (termed as the safe region) from the samples drawn from the standard IS distribution. With numerical examples, we demonstrate that the proposed method can yield a rather reliable estimate when the standard IS fails, and it also outperforms the defensive IS, a popular method to prevent unbounded variance.
10:00
Conditional reliability estimation with importance sampling based on information reuse
Max Ehre | Technical University of Munich | Germany
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Authors:
Max Ehre | Technical University of Munich | Germany
Iason Papaioannou | Technical University of Munich | Germany
Karen Willcox | University of Texas at Austin | United States
Daniel Straub | Technical University of Munich | Germany
In reliability analysis of engineering systems, the quantity of interest is the system’s probability of failure. The influence of selected uncertainties (input random variables B) on the failure probability is a useful information for decision support. One can assess this influence through conditioning the failure probability on B. Sampling from the conditional failure probability requires solving a reliability problem at each sample of B; this increases the computational cost by several orders of magnitude compared to classical reliability analysis.
To address this challenge, we solve the resulting series of reliability problems associated with a set of samples of B through an information reuse strategy, which is based on the improved cross entropy (iCE) method combined with either mixed (mIS) or controlled (cIS) importance sampling. iCE solves a series of optimisation problems to sequentially approximate the optimal importance sampling density. We reuse selected biasing densities from previous reliability computations based on which we estimate the current failure probability with mIS/cIS. The performance of the two IS strategies is investigated with numerical examples.