On the optimal importance process for piecewise deterministic Markov process
ESAIM: Probability and Statistics, Tome 23 (2019), pp. 893-921.

In order to assess the reliability of a complex industrial system by simulation, and in reasonable time, variance reduction methods such as importance sampling can be used. We propose an adaptation of this method for a class of multi-component dynamical systems which are modeled by piecewise deterministic Markovian processes (PDMP). We show how to adapt the importance sampling method to PDMP, by introducing a reference measure on the trajectory space. This reference measure makes it possible to identify the admissible importance processes. Then we derive the characteristics of an optimal importance process, and present a convenient and explicit way to build an importance process based on theses characteristics. A simulation study compares our importance sampling method to the crude Monte-Carlo method on a three-component systems. The variance reduction obtained in the simulation study is quite spectacular.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2019015
Classification : 60K10, 90B25, 62N05
Mots-clés : Monte-Carlo acceleration, importance sampling, hybrid dynamic system, piecewise deterministic Markovian process, cross-entropy, reliability
Chraibi, H. 1 ; Dutfoy, A. 1 ; Galtier, T. 1 ; Garnier, J. 1

1 
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     author = {Chraibi, H. and Dutfoy, A. and Galtier, T. and Garnier, J.},
     title = {On the optimal importance process for piecewise deterministic {Markov} process},
     journal = {ESAIM: Probability and Statistics},
     pages = {893--921},
     publisher = {EDP-Sciences},
     volume = {23},
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     zbl = {1506.60102},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2019015/}
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Chraibi, H.; Dutfoy, A.; Galtier, T.; Garnier, J. On the optimal importance process for piecewise deterministic Markov process. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 893-921. doi : 10.1051/ps/2019015. http://www.numdam.org/articles/10.1051/ps/2019015/

[1] C.-E. Bréhier, T. Lelièvre and M. Rousset, Analysis of adaptive multilevel splitting algorithms in an idealized case. ESAIM: PS 19 (2015) 361–394. | DOI | Numdam | MR | Zbl

[2] J. Bucklew, Introduction to rare event simulation. Springer Science & Business Media (2013). | Zbl

[3] V. Caron, A. Guyader, M. Zuniga and B. Tuffin, Some recent results in rare event estimation. ESAIM: Proc. 44 (2014) 239–259. | DOI | MR | Zbl

[4] M. Čepin, Assessment of power system reliability: methods and applications. Springer Science & Business Media (2011). | DOI

[5] F. Cérou, P. Del Moral, F. Le Gland and P. Lezaud, Genetic genealogical models in rare event analy-sis. ALEA Latin Am. J. Probab. Math. Stat. 1 (2006) 181–203. | MR | Zbl

[6] F. Cérou, B. Delyon, A. Guyader and M. Rousset, On the asymptotic normality of adaptive multilevel splitting. SIAM/ASA J. Uncert. Quant. 7 (2019) 1–30. | MR | Zbl

[7] J.C. Chan, P.W. Glynn, D.P. Kroese et al., A comparison of cross-entropy and variance minimization strategies. J. Appl. Probab. 48 (2011) 183–194. | DOI | MR | Zbl

[8] H. Chraibi, Dynamic reliability modeling and assessment with PyCATSHOO: application to a test case. PSAM congress (2013).

[9] H. Chraibi, A. Dutfoy, T. Galtier and J. Garnier, Optimal input potential functions in the interacting particle system method. Preprint (2018). | arXiv | MR

[10] H. Chraibi, J.-C. Houbedine and A. Sibler, Pycatshoo: Toward a new platform dedicated to dynamic reliability assessments of hybrid systems. PSAM congress (2016).

[11] M.H. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models. J. Roy. Stat. Soc. Ser. B (Methodological) 46 (1984) 353–388. | MR | Zbl

[12] M.H. Davis, Markov Models & Optimization. In Vol. 49 of Monogaphs on Statistics and Applied Probability. CRC Press (1993). | MR | Zbl

[13] P.-T. De Boer, D.P. Kroese, S. Mannor and R.Y. Rubinstein, A tutorial on the cross-entropy method. Ann. Oper. Res. 134 (2005) 19–67. | DOI | MR | Zbl

[14] P. Del Moral and J. Garnier, Genealogical particle analysis of rare events. Ann. Appl. Probab. 15 (2005) 2496–2534. | DOI | MR | Zbl

[15] F. Dufour, H. Zhang and B. De Saporta, Numerical methods for simulation and optimization of piecewise deterministic Markov processes: application to reliability. John Wiley & Sons (2015). | MR

[16] P. Dupuis and H. Wang, Importance sampling, large deviations, and differential games. Stochastics 76 (2004) 481–508. | MR | Zbl

[17] P. Heidelberger, Fast simulation of rare events in queueing and reliability models. ACM Trans. Model. Comput. Simul. 5 (1995) 43–85. | DOI | Zbl

[18] I. Kuruganti, Importance sampling for markov chains: computing variance and determining optimal measures. In Proceedings of the 1996 Winter Simulation Conference. IEEE (1996) 273–280.

[19] P.-E. Labeau, A Monte-Carlo estimation of the marginal distributions in a problem of probabilistic dynamics. Reliab. Eng. Syst. Safety 52 (1996) 65–75. | DOI

[20] P.-E. Labeau, Probabilistic dynamics: estimation of generalized unreliability through efficient Monte-Carlo simulation. Ann. Nucl. Energy 23 (1996) 1355–1369. | DOI

[21] E. Lewis and F. Böhm, Monte-Carlo simulation of Markov unreliability models. Nucl. Eng. Des. 77 (1984) 49–62. | DOI

[22] M. Marseguerra and E. Zio, Monte-Carlo approach to psa for dynamic process systems. Reliab. Eng. Syst. Safety 52 (1996) 227–241. | DOI

[23] P. Metzner, C. Schütte and E. Vanden-Eijnden, Transition path theory for markov jump processes. Multis. Model. Simul. 7 (2009) 1192–1219. | DOI | MR | Zbl

[24] J. Morio, M. Balesdent, D. Jacquemart and C. Vergé, A survey of rare event simulation methods for static input–output models. Simul. Model. Pract. Theory 49 (2014) 287–304. | DOI

[25] M. Ramakrishnan, Unavailability estimation of shutdown system of a fast reactor by Monte-Carlo simulation. Ann. Nucl. Energy 90 (2016) 264–274. | DOI

[26] T. Rolski, H. Schmidli, V. Schmidt and J.L. Teugels, Stochastic processes for insurance and finance. In Vol. 505 of Wiley Series in Probability and Statistics. John Wiley & Sons (2009). | Zbl

[27] D. Siegmund, Importance sampling in the Monte-Carlo study of sequential tests. Ann. Stat. 4 (1976) 673–684. | DOI | MR | Zbl

[28] N. Whiteley, A.M. Johansen and S. Godsill, Monte carlo filtering of piecewise deterministic processes. J. Comput. Graph. Stat. 20 (2011) 119–139. | DOI | MR

[29] H. Zhang, F. Dufour, Y. Dutuit and K. Gonzalez, Piecewise deterministic Markov processes and dynamic reliability. Proc. Inst. Mech. Eng., Part O: J. Risk Reliab. 222 (2008) 545–551.

[30] E. Zio, The Monte-Carlo simulation method for system reliability and risk analysis. Springer (2013). | DOI | MR

[31] P. Zuliani, C. Baier and E.M. Clarke, Rare-event verification for stochastic hybrid systems. In Proceedings of the 15th ACM international conference on Hybrid Systems: Computation and Control. ACM (2012) 217–226. | DOI | MR | Zbl

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