Barrera, J., Homem-De-Mello, T., Moreno, E., Pagnoncelli, B. K., & Canessa, G. (2016). Chance-constrained problems and rare events: an importance sampling approach. Math. Program., 157(1), 153–189.
Abstract: We study chance-constrained problems in which the constraints involve the probability of a rare event. We discuss the relevance of such problems and show that the existing sampling-based algorithms cannot be applied directly in this case, since they require an impractical number of samples to yield reasonable solutions. We argue that importance sampling (IS) techniques, combined with a Sample Average Approximation (SAA) approach, can be effectively used in such situations, provided that variance can be reduced uniformly with respect to the decision variables. We give sufficient conditions to obtain such uniform variance reduction, and prove asymptotic convergence of the combined SAA-IS approach. As it often happens with IS techniques, the practical performance of the proposed approach relies on exploiting the structure of the problem under study; in our case, we work with a telecommunications problem with Bernoulli input distributions, and show how variance can be reduced uniformly over a suitable approximation of the feasibility set by choosing proper parameters for the IS distributions. Although some of the results are specific to this problem, we are able to draw general insights that can be useful for other classes of problems. We present numerical results to illustrate our findings.
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Jerez, D. J., Jensen, H. A., Valdebenito, M. A., Misraji, M. A., Mayorga, F., & Beer, M. (2022). On the use of Directional Importance Sampling for reliability-based design and optimum design sensitivity of linear stochastic structures. Probabilistic Eng. Mech., 70, 103368.
Abstract: This contribution focuses on reliability-based design and optimum design sensitivity of linear dynamical structural systems subject to Gaussian excitation. Directional Importance Sampling (DIS) is implemented for reliability assessment, which allows to obtain first-order derivatives of the failure probabilities as a byproduct of the sampling process. Thus, gradient-based solution schemes can be adopted by virtue of this feature. In particular, a class of feasible-direction interior point algorithms are implemented to obtain optimum designs, while a direction-finding approach is considered to obtain optimum design sensitivity measures as a post -processing step of the optimization results. To show the usefulness of the approach, an example involving a building structure is studied. Overall, the reliability sensitivity analysis framework enabled by DIS provides a potentially useful tool to address a practical class of design optimization problems.
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Valdebenito, M. A., Misraji, M. A., Jensen, H. A., & Mayorga, C. F. (2021). Sensitivity estimation of first excursion probabilities of linear structures subject to stochastic Gaussian loading. Comput. Struct., 248, 106482.
Abstract: This contribution focuses on evaluating the sensitivity associated with first excursion probabilities of linear structural systems subject to stochastic Gaussian loading. The sensitivity measure considered is the partial derivative of the probability with respect to parameters that affect the structural response, such as dimensions of structural elements. The actual calculation of the sensitivity demands solving high dimensional integrals over hypersurfaces, which can be challenging from a numerical viewpoint. Hence, sensitivity evaluation is cast within the context of a reliability analysis that is conducted with Directional Importance Sampling. In this way, the sought sensitivity is obtained as a byproduct of the calculation of the failure probability, where the post-processing step demands performing a sensitivity analysis of the unit impulse response functions of the structure. Thus, the sensitivity is calculated using sampling by means of an estimator, whose precision can be quantified in terms of its standard deviation. Numerical examples involving both small- and large-scale structural models illustrate the procedure for probability sensitivity estimation. (C) 2021 Elsevier Ltd. All rights reserved.
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