Multifidelity support vector machines classifiers exploiting discretization error estimators for structural reliability

Abstract

In reliability analysis, the estimation of probability of failure is computionally challenging. The deterministic numerical simulation of large civil engineering structures using discretization techniques such as the Finite Element Method (FEM) may be expensive. Moreover, uncertainty may be modeled by dozens of random variables. As a consequence, crude Monte Carlo estimator becomes unaffordable.

Recently, numerous methods were developped to tackle this issue while keeping the appealing non intrusivity of Monte Carlo estimators : variance reduction [1], Multi Level Monte Carlo [2], surrogate models [3], ... Since the definition of the failure domain only relies on the sign of a performance function representing the failure scenario, Support Vector Machines (SVM) are interesting candidates for the construction of a meta-model. Indeed, the classifier enables to separate the Monte Carlo population into 2 subpopulations : failure population and safe population. Adaptive construction of such meta-model have been proposed in the context of reliability analysis [4]. The learning functions and learning criteria enable to control the error introduced by the meta-modelization. They ensure that only few calls to the FE solver are done close to the limit state. Once the surrogate model is built, estimating the probability of failure is cheap and straightforward.

It is well known the discretization techniques lead to error on the outputs of the FE solving (displacement, strain, stress, ...). This error therefore pollutes the estimation of the probability of failure. In this paper, we propose to exploit a posteriori discretization error estimator to guarantee that observations are in the correct domain (safety of failure). Two approaches are presented: the first one aims at building a multi-fidelity classifier using calls to the FE solver on different mesh sizes which enables to focus the numerical cost close to the limit state. The second approach consists in building two classifiers in parallel bounding the exact unknown limit state.

References

[1] M. Rashki, A. Ghavidel, H. Arab, S. Mousavi, Low-cost finite element method-based reliability analysis using adjusted control variate technique, Structural Safety 75 (2018) 133ﾖ142.

[2] M. Giles, Multilevel monte carlo path simulation, Operations Research 56 (3) (2008) 607ﾖ617.

[3] J. N. Fuhg, A. Fau, U. Nackenhorst, State-of-the-art and comparative review of adaptive sampling methods for kriging, Archives of Computational Methods in Engineering 28 (4) (2021) 2689ﾖ2747.

[4] Q. Pan, D. Dias, An efficient reliability method combining adaptive support vector machine and Monte Carlo simulation, Structural Safety 67 (2017) 85ﾖ95