Mean field games with nonlinear mobilities in pedestrian dynamics

Martin Burger; Marco Di Francesco; Peter A. Markowich; Marie-Therese Wolfram

doi:10.3934/dcdsb.2014.19.1311

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July 2014, 19(5): 1311-1333. doi: 10.3934/dcdsb.2014.19.1311

Mean field games with nonlinear mobilities in pedestrian dynamics

Martin Burger ^1, , Marco Di Francesco ^2, , Peter A. Markowich ^3, and Marie-Therese Wolfram ^4,

1.	Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany
2.	Department of Mathematical Sciences, 4W, 1.14, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom
3.	King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
4.	Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria

Received April 2013 Revised November 2013 Published April 2014

Abstract
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In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.

Keywords: mean field limit, optimal control, numerical simulations., Pedestrian dynamics, calculus of variations.

Mathematics Subject Classification: 49J20, 35K55, 65K1.

Citation: Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311

References:

[1]	D. Amadori and M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions,, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 259. doi: 10.1016/S0252-9602(12)60016-2. Google Scholar
[2]	L. Ambrosio, S. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps,, Manuscripta Math., 121 (2006), 1. doi: 10.1007/s00229-006-0003-0. Google Scholar
[3]	C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors,, Netw. Heterog. Media, 6 (2011), 351. doi: 10.3934/nhm.2011.6.351. Google Scholar
[4]	C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions,, Comm. Partial Differential Equations, 4 (1979), 1017. doi: 10.1080/03605307908820117. Google Scholar
[5]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375. doi: 10.1007/s002110050002. Google Scholar
[6]	V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways,, Transportation Research Part B: Methodological, 35 (2001), 293. doi: 10.1016/S0191-2615(99)00052-1. Google Scholar
[7]	L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149. doi: 10.1016/0022-1236(89)90005-0. Google Scholar
[8]	C. Brune, 4D Imaging in Tomography and Optimal Nanoscopy,, PhD thesis, (2010). Google Scholar
[9]	M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961. doi: 10.1088/0951-7715/25/4/961. Google Scholar
[10]	M. Burger, M. Di Francesco, J.-F. Pietschmann and B. Schlake, Nonlinear cross-diffusion with size exclusion,, SIAM J. Math. Anal., 42 (2010), 2842. doi: 10.1137/100783674. Google Scholar
[11]	M. Burger, P. A. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations,, Kinet. Relat. Models, 4 (2011), 1025. doi: 10.3934/krm.2011.4.1025. Google Scholar
[12]	C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar
[13]	R. Carmona and F. Delarue, Probabilistic analysis of mean-field games,, SIAM J. Control Optim., 51 (2013), 2705. doi: 10.1137/120883499. Google Scholar
[14]	M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics,, Netw. Heterog. Media, 6 (2011), 425. doi: 10.3934/nhm.2011.6.425. Google Scholar
[15]	R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202511500230. Google Scholar
[16]	R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations,, in Current advances in nonlinear analysis and related topics, (2010), 255. Google Scholar
[17]	M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case,, J. Differential Equations, 250 (2011), 1334. doi: 10.1016/j.jde.2010.10.015. Google Scholar
[18]	C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Math. Comput. Modelling, 52 (2010), 1506. doi: 10.1016/j.mcm.2010.06.012. Google Scholar
[19]	L. Dyson, P. Maini and R. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 86 (2012). doi: 10.1103/PhysRevE.86.031903. Google Scholar
[20]	H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems,, IMA J. Numer. Anal., 30 (2010), 1206. doi: 10.1093/imanum/drn083. Google Scholar
[21]	N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes' model for pedestrian motion,, Z. Angew. Math. Phys., 64 (2013), 223. doi: 10.1007/s00033-012-0232-x. Google Scholar
[22]	P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes' model of pedestrian motion,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/120898863. Google Scholar
[23]	D. Gomes and J. Saúde, Mean Field Games - a Brief Survey,, Technical report, (2013). Google Scholar
[24]	O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in Paris-Princeton Lectures on Mathematical Finance 2010, (2010), 205. doi: 10.1007/978-3-642-14660-2_3. Google Scholar
[25]	D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic,, Nature, 407 (2000), 487. Google Scholar
[26]	D. Helbing and P. Molnar, Social force model for pedestrian dynamics,, Physical Review E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar
[27]	S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian route-choice and activity scheduling theory and models,, Transportation Research Part B: Methodological, 38 (2004), 169. doi: 10.1016/S0191-2615(03)00007-9. Google Scholar
[28]	R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar
[29]	H. Ishii, Asymptotic solutions for large time of hamilton-jacobi equations in euclidean n space,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 25 (2008), 231. doi: 10.1016/j.anihpc.2006.09.002. Google Scholar
[30]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics),, 2nd edition, (1991), 07. doi: 10.1007/978-1-4612-0949-2. Google Scholar
[31]	A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds,, Transportation Research Part B: Methodological, 45 (2011), 1572. doi: 10.1016/j.trb.2011.07.011. Google Scholar
[32]	J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. Google Scholar
[33]	M. Moussad, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettr, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds,, PLoS Comput. Biol., 8 (2012). doi: 10.1371/journal.pcbi.1002442. Google Scholar
[34]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar
[35]	R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997). Google Scholar
[36]	M. Simpson, B. Hughes and K. Landman, Diffusing populations: Ghosts or folks,, Australasian Journal of Engineering Education, 15 (2009), 59. Google Scholar
[37]	J. van den Berg, S. Patil, J. Sewall, D. Manocha and M. Lin, Interactive navigation of multiple agents in crowded environments,, in Proceedings of the 2008 symposium on Interactive 3D graphics and games, (2008), 139. Google Scholar

show all references

References:

[1]	D. Amadori and M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions,, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 259. doi: 10.1016/S0252-9602(12)60016-2. Google Scholar
[2]	L. Ambrosio, S. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps,, Manuscripta Math., 121 (2006), 1. doi: 10.1007/s00229-006-0003-0. Google Scholar
[3]	C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors,, Netw. Heterog. Media, 6 (2011), 351. doi: 10.3934/nhm.2011.6.351. Google Scholar
[4]	C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions,, Comm. Partial Differential Equations, 4 (1979), 1017. doi: 10.1080/03605307908820117. Google Scholar
[5]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375. doi: 10.1007/s002110050002. Google Scholar
[6]	V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways,, Transportation Research Part B: Methodological, 35 (2001), 293. doi: 10.1016/S0191-2615(99)00052-1. Google Scholar
[7]	L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149. doi: 10.1016/0022-1236(89)90005-0. Google Scholar
[8]	C. Brune, 4D Imaging in Tomography and Optimal Nanoscopy,, PhD thesis, (2010). Google Scholar
[9]	M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961. doi: 10.1088/0951-7715/25/4/961. Google Scholar
[10]	M. Burger, M. Di Francesco, J.-F. Pietschmann and B. Schlake, Nonlinear cross-diffusion with size exclusion,, SIAM J. Math. Anal., 42 (2010), 2842. doi: 10.1137/100783674. Google Scholar
[11]	M. Burger, P. A. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations,, Kinet. Relat. Models, 4 (2011), 1025. doi: 10.3934/krm.2011.4.1025. Google Scholar
[12]	C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar
[13]	R. Carmona and F. Delarue, Probabilistic analysis of mean-field games,, SIAM J. Control Optim., 51 (2013), 2705. doi: 10.1137/120883499. Google Scholar
[14]	M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics,, Netw. Heterog. Media, 6 (2011), 425. doi: 10.3934/nhm.2011.6.425. Google Scholar
[15]	R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202511500230. Google Scholar
[16]	R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations,, in Current advances in nonlinear analysis and related topics, (2010), 255. Google Scholar
[17]	M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case,, J. Differential Equations, 250 (2011), 1334. doi: 10.1016/j.jde.2010.10.015. Google Scholar
[18]	C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Math. Comput. Modelling, 52 (2010), 1506. doi: 10.1016/j.mcm.2010.06.012. Google Scholar
[19]	L. Dyson, P. Maini and R. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 86 (2012). doi: 10.1103/PhysRevE.86.031903. Google Scholar
[20]	H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems,, IMA J. Numer. Anal., 30 (2010), 1206. doi: 10.1093/imanum/drn083. Google Scholar
[21]	N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes' model for pedestrian motion,, Z. Angew. Math. Phys., 64 (2013), 223. doi: 10.1007/s00033-012-0232-x. Google Scholar
[22]	P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes' model of pedestrian motion,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/120898863. Google Scholar
[23]	D. Gomes and J. Saúde, Mean Field Games - a Brief Survey,, Technical report, (2013). Google Scholar
[24]	O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in Paris-Princeton Lectures on Mathematical Finance 2010, (2010), 205. doi: 10.1007/978-3-642-14660-2_3. Google Scholar
[25]	D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic,, Nature, 407 (2000), 487. Google Scholar
[26]	D. Helbing and P. Molnar, Social force model for pedestrian dynamics,, Physical Review E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar
[27]	S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian route-choice and activity scheduling theory and models,, Transportation Research Part B: Methodological, 38 (2004), 169. doi: 10.1016/S0191-2615(03)00007-9. Google Scholar
[28]	R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar
[29]	H. Ishii, Asymptotic solutions for large time of hamilton-jacobi equations in euclidean n space,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 25 (2008), 231. doi: 10.1016/j.anihpc.2006.09.002. Google Scholar
[30]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics),, 2nd edition, (1991), 07. doi: 10.1007/978-1-4612-0949-2. Google Scholar
[31]	A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds,, Transportation Research Part B: Methodological, 45 (2011), 1572. doi: 10.1016/j.trb.2011.07.011. Google Scholar
[32]	J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. Google Scholar
[33]	M. Moussad, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettr, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds,, PLoS Comput. Biol., 8 (2012). doi: 10.1371/journal.pcbi.1002442. Google Scholar
[34]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar
[35]	R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997). Google Scholar
[36]	M. Simpson, B. Hughes and K. Landman, Diffusing populations: Ghosts or folks,, Australasian Journal of Engineering Education, 15 (2009), 59. Google Scholar
[37]	J. van den Berg, S. Patil, J. Sewall, D. Manocha and M. Lin, Interactive navigation of multiple agents in crowded environments,, in Proceedings of the 2008 symposium on Interactive 3D graphics and games, (2008), 139. Google Scholar

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