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We propose to investigate traffic phenomena from the macroscopic point of view, using models derived from fluid-dynamics consisting in hyperbolic conservation laws. In fact, even if the continuum hypothesis is clearly not physically satisfied, macroscopic quantities can be regarded as measures of traffic features and allow to depict the spatio-temporal evolution of traffic waves. Continuum models have shown to be in good agreement with empirical data. Moreover, they are suitable for analytical investigations and very efficient from the numerical point of view. Therefore, they provide the right framework to state and solve control and optimization problems, and we believe that the use of macroscopic models can open new horizons in traffic management.
The major mathematical difficulties related to this study follow from the mandatory use of weak (possibly discontinuous) solutions in distributional sense. Indeed, due to the presence of shock waves and interactions among them, standard techniques are generally useless for solving optimal control problems, and the available results are scarce and restricted to particular and unrealistic cases. This strongly limits their applicability.
Our scope is to develop a rigorous analytical framework and fast and efficient numerical tools for solving optimization and control problems, such as queues lengths control or buildings exits design. This will allow to elaborate reliable predictions and to optimize traffic fluxes. To achieve this goal, we will move from the detailed structure of the solutions in order to construct ad hoc methods to tackle the analytical and numerical difficulties arising in this study. The foreseen applications target the sustainability and safety issues of modern society.
People
Paola Goatin (PI), Researcher, INRIA Sophia Antipolis - MéditerranéeJean-Antoine Désidéri, Senior Researcher, INRIA Sophia Antipolis - Méditerranée:
specialist in numerical methods for fluid-dynamics equations as well as hierarchical optimization and multi-objective strategies.
Jean-Paul Zolésio, Senior Researcher, CNRS:
shape optimization, control of large flexible structures, free boundary problems in plasma physics, and control of non-cylindrical problems.
Rinaldo M. Colombo, Full Professor, Brescia University:
specialist in hyperbolic systems of conservation laws (basic theory, applications and control), currently involved in the study of models concerning crowd dynamics, traffic flow, gas dynamics, granular matter, phase transitions and combustion.
Mauro Garavello, Research Associate, University of Milano Bicocca:
hyperbolic systems of conservation laws, Hamilton-Jacobi- Bellmann equations and control theory.
Nader El-Khatib, post-doc fellow at INRIA Sophia Antipolis (January - August 2011):
numerical study of 2D models for crowd motion and implementation on Num3sis (co-supervision with Régis Duvigneau).
Jihed Joobeur, Internship Program, INRIA Sophia Antipolis (March - September 2011):
crowd data collection from video recordings (co-supervision with François Brémond, PULSAR project-team).
Maria Laura Delle Monache, PhD student at INRIA Sophia Antipolis (October 2011 - September 2014):
traffic flow modeling by conservation laws (defended on September 18, 2014).
Monika Twarogowska, post-doc fellow at INRIA Sophia Antipolis (March 2012 - September 2013):
shape optimization problems in crowd management (co-supervision with Régis Duvigneau).
Matthias Mimault, PhD student at INRIA Sophia Antipolis (October 2012 - September 2015):
crowd motion modeling by conservation laws (defended on December 14, 2015).
Enrico Bertino, (Intern at INRIA Sophia Antipolis May - November 2015):
data-driven uncertainty quantification in macroscopic traffic flow models.
Aekta Aggarwal, post-doc fellow at INRIA Sophia Antipolis (March 2014 - August 2015):
Analytical and numerical study of macroscopic traffic flow models (co-supervision with Rinaldo M. Colombo).
Publications
• S. Villa, P. Goatin and C. Chalons, Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model, submitted.• P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Comm. Math. Sci., to appear.
• P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11(1) (2016), 107-121.
• A. Aggarwal and P. Goatin, Crowd dynamics through non-Local conservation laws, Bulletin of the Brazilian Mathematical Society, New Series, 47(1) (2016), 37-50. Proceedings of the XV international conference on hyperbolic problems: Theory, numerics, applications.
• M.L. Delle Monache and P. Goatin, A numerical scheme for moving bottlenecks in traffic flow, Bulletin of the Brazilian Mathematical Society, New Series, 47(2) (2016), 605-617. Proceedings of the XV international conference on hyperbolic problems: Theory, numerics, applications.
• P. Goatin and S. Scialanga, The Lighthill-Whitham- Richards traffic flow model with non-local velocity: analytical study and numerical results, INRIA Research Report no. 8685, February 2015.
• S. Samaranayake, J. Reilly, W. Krichene, M.L. Delle Monache, P. Goatin and A. Bayen, Discrete-time system optimal dynamic traffic assignment (SO-DTA) with partial control for horizontal queuing networks, submitted.
• M. Twarogowska, P. Goatin and R. Duvigneau, Comparative study of macroscopic pedestrian models, Transportation Research Procedia 2 ( 2014 ) 477 – 485. The Conference on Pedestrian and Evacuation Dynamics 2014 (PED2014).
• C. Chalons, M.L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, submitted.
• A. Aggarwal, R.M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53(2) (2015), 963-983.
• M.L. Delle Monache and P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J. Differential Equations, 257 (2014), 4015-4029.
• P. Goatin and M. Mimault, A mixed system modeling two-directional pedestrian flows, Math. Biosci. Eng., 12(2) (2015), 375-392.
•S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math.,132(2) (2016), 217-241.
• L.L. Obsu, M.L. Delle Monache, P. Goatin and S.M. Kassa, Traffic flow optimization on roundabouts, Math. Methods Appl. Sci., 38(14) (2015), 3075-3096
• L.L. Obsu, P. Goatin and S.M. Kassa, Gradient-based Instantaneous Traffic Flow Optimization on a Roundabout, preprint (2014).
• J. Reilly, W. Krichene, M.L. Delle Monache, S. Samaranayake, P. Goatin and A. Bayen, Adjoint-based optimization on a network of discretized scalar conservation law PDEs with applications to coordinated ramp metering, J. Optim. Theory Appl., 167(2) (2015), 733-760.
• D. Amadori, P. Goatin and M.D. Rosini, Existence results for Hughes' model for pedestrian flows, J. Math. Anal. Appl., 420(1) (2014), 387-406.
• M. Twarogowska, P. Goatin and R. Duvigneau, Macroscopic modeling and simulations of room evacuation, Appl. Math. Model., 38(24) (2014), 5781-5795.
• M.L. Delle Monache and P. Goatin, A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7(3) (2014), 435-447.
• M.L. Delle Monache, L.L. Obsu, P. Goatin and S.M. Kasa, Traffic flow optimization on roundabouts, Procedia - Social and Behavioral Sciences, (2014), 127-136. Proceedings of EWGT2013 - 16th Meeting of the EURO Working Group on Transportation, September 2013, Porto.
• M. Twarogowska, P. Goatin and R. Duvigneau, Numerical study of macroscopic pedestrian flow models, INRIA Research Report no. 8340, July 2013.
• M.L. Delle Monache and P. Goatin, A strongly coupled PDE-ODE system modeling moving density constraints in traffic flow, in "Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the fourteenth international conference in Padova, June 2012", to appear.
• L.L. Obsu, M.L. Delle Monache, P. Goatin and S.M. Kasa, Macroscopic traffic flow optimization on roundabouts, INRIA Research Report no. 8291, April 2013.
• M.L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin and A. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM J. Appl. Math., 74(1) (2014), 22-39.
• P. Goatin and J.P. Zolesio, Crowd Modeling, in System Modeling and Optimization, IFIP AICT 391, Hömberg, Dietmar; Tröltzsch, Fredi (Eds.), Springer, Heidelberg; IX, p. 89-107, 2013.
• P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes' model of pedestrian motion, SIAM J. Sci. Comput., 35(3) (2013), B606-B622.
• M.L. Delle Monache and P. Goatin, Scalar conservation laws with moving density constraints arising in traffic flow modeling, INRIA Research Report no. 8119, October 2012.
• C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling., Netw. Heterog. Media, to appear.
• N. El-Khatib, P. Goatin and M.D. Rosini, On entropy weak solutions of Hughes'model for pedestrian motion, Z. Angew. Math. Phys. 64(2) (2013), 223-251 (published online on June, 2012).
• M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A 32(6) (2012), 1915-1938.
• M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl. 378(2) (2011), 634-648.
• R.M. Colombo, P. Goatin and M.D. Rosini, On the modeling and management of traffic, ESAIM: M2AN 45 (2011), 853-872.
Dissemination
• P. Goatin: Mathematical modeling of crowds. Workshop "Crowd Flow Dynamics, Modeling and Management", Transportation Research Board 93rd Annual Meeting, Washington, D.C. (USA). (INVITED TALK)• P. Goatin: Macroscopic models of road and pedestrian traffic. Congrès SMAI, Seignosse Le Penon (France), May 2013. (PLENARY TALK)
• M.L. Delle Monache: Scalar conservation laws with moving density constraints arising in traffic flow modeling. Congrès SMAI, Seignosse Le Penon (France), May 2013. (POSTER)
• M. Twarogowska, P. Goatin, R. Duvigneau, N. Aissiouene: On the Hughes' models of crowd dynamics. ESF Research Conference on Applied Partial Differential Equations in Physics, Biology and Social Sciences: Classical and Modern Perspectives, Bellaterra (Spain), September 2012. (POSTER)
• M.L. Delle Monache: Scalar conservation laws with moving density constraints arising in traffic flow modeling. 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Padova (Italy), June 2012. (CONTRIBUTED TALK)
• P. Goatin: A general phase transition model for vehicular traffic. 12th International Conference on Free Boundary Problems, Theory and Application, Frauenchiemsee (Germany), June 2012. (INVITED TALK)
• P. Goatin: Locally constrained conservation laws in traffic management. Conference of the European GDR Control of PDEs, Marseille (France), November 2011. (INVITED TALK)
• P. Goatin: Vehicular traffic management by conservation laws. Workshop "Hyperbolic systems and control in networks", Institut HenriPoincaré, Paris (France), October 2010. (INVITED TALK)
Software
In collaboration with the OPALE project-team (Régis Duvigneau and Nora Aissiouene) we are developing some numerical codes on the Num3sis platform, for simulating crowd movements:Besides, M. Mimaut has developed a MATLAB code for solving numerically a 1D system coupling a conservation law and an eikonal equation, which models crowd movements. It can bee freely downloaded HERE.
Meetings
Workshop TRAM3 Terminus,Sophia Antipolis (France), January 6-8, 2016.
Workshop Traffic Modeling and Management: Trends and Perspectives,
Sophia Antipolis (France), March 20-22, 2013.
HYP2012 - 14th International Conference on Hyperbolic Problems,
Padova (Italy), June 25-29, 2012.