Abstract
In this work we investigate the ability of a kinetic approach for traffic dynamics to predict speed distributions obtained through rough data. The present approach adopts the formalism of uncertainty quantification, since reaction strengths are uncertain and linked to different types of driver behaviour or different classes of vehicles present in the flow. Therefore, the calibration of the expected speed distribution has to face the reconstruction of the distribution of the uncertainty. We adopt experimental microscopic measurements recorded on a German motorway, whose speed distribution shows a multimodal trend. The calibration is performed by extrapolating the uncertainty parameters of the kinetic distribution via a constrained optimisation approach. The results confirm the validity of the theoretical set-up.
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References
D.S. Berry, D.M. Belmont, Distribution of vehicle speeds and travel times, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, 1951), pp. 589–602
S. Cordier, L. Pareschi, G. Toscani, On a kinetic model for a simple market economy. J. Stat. Phys. 120(1), 253–277 (2005)
G. Dimarco, L. Pareschi, M. Zanella, Uncertainty quantification for kinetic models in socio-economic and life sciences, in Uncertainty Quantification for Hyperbolic and Kinetic Equations. SEMA-SIMAI Springer Series, vol. 14, ed. by S. Jin, L. Pareschi (Springer, Berlin, 2017), pp. 151–191
D.C. Gazis, R. Herman, R.W. Rothery, Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9(4), 545–567 (1961)
Global Status Report on Road Safety. Technical report (World Health Organization, Switzerland, 2018)
M. Günter, A. Klar, T. Materne, R. Wegener, An explicitly solvable kinetic model for vehicular traffic and associated macroscopic equations. Math. Comp. Model. 35(5–6), 591–606 (2002)
M. Herty, L. Pareschi, Fokker-Planck asymptotics for traffic flow models. Kinet. Relat. Mod. 3(1), 165–179 (2010)
M. Herty, A. Fazekas, G. Visconti, A two-dimensional data-driven model for traffic flow on highways. Netw. Heterog. Media 13(2), 217–240 (2018)
M. Herty, A. Tosin, G. Viconti, M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics. SIAM J. Appl. Math. 78(5), 2737–2762 (2018)
R.W. Hockney, J.W. Eastwook, Computer Simulation using Particles (McGraw Hill International Book Co., New York, 1981)
S.P. Hoogendoorn, Traffic flow theory and simulation, in Lecture notes CT4821 (Delft University of Technology, New York, 2007)
J. Hu, S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Hyperbolic and Kinetic Equations. SEMA-SIMAI Springer Series, vol. 14, ed. by S. Jin, L. Pareschi (Springer, Berlin, 2017), pp. 193–229
R. Illner, A. Klar, H. Lange, A. Unterreiter, R. Wegener, A kinetic model for vehicular traffic: existence of stationary solutions. J. Math. Anal. Appl. 237, 622–643 (1999)
E. Kallo, A. Fazekas, S. Lamberty, M. Oeser, Microscopic traffic data obtained from videos recorded on a German motorway. Mendeley Data v1, 7 (2019)
A. Klar, R. Wegener, Enskog-like models for vehicular traffic. J. Stat. Phys. 87(1–2), 91–114 (1997)
A.K. Maurya, S. Das, S. Dey, S. Nama, Study on speed and time-headway distributions on two-lane bidirectional road in heterogeneous traffic condition. Transp. Res. Proc. 17, 428–437 (2016)
C.R. Munigety, Modelling behavioural interactions of drivers in mixed traffic conditions. J. Traffic Transp. Eng. 5(4), 284–295 (2018)
C.R. Munigety, T.V. Mathew, Towards behavioral modeling of drivers in mixed traffic conditions. Transp. Dev. Econom. 2(6) (2016)
D. Ni, H.K. Hsieh, T. Jiang, Modeling phase diagrams as stochastic processes with application in vehicular traffic flow. Appl. Math. Model. 53, 106–117 (2018)
L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford University, Oxford, 2013)
B. Piccoli, A. Tosin, M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory. Preprint arXiv:1911.04911
G. Puppo, M. Semplice, A. Tosin, G. Visconti, Fundamental diagrams in traffic flow: the case of heterogeneous kinetic models. Commun. Math. Sci. 14(3), 643–669 (2016)
G. Puppo, M. Semplice, A. Tosin, G. Visconti, Analysis of a multi-population kinetic model for traffic flow. Commun. Math. Sci. 15(2), 379–412 (2017)
B. Seibold, M.R. Flynn, A.R. Kasimov, R.R. Rosales, Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Netw. Heterog. Media 8(3), 745–772 (2013)
G. Toscani, Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)
A. Tosin, M. Zanella, Boltzmann-type models with uncertain binary interactions. Commun. Math. Sci. 16(4), 963–985 (2018)
A. Tosin, M. Zanella, Control strategies for road risk mitigation in kinetic traffic modelling. IFAC-PapersOnLine 51(9), 67–72 (2018)
A. Tosin, M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls. Math. Control Relat. Fields (2019). Preprint arXiv:1904.00257
A. Tosin, M. Zanella, Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles. Multiscale Model. Simul. 17(2), 716–749 (2019)
G. Visconti, M. Herty, G. Puppo, A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit. Multiscale Model. Simul. 15, 1267–1293 (2017)
D. Xiu, Numerical Methods for Stochastic Computations. (Princeton University, Princeton, 2010)
Acknowledgements
This research was partially supported by the Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018–2022)—Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino (CUP:E11G18000350001) and Department of Mathematics “F. Casorati”, University of Pavia; and through the PRIN 2017 project (No. 2017KKJP4X) “Innovative numerical methods for evolutionary partial differential equations and applications”.
This work is also part of the activities of the Starting Grant “Attracting Excellent Professors” funded by “Compagnia di San Paolo” (Torino) and promoted by Politecnico di Torino.
A.T. and M.Z. are members of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy.
The research of M.H. and G.V. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2023 Internet of Production—390621612.
M.H. and G.V. acknowledge the ISAC institute at RWTH Aachen, Prof. M. Oeser, Dr. A. Fazekas, MSc. M. Berghaus and MSc. E. Kalló for kindly providing the trajectory data within the DFG project “Basic Evaluation for Simulation-Based Crash-Risk-Models: Multi-Scale Modelling Using Dynamic Traffic Flow States”.
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Herty, M., Tosin, A., Visconti, G., Zanella, M. (2021). Reconstruction of Traffic Speed Distributions from Kinetic Models with Uncertainties. In: Puppo, G., Tosin, A. (eds) Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models. SEMA SIMAI Springer Series(), vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-66560-9_1
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