We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models.
At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction pressures, and that are set on the contact network. At the macroscopic level, a similar problem is obtained, that is set on the congested zone.
We emphasize the differences between the two settings: at the macroscopic level, a straight use of the maximum principle shows that congestion actually favors evacuation, which is in contradiction with experimental evidence. On the contrary, in the microscopic setting, the very particular structure of the discrete differential operators makes it possible to reproduce observed "Stop and Go waves", and the so called "Faster is Slower" effect.[-]

This minicourse aims at providing tentative explanations of some specific phenomena observed in the motion of crowds, or more generally collections of living entities. The first lecture shall focus on the so-called Stop and Go Waves, which sometimes spontaneously emerge and persist in crowds in motion. We shall present a general class of dynamical systems which are likely to exhibit this type of instabilities, and emphasize the critical role of two basic ingredients: the asymmetry of interactions, and any sort of delay in the transmission of information through the network of entities. The second lecture will address the Capacity Drop Phenomenon (decrease of the flux though a bottleneck when the upstream density becomes too high), and the more paradoxical Faster is Slower Effect (in some regimes, attempts to go quicker may slow down the overall process). We shall in particular detail how an accurate description of the relative position of entities (at the microscopic level) is crucial to recover and understand those effects.[-]

Optimal transport, a mathematical theory which developed out of a problem raised by Gaspard Monge in the 18th century and of the reformulation that Leonid Kantorovich gave of it in the 20th century in connection with linear programming, is now a very lively branch of mathematics at the intersection of analysis, PDEs, probability, optimization and many applications, ranging from fluid mechanics to economics, from differential geometry to data sciences. In this short course we will have a very basic introduction to this field. The first lecture (2h) will be mainly devoted to the problem itself: given two distributions of mass, find the optimal displacement transforming the first one into the second (studying existence of such an optimal solution and its main properties). The second one (2h) will be devoted to the distance on mass distributions (probability measures) induced by the optimal cost, looking at topological questions (which is the induced topology?) as well as metric ones (which curves of measures are Lipschitz continuous for such a distance? what can we say about their speed, and about geodesic curves?) in connection with very natural PDEs such as the continuity equation deriving from mass conservation.[-]

In this course, we will consider the development and the analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. In the first part, we will review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In the second part, we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes. Since, in all models a degree of uncertainty is implicitly embedded which can be due to the lack of knowledge about the microscopic interaction details, incomplete informations on the initial state or at the boundaries, a last part will be dedicated to an overview of numerical methods to deal with the quantification of the uncertainties in kinetic equations. Applications of the models and the numerical methods to different fields ranging from physics to biology and social sciences will be discussed as well.[-]

What would be the impact of an environment change on the persistence and the genetic/phenotypic distribution of a population? We present some integro-differential models describing the evolutionary adaptation of asexual phenotypically structured populations subject to mutation and selection in changing environments. Using an approach based on Hamilton-Jacobi equations, we provide an asymptotic analysis of such equations in the regime of small mutational variance. This analysis allows us to characterize different evolutionary outcomes depending on the type of the environmental change.[-]

We consider statistical and computation methods to infer trajectories of a stochastic process from snapshots of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data. The goal of this mini-course is to present and understand the estimator proposed by [Lavenant et al. 2020] which searches for the diffusion process that fits the observations with minimal entropy relative to a Wiener process. This estimator comes with consistency guarantees—for a suitable class of ground truth processes—and lends itself to computational methods with global optimality guarantees. Its analysis is the occasion to review important tools from optimal transport and diffusion process theory.[-]

Optimal transport, a mathematical theory which developed out of a problem raised by Gaspard Monge in the 18th century and of the reformulation that Leonid Kantorovich gave of it in the 20th century in connection with linear programming, is now a very lively branch of mathematics at the intersection of analysis, PDEs, probability, optimization and many applications, ranging from fluid mechanics to economics, from differential geometry to data sciences. In this short course we will have a very basic introduction to this field. The first lecture (2h) will be mainly devoted to the problem itself: given two distributions of mass, find the optimal displacement transforming the first one into the second (studying existence of such an optimal solution and its main properties). The second one (2h) will be devoted to the distance on mass distributions (probability measures) induced by the optimal cost, looking at topological questions (which is the induced topology?) as well as metric ones (which curves of measures are Lipschitz continuous for such a distance? what can we say about their speed, and about geodesic curves?) in connection with very natural PDEs such as the continuity equation deriving from mass conservation.[-]

In this course, we will consider the development and the analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. In the first part, we will review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In the second part, we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes. Since, in all models a degree of uncertainty is implicitly embedded which can be due to the lack of knowledge about the microscopic interaction details, incomplete informations on the initial state or at the boundaries, a last part will be dedicated to an overview of numerical methods to deal with the quantification of the uncertainties in kinetic equations. Applications of the models and the numerical methods to different fields ranging from physics to biology and social sciences will be discussed as well.[-]

What would be the impact of an environment change on the persistence and the genetic/phenotypic distribution of a population? We present some integro-differential models describing the evolutionary adaptation of asexual phenotypically structured populations subject to mutation and selection in changing environments. Using an approach based on Hamilton-Jacobi equations, we provide an asymptotic analysis of such equations in the regime of small mutational variance. This analysis allows us to characterize different evolutionary outcomes depending on the type of the environmental change.[-]

We consider statistical and computation methods to infer trajectories of a stochastic process from snapshots of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data. The goal of this mini-course is to present and understand the estimator proposed by [Lavenant et al. 2020] which searches for the diffusion process that fits the observations with minimal entropy relative to a Wiener process. This estimator comes with consistency guarantees—for a suitable class of ground truth processes—and lends itself to computational methods with global optimality guarantees. Its analysis is the occasion to review important tools from optimal transport and diffusion process theory.[-]