Quantification of Polymorphic Uncertainties: A Quasi-Monte Carlo Approach
Publication: ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 10, Issue 3
Abstract
A novel methodology for the quantification of mixed and nested polymorphic uncertainties has been developed. It is designed to be applied to moderately computationally expensive deterministic models, and it preserves the distinction between aleatory and epistemic uncertainties throughout the entire process. Quasi-Monte Carlo sampling is used to efficiently represent the local and global behavior of the models. By decoupling uncertainty propagation and processing, the methodology achieves efficient reuse of samples and can support multiple outputs. In addition, simultaneous estimation of sensitivity indices is possible to facilitate decisions on where to reduce epistemic uncertainties. It is demonstrated on a structural dynamics example and compared with a fully stochastic approach using the pignistic transform. The proposed methodology has been demonstrated to be significantly more efficient than a naive implementation, but adds computational cost compared with a fully stochastic approach.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The financial support of the German Research Foundation (DFG) for the research project “Assessment and Reduction of Uncertainties in Operational Modal Analysis” (RUN-OMA) is gratefully acknowledged. Additionally, the authors thank the DFG priority program 1886 for inspiring the work and gratefully acknowledge the computational resources provided by the MaPaCC4 cluster at the TU Ilmenau, Germany.
References
Alvarez, D. A., F. Uribe, and J. E. Hurtado. 2018. “Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory.” Mech. Syst. Signal Process. 100 (Feb): 782–801. https://doi.org/10.1016/j.ymssp.2017.07.040.
Arlot, S., and A. Celisse. 2010. “A survey of cross-validation procedures for model selection.” Stat. Surv. 4 (Mar): 40–79. https://doi.org/10.1214/09-SS054.
Beer, M., S. Ferson, and V. Kreinovich. 2013. “Imprecise probabilities in engineering analyses.” Mech. Syst. Sig. Process. 37 (1–2): 4–29. https://doi.org/10.1016/j.ymssp.2013.01.024.
Biehler, J., M. Mäck, J. Nitzler, M. Hanss, P.-S. Koutsourelakis, and W. A. Wall. 2019. “Multifidelity approaches for uncertainty quantification.” GAMM-Mitt. 42 (2): e201900008. https://doi.org/10.1002/gamm.201900008.
Byrd, R. H., P. Lu, J. Nocedal, and C. Zhu. 1995. “A limited memory algorithm for bound constrained optimization.” SIAM J. Sci. Comput. 16 (5): 1190–1208. https://doi.org/10.1137/0916069.
Caylak, I., E. Penner, and R. Mahnken. 2021. “Mean-field and full-field homogenization with polymorphic uncertain geometry and material parameters.” Comput. Methods Appl. Mech. Eng. 373 (Mar): 113439. https://doi.org/10.1016/j.cma.2020.113439.
De Angelis, M., S. Ferson, E. Patelli, and V. Kreinovich. 2019. “Black-box propagation of failure probabilities under epistemic uncertainty.” In Proc., 3rd Int. Conf. on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Athens, Greece: National Technical Univ. of Athens. https://doi.org/10.7712/120219.6373.18699.
Dempster, A. P. 1967. “Upper and lower probabilities induced by a multivalued mapping.” Ann. Math. Stat. 38 (2): 325–339. https://doi.org/10.1214/aoms/1177698950.
DIN (Deutsches Institut für Normung e.V). 2019. Nationaler Anhang-National festgelegte Parameter-Eurocode 1: Einwirkungen auf Tragwerke-Teil 1–3: Allgemeine Einwirkungen. DIN EN 1991-1-3/NA:2019-04. Heidelberg, Germany: DIN. https://doi.org/10.31030/3041194.
Drieschner, M., Y. Petryna, S. Freitag, P. Edler, A. Schmidt, and T. Lahmer. 2021. “Decision making and design in structural engineering problems under polymorphic uncertainty.” Eng. Struct. 231 (Mar): 111649. https://doi.org/10.1016/j.engstruct.2020.111649.
Du, X. 2006. “Uncertainty analysis with probability and evidence theories.” In Proc., Int. Design Engineering Technical Conf. and Computers and Information in Engineering Conf. New York: ASME Digital Collection. https://doi.org/10.1115/DETC2006-99078.
Dubois, D., and H. Prade. 2009. “Formal representations of uncertainty.” In Decision-making process, 85–156. Melvisharam, Tamil Nadu, India: Indian Society for Technical Education. https://doi.org/10.1002/9780470611876.ch3.
DWD CDC (Deutscher Wetterdienst Climate Data Center). 2022. Jährliche Gebietsmittel der Eistage (Jährliche Anzahl) (). Offenbach am Main, Germany: DWD CDC.
Eldred, M., L. Swiler, and G. Tang. 2011. “Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation.” Reliab. Eng. Syst. Saf. 96 (9): 1092–1113. https://doi.org/10.1016/j.ress.2010.11.010.
Faes, M. G., M. Daub, S. Marelli, E. Patelli, and M. Beer. 2021. “Engineering analysis with probability boxes: A review on computational methods.” Struct. Saf. 93 (Nov): 102092. https://doi.org/10.1016/j.strusafe.2021.102092.
Feldmann, M. 2022. “Türme und maste.” In Petersen Stahlbau, 1197–1297. New York: Springer.
Ferson, S., and L. R. Ginzburg. 1996. “Different methods are needed to propagate ignorance and variability.” Reliab. Eng. Syst. Saf. 54 (2–3): 133–144. https://doi.org/10.1016/S0951-8320(96)00071-3.
Götz, M., W. Graf, and M. Kaliske. 2015. “Structural design with polymorphic uncertainty models.” Int. J. Reliab. Saf. 9 (2/3): 112. https://doi.org/10.1504/IJRS.2015.072715.
Gray, A., A. Wimbush, M. de Angelis, P. Hristov, D. Calleja, E. Miralles-Dolz, and R. Rocchetta. 2022. “From inference to design: A comprehensive framework for uncertainty quantification in engineering with limited information.” Mech. Syst. Signal Process. 165 (Feb): 108210. https://doi.org/10.1016/j.ymssp.2021.108210.
Helton, J., J. Johnson, and W. Oberkampf. 2004. “An exploration of alternative approaches to the representation of uncertainty in model predictions.” Reliab. Eng. Syst. Saf. 85 (1–3): 39–71. https://doi.org/10.1016/j.ress.2004.03.025.
Hoffman, F. O., and J. S. Hammonds. 1994. “Propagation of uncertainty in risk assessments: The need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability.” Risk Anal. 14 (5): 707–712. https://doi.org/10.1111/j.1539-6924.1994.tb00281.x.
Jiang, C., J. Zheng, and X. Han. 2017. “Probability-interval hybrid uncertainty analysis for structures with both aleatory and epistemic uncertainties: A review.” Struct. Multidiscip. Optim. 57 (6): 2485–2502. https://doi.org/10.1007/s00158-017-1864-4.
Kausel, E. 2017. Advanced structural dynamics, 8. Cambridge, UK: Cambridge University Press.
Kiureghian, A. D., and O. Ditlevsen. 2009. “Aleatory or epistemic? Does it matter?” Struct. Saf. 31 (2): 105–112. https://doi.org/10.1016/j.strusafe.2008.06.020.
Kleiner, A., A. Talwalkar, P. Sarkar, and M. I. Jordan. 2014. “A scalable bootstrap for massive data.” J. R. Stat. Soc. B 76 (4): 795–816. https://doi.org/10.1111/rssb.12050.
Limbourg, P., and E. de Rocquigny. 2010. “Uncertainty analysis using evidence theory - confronting level-1 and level-2 approaches with data availability and computational constraints.” Reliab. Eng. Syst. Saf. 95 (5): 550–564. https://doi.org/10.1016/j.ress.2010.01.005.
Marques, M., H. Ynuhui, and A. Marrel. 2018. “Propagation of epistemic uncertainties using dempster-shafer theory in bepu evaluation.” In Proc., ANS Best Estimate Plus Uncertainty Int. Conf. (BEPU 2018). Springfield, IL: American Nuclear Society.
Marzban, S., and T. Lahmer. 2016. “Conceptual implementation of the variance-based sensitivity analysis for the calculation of the first-order effects.” J. Stat. Theory Pract. 10 (4): 589–611. https://doi.org/10.1080/15598608.2016.1207578.
Möller, B., and M. Beer. 2008. “Engineering computation under uncertainty—Capabilities of non-traditional models.” Comput. Struct. 86 (10): 1024–1041. https://doi.org/10.1016/j.compstruc.2007.05.041.
Möller, B., W. Graf, and M. Beer. 2000. “Fuzzy structural analysis using-level optimization.” Comput. Mech. 26 (6): 547–565. https://doi.org/10.1007/s004660000204.
Most, T. 2011. “Efficient structural reliability methods considering incomplete knowledge of random variable distributions.” Probab. Eng. Mech. 26 (2): 380–386. https://doi.org/10.1016/j.probengmech.2010.09.003.
Niederreiter, H. 1992. Random number generation and Quasi-Monte Carlo methods. Philadelphia: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611970081.
Oberkampf, W. L., J. C. Helton, C. A. Joslyn, S. F. Wojtkiewicz, and S. Ferson. 2004. “Challenge problems: Uncertainty in system response given uncertain parameters.” Reliab. Eng. Syst. Saf. 85 (1–3): 11–19. https://doi.org/10.1016/j.ress.2004.03.002.
Pannier, S., M. Waurick, W. Graf, and M. Kaliske. 2013. “Solutions to problems with imprecise data - an engineering perspective to generalized uncertainty models.” Mech. Syst. Sig. Process. 37 (1–2): 105–120. https://doi.org/10.1016/j.ymssp.2012.08.002.
Petersen, C. 2001. Schwingungsdämpfer im Ingenieurbau. München, Germany: Maurer SE.
Petersen, C., and H. Werkle. 2017. Dynamik der Baukonstruktionen. New York: Springer.
Pivovarov, D., et al. 2019. “Challenges of order reduction techniques for problems involving polymorphic uncertainty.” GAMM-Mitt. 42 (2): e201900011. https://doi.org/10.1002/gamm.201900011.
Ristic, B., and P. Smets. 2006. “Belief function theory on the continuous space with an application to model based classification.” In Modern information processing, 11–24. New York: Elsevier.
Schmelzer, B., M. Oberguggenberger, and C. Adam. 2010. “Efficiency of tuned mass dampers with uncertain parameters on the performance of structures under stochastic excitation.” Proc. Inst. Mech. Eng., Part O: J. Risk Reliab. 224 (4): 297–308.
Shafer, G. 1976. Vol. 42 of A mathematical theory of evidence. Princeton, NJ: Princeton University Press.
Shah, H., S. Hosder, and T. Winter. 2015. “A mixed uncertainty quantification approach using evidence theory and stochastic expansions.” Int. J. Uncertainty Quantif. 5 (1): 21–48. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2015010941.
Smets, P. 2005. “Decision making in the TBM: The necessity of the pignistic transformation.” Int. J. Approximate Reasoning 38 (2): 133–147. https://doi.org/10.1016/j.ijar.2004.05.003.
Tonon, F. 2004. “Using random set theory to propagate epistemic uncertainty through a mechanical system.” Reliab. Eng. Syst. Saf. 85 (1–3): 169–181. https://doi.org/10.1016/j.ress.2004.03.010.
Troffaes, M. C. 2018. “Imprecise monte carlo simulation and iterative importance sampling for the estimation of lower previsions.” Int. J. Approximate Reasoning 101 (Oct): 31–48. https://doi.org/10.1016/j.ijar.2018.06.009.
VDI (Verein Deutscher Ingenieure e. V). 2013. VDI 2038-2 Gebrauchstauglichkeit von Bauwerken bei dynamischen Einwirkungen. Düsseldorf, Germany: VDI.
Wang, Q., Y. Feng, D. Wu, G. Li, Z. Liu, and W. Gao. 2022. “Polymorphic uncertainty quantification for engineering structures via a hyperplane modelling technique.” Comput. Methods Appl. Mech. Eng. 398 (May): 115250. https://doi.org/10.1016/j.cma.2022.115250.
Xiao, Z., Q. Zhang, Z. Zhang, W. Bai, and H. Liu. 2023. “A collaborative quasi-monte carlo uncertainty propagation analysis method for multiple types of epistemic uncertainty quantified by probability boxes.” Struct. Multidiscip. Optim. 66 (5): 109. https://doi.org/10.1007/s00158-023-03564-2.
Zabel, V., S. Marwitz, and A. Habtemariam. 2020. “Bestimmung von modalen Parametern seilabgespannter Rohrmasten.” In Vol. 14 of Berichte der Fachtagung Baustatik-Baupraxis, edited by M. Bischoff, M. vonScheven, and B. Oesterle, 519–526. Stuttgart, Germany: Institut für Baustatik und Baudynamik, Universität Stuttgart.
Zadeh, L. 1965. “Fuzzy sets.” Inf. Control 8 (3): 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X.
Zadeh, L. 1999. “Fuzzy sets as a basis for a theory of possibility.” Fuzzy Sets Syst. 1 (1): 3–38. https://doi.org/10.1016/S0165-0114(99)80004-9.
Zhang, H., H. Dai, M. Beer, and W. Wang. 2013. “Structural reliability analysis on the basis of small samples: An interval quasi-Monte Carlo method.” Mech. Syst. Sig. Process. 37 (1–2): 137–151. https://doi.org/10.1016/j.ymssp.2012.03.001.
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ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 10 • Issue 3 • September 2024
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© 2024 American Society of Civil Engineers.
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Received: Jul 15, 2023
Accepted: Dec 23, 2023
Published online: Apr 25, 2024
Published in print: Sep 1, 2024
Discussion open until: Sep 25, 2024
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