Approximation of Limit State Surfaces in Monotonic Monte Carlo Settings, with Applications to Classification

作者：Bousquet N^*; Klein T; Moutoussamy V

来源：SIAM-ASA Journal on Uncertainty Quantification, 2018, 6(1): 1-33.

DOI：10.1137/15M1015091

• limit state surface
• computer model
• monotonicity
• convexity
• structural reliability
• rare event probability
• random sets
• support vector machine

This article investigates the theoretical convergence properties of the estimators produced by a numerical exploration of a monotonic function, with multivariate random inputs, in a structural reliability framework. The quantity to be estimated is a probability, typically associated with an undesirable (unsafe) event, and the function is usually implemented as a computer model. The estimators produced by a Monte Carlo numerical design are two subsets of inputs leading to safe and unsafe situations, the measures of which define two deterministic bounds for the probability. Several situations are considered, including when the design is independent, identically distributed (or not), and sequential. A major consequence is that a consistent estimator of the (limit state) surface separating the subsets under isotonicity and regularity arguments can be established, along with its rate of convergence. This estimator is built by aggregating semisupervised binary classifiers chosen as constrained support vector machines. Numerical experiments, conducted on toy examples, show that this works faster than recently developed monotonic neural networks, with slightly better predictable power. These are therefore easier to handle when computational time is a key issue. Furthermore, they offer more stable results when the number of dimensions increases.

• 出版日期2018