Rare Event Probability Estimation in the Presence of Epistemic Uncertainty on Input Probability Distribution Parameters
Tóm tắt
The accurate estimation of rare event probabilities is a crucial problem in engineering to characterize the reliability of complex systems. Several methods such as Importance Sampling or Importance Splitting have been proposed to perform the estimation of such events more accurately (i.e., with a lower variance) than crude Monte Carlo method. However, these methods assume that the probability distributions of the input variables are exactly defined (e.g., mean and covariance matrix perfectly known if the input variables are defined through Gaussian laws) and are not able to determine the impact of a change in the input distribution parameters on the probability of interest. The problem considered in this paper is the propagation of the input distribution parameter uncertainty defined by intervals to the rare event probability. This problem induces intricate optimization and numerous probability estimations in order to determine the upper and lower bounds of the probability estimate. The calculation of these bounds is often numerically intractable for rare event probability (say 10−5), due to the high computational cost required. A new methodology is proposed to solve this problem with a reduced simulation budget, using the adaptive Importance Sampling. To this end, a method for estimating the Importance Sampling optimal auxiliary distribution is proposed, based on preceding Importance Sampling estimations. Furthermore, a Kriging-based adaptive Importance Sampling is used in order to minimize the number of evaluations of the computationally expensive simulation code. To determine the bounds of the probability estimate, an evolutionary algorithm is employed. This algorithm has been selected to deal with noisy problems since the Importance Sampling probability estimate is a random variable. The efficiency of the proposed approach, in terms of accuracy of the found results and computational cost, is assessed on academic and engineering test cases.
Tài liệu tham khảo
Ackley DH (1987) A connectionist machine for genetic hillclimbing, vol 28. Kluwer Academic Publisher
Arnold D (2002) Noisy optimization with evolution strategies. Springer
Auger A, Finck S, Hansen N, Ros R (2010) Comparison tables: BBOB 2009 noisy testbed. INRIA technical report N∘384
Balesdent M, Morio J, Marzat J (2013) Kriging-based adaptive importance sampling algorithms for rare event estimation. Struct Saf 44:1–10
Bucklew JA (2004) Introduction to rare event simulation. Springer
Choi J, An D, Won J (2010) Bayesian approach for structural reliability analysis and optimization using the Kriging Dimension Reduction Method. J Mech Des 132:051003
Daniels HE (1954) Saddlepoint approximations in statistics. Ann Math Stat:631–650
Davison A, Smith R (1990) Models for exceedances over high thresholds (with discussion). J R Stat Soc Ser B 52:393–442
De Boer PT, Kroese DP, Mannor S, Rubinstein RY (2005) A tutorial on the cross-entropy method. Ann Oper Res 134(1):19–67
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339
Deng G, Ferris M (2007) Extension of the direct optimization algorithm for noisy functions. In: Proceedings of the 2007 winter simulation conference. Winter Simulation Conference
Der Kiureghian A, Ditlevsen O (2009) Aleatory or epistemic? Does it matter? Struct Saf 31(2):105–112
Ditlevsen O, Madsen HO (1996) Structural reliability methods. Wiley
Dubourg V, Deheeger E, Sudret B (2011a) Metamodel-based importance sampling for the simulation of rare events. In: Faber, Kohler M J, Nishilima K (eds) Proceedings of the 11th international conference of statistics and probability in civil engineering (2011), Zurich, Switzerland
Dubourg V, Sudret B, Bourinet J-M (2011b) Reliability-based design optimization using Kriging surrogates and subset simulation. Struct Multidiscip Optim 44(5):673–690
Ferson S, Oberkampf WL (2009) Validation of imprecise probability models. Int J Reliab Saf 3(1):3–22
Glasserman P, Heidelberger P, Shahabuddin P, Zajic T (1996) Splitting for rare event simulation: analysis of simple cases. In: Proceedings of the 1996 winter simulation conference. Winter Simulation Conference
Glynn P (1996) Importance sampling for Monte Carlo estimation of quantiles. Publishing House of Saint Petersburg University
Gomes H, Awruch A (2004) Comparison of response surface and neural network with other methods for structural reliability analysis. Struct Saf 26(1):49–67
Hansen N (2006) The CMA Evolution Strategy: a comparing review. In: Lozano J A, Larranaga P, Inza I, Bengoetxea E (eds) Towards a new evolutionary computation. Advances on estimation of distribution algorithms, pp 75–102
Hansen N, Finck S, Ros R, Auger A (2009a) Real-parameter black-box optimization benchmarking 2009: noisy functions definitions. INRIA technical report N∘6869
Hansen N, Niederberger A, Guzzella L, Koumoutsakos P (2009b) A method for handling uncertainty in evolutionary optimization with an application to feedback control of combustion. IEEE Trans Evol Comput 13(1):180–197
Helton JC (2011) Quantification of margins and uncertainties: conceptual and computational basis. Reliab Eng Syst Saf 96(9):976–1013
Helton JC, Johnson JD, Oberkampf WL (2004) An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliab Eng Syst Saf 85(1):39–71
Janusevskis J, Le Riche R (2012) Simultaneous Kriging-based estimation and optimization of mean response. J Glob Optim 55(2):313–336
Kruisselbrink JW, Reehuis E, Deutz A, Bäck T, Emmerich M (2011) Using the uncertainty handling cma-es for finding robust optima. In: Proceedings of the 13th annual conference on Genetic and evolutionary computation, GECCO ’11. ACM, New York
Larson J (2012) Derivative-free optimization of noisy functions. Ph.D. thesis, University of Colorado
L’Écuyer P, Demers V, Tuffin B (2006) Splitting for rare event simulation. In: Proceedings of the 2006 winter simulation conference. Winter Simulation Conference
Li L, Bect J, Vazquez E (2010) A numerical comparison of two sequential Kriging-based algorithms to estimate a probability of failure. In: Uncertainty in computer model conference, Sheffield, UK, July, 12-14, 2010
Limbourg P, De Rocquigny E, Andrianov G (2010) Accelerated uncertainty propagation in two-level probabilistic studies under monotony. Reliab Eng Syst Saf 95(9):998–1010
Lophaven S, Nielsen H, Songdergaard J (2002) DACE a MATLAB Kriging Toolbox. Technical Report IMM-TR-2002-12, Technical University of Denmark
Lugannani R, Rice S (1980) Saddlepoint approximation for the distribution of the sum of independent random variables. Adv Appl Probab 12:475–490
Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246
Moore RE, Cloud MJ, Kearfott RB (2009) Introduction to interval analysis. SIAM
Morio J (2010) How to approach the importance sampling density. Eur J Phys 31(2):41–48
Morio J (2011) Non parametric adaptive importance sampling of the probability estimation of launcher impact position. Reliab Eng Syst Saf 96(1):178–183
Niederreiter H, Spanier J (2000) Monte Carlo and Quasi-Monte Carlo methods. Springer
Picheny V (2009) Improving accuracy and compensating for uncertainty in surrogate modeling. Ph.D. thesis, University of Florida
Pickands J (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131
Rackwitz R (2001) Reliability analysis: a review and some perspectives. Struct Saf 23(4):365–395
Rajashekhar M, Ellingwood B (1993) A new look at the response surface approach for reliability analysis. Struct Saf 12(3):205–220
Rubinstein RY (1997) Optimization of computer simulation models with rare events. Eur J Oper Res 99(1):89–112
Rubinstein RY, Kroese DP (2004) The cross-entropy method : a unified approach to combinatorial optimization, Monte-Carlo simulation and machine learning (Information Science and Statistics). Springer
Sasena M (2002) Flexibility and Efficiency enhancements for constrained global design optimization with Kriging approximation. Ph.D. thesis, University of Michigan
Shafer G (1976) A mathematical theory of evidence, vol 1. Princeton University Press
Sudret B (2012) Meta-models for structural reliability and uncertainty quantification. In: Asian-Pacific Symposium on Structural Reliability and its Applications, Singapore, Singapore 2012
Thunnissen DP (2003) Uncertainty classification for the design and development of complex systems. In: 3rd annual predictive methods conference, Newport Beach, California, 2003
Tomixk J, Arnold S, Barton R (1995) Sample size selection for improved Nelder-Mead performance. In: Proceedings of the 1995 winter simulation conference. Winter Simulation Conference
Yao W, Chen X, Huang Y, Gurdal Z, van Tooren M (2013) Sequential optimization and mixed uncertainty analysis method for reliability-based optimization. AIAA J 51(9):2266–2277
Zadeh LA (1999) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 100:9–34
Zhang P (1996) Nonparametric importance sampling. J Am Stat Assoc 91(434):1245–1253