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We propose an algorithm for an optimal adaptive selection of points from thedesign domain of input random variables that are needed for an accurateestimation of failure probability and the determination of the boundary betweensafe and failure domains. The method is particularly useful when eachevaluation of the performance function g(x) is very expensive and the functioncan be characterized as either highly nonlinear, noisy, or even discrete-state(e.g., binary). In such cases, only a limited number of calls is feasible, andgradients of g(x) cannot be used. The input design domain is progressivelysegmented by expanding and adaptively refining mesh-like lock-free geometricalstructure. The proposed triangulation-based approach effectively combines thefeatures of simulation and approximation methods. The algorithm performs twoindependent tasks: (i) the estimation of probabilities through an ingeniouscombination of deterministic cubature rules and the application of thedivergence theorem and (ii) the sequential extension of the experimental designwith new points. The sequential selection of points from the design domain forfuture evaluation of g(x) is carried out through a new learning function, whichmaximizes instantaneous information gain in terms of the probabilityclassification that corresponds to the local region. The extension may behalted at any time, e.g., when sufficiently accurate estimations are obtained.Due to the use of the exact geometric representation in the input domain, thealgorithm is most effective for problems of a low dimension, not exceedingeight. The method can handle random vectors with correlated non-Gaussianmarginals. The estimation accuracy can be improved by employing a smoothsurrogate model. Finally, we define new factors of global sensitivity tofailure based on the entire failure surface weighted by the density of theinput random vector.

Failure Probability Estimation and Detection of Failure Surfaces via Adaptive Sequential Decomposition of the Design Domain