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By Robby Caspeele, Luc Taerwe, Dirk Proske

This e-book offers the court cases of the 14th foreign Probabilistic Workshop that was once held in Ghent, Belgium in December 2016. Probabilistic tools are at present of an important value for study and advancements within the box of engineering, which face demanding situations provided by means of new fabrics and applied sciences and swiftly altering societal wishes and values. modern wishes on the topic of, for instance, performance-based layout, service-life layout, life-cycle research, product optimization, review of latest buildings and structural robustness supply upward push to new advancements in addition to exact and virtually appropriate probabilistic and statistical engineering easy methods to help those advancements. those complaints are a important source for a person attracted to modern advancements within the box of probabilistic engineering applications.

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For the Bessel function one has the following asymptotic√approximation (see Abramowitz and Stegun 1965, Eq. 2, p. 378): K0 (z) ∼ ????∕z ⋅ e−z , z → ∞. Inserted into Eq. 5 and using Mill’s ratio (see Abramowitz and Stegun 1965, Eq. 12, p. 932) this gives P(g???? (u1 , u2 ) < 0) ∼ √ 2 ⋅ Φ(−????) ???? → ∞ (6) 36 K. Breitung Consider now the SORM approach. √ Here √using the Lagrange√multiplier √ method two beta points are found u1 = (????∕ 2, ????∕ 2) and u2 = (−????∕ 2, −????∕ 2). At these points one has (In denoting the n-dimensional unit matrix): ( ∇g???? (u) = − √???? ) I2 − P = 0 −1 −1 0 , ∇ g???? (u) = 2 − √???? 2 ( ( 2 1 2 − 12 − 12 1 2 (1 ) 1 2 2 1 1 2 2 , P= ) ) (7) ̃ = I2 + ????|∇???? g(u)|−1 ∇2 g???? (u) , H (8) The SORM approximation is found adding the equal contributions from the two beta points.

4 Design of Experiment The individual steps of AS exploit sets of simulations (design plans) that may be prepared via various deterministic or stochastic procedures. This article compares the results of AS performed on design plans prepared via the crude Monte Carlo method with AS performed using LHS designs optimized with respect to the PAE criterion (Vořechovský and Eliáš 2015) and AS applied to a design plan prepared using the QMC sampling method called the Sobol sequence. The following sections describe the sampling methods used in the AS procedure.

9 Conclusions The subset sampling method gives often good failure probability estimates, but it has limitations as outlined here. These were illustrated by simple examples to give an intuitive idea of the possible shortcomings to aid to a better understanding of SuS. 42 K. Breitung The examples show that the points chosen by SuS move in the directions of steepest descent of the LSF’s near the origin, but changes in the descent speed of the LSF’s and changes in the topological structure of the failure domains may lead the SuS method in wrong directions.

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