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

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 to 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 can be traduced as deterministic bounds for the probability. Several situations are considered, when the design is independent, identically distributed or not, or sequential. As a major consequence, a consistent estimator of the (limit state) surface separating the subsets under isotonicity and regularity arguments can be built, and its convergence speed can be exhibited. This estimator is built by aggregating semi-supervized binary classifiers chosen as constrained Support Vector Machines. Numerical experiments conducted on toy examples highlight that they work faster than recently developed monotonic neural networks with comparable predictable power. They are therefore more adapted when the computational time is a key issue.

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