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Adaptive Monte Carlo analysis for strongly nonlinear stochastic systems

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  • Shields, Michael D.

Abstract

This paper compares space-filling and importance sampling (IS)-based Monte Carlo sample designs with those derived for optimality in the error of stratified statistical estimators. Space-filling designs are shown to be optimal for systems whose response depends linearly on the input random variables. They are, however, shown to be far from optimal when the system is nonlinear. To achieve optimality, it is shown that samples should be placed densely in regions of large variation (sparsely in regions of small variation). This notion is shown to be subtly, but importantly, different from other non-space-filling designs, particularly IS. To achieve near-optimal sample designs, the adaptive Gradient Enhanced Refined Stratified Sampling (GE-RSS) is proposed that sequentially refines the probability space in accordance with stratified sampling. The space is refined according to the estimated local variance of the system computed from gradients using a surrogate model. The method significantly reduces the error in stratified Monte Carlo estimators for strongly nonlinear systems, outperforms both space-filling methods and IS-based methods, and is simple to implement. Numerical examples on strongly nonlinear systems illustrate the improvement over space-filling and IS designs. The method is applied to study the probability of shear band formation in a bulk metallic glass.

Suggested Citation

  • Shields, Michael D., 2018. "Adaptive Monte Carlo analysis for strongly nonlinear stochastic systems," Reliability Engineering and System Safety, Elsevier, vol. 175(C), pages 207-224.
    • Handle: RePEc:eee:reensy:v:175:y:2018:i:c:p:207-224
    DOI: 10.1016/j.ress.2018.03.018
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    References listed on IDEAS

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    1. Jean-Marie Cornuet & Jean-Michel Marin & Antonietta Mira & Christian P. Robert, 2012. "Adaptive Multiple Importance Sampling," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(4), pages 798-812, December.
    2. Shields, Michael D. & Teferra, Kirubel & Hapij, Adam & Daddazio, Raymond P., 2015. "Refined Stratified Sampling for efficient Monte Carlo based uncertainty quantification," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 310-325.
    3. Janssen, Hans, 2013. "Monte-Carlo based uncertainty analysis: Sampling efficiency and sampling convergence," Reliability Engineering and System Safety, Elsevier, vol. 109(C), pages 123-132.
    4. Michael D. Shields & V.S. Sundar, 2015. "Targeted random sampling: a new approach for efficient reliability estimation for complex systems," International Journal of Reliability and Safety, Inderscience Enterprises Ltd, vol. 9(2/3), pages 174-190.
    5. repec:dau:papers:123456789/10690 is not listed on IDEAS
    6. Shields, Michael D. & Zhang, Jiaxin, 2016. "The generalization of Latin hypercube sampling," Reliability Engineering and System Safety, Elsevier, vol. 148(C), pages 96-108.
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