Computational mean-field games and application

In 2006 J.-M. Lasry and P.-L. Lions introduced the concept of mean-field games to describe strategic interactions of agents in the economic market. The basic assumption (specific for mean-field principles) is that the decision making process of a single agent is not based on every interaction he or she is going to have with others, but on some statistics of the overall community of agents. Mean-field principles were originally introduced in statistical physics, but became a powerful approach in biology and economy. This project is concerned with the development of numerical methods for mean-field games and their applications in optimal transport and life sciences. We discuss the connections between mean-field games and optimal transportation problems as well as the link to optimal control problems. The first objective of this proposal is the development of efficient optimization techniques, like gradient based methods or Kaczmarz-type algorithms. These methods can be applied to different problems in socio- and econophysics, like portfolio management or population dynamics. The second objective is the development of numerical methods for optimal transportation problems. Many well known linear and nonlinear partial differential equations can be recast in the formalism of optimal transport. We propose different techniques like Newton-type methods and semi-implicit schemes for the discretization in time and mixed finite element methods as well as discontinuous Galerkin methods for the discretization in space. In addition we discuss a variational particle scheme, which can be applied to mean-field games and in particular optimal transportation problems. The third objective is the development of numerical methods for Poisson-Fokker-Planck systems with nonlinear diffusion. These systems can be found in different mathematical models in biology, describing chemotactical movement or the flow of ions through a biological channel. Here the proposed methods for the optimal transportation problems will serve as a starting point for the development of efficient numerical schemes for such systems.

The motion of large pedestrian crowds, animal herds or charged particles in biological and synthetic channels can be described on various scales. On the microscopic level the trajectory of each particle is modeled by a set of equations. For a large number of particles it is reasonable to zoom out and look at the distribution of the particles in space and time. Then the evolution of this density can be described by the mean-field limit of the microscopic equations. There are still a lot of open questions in the transition from the microscopic to the macroscopic level, but mean field games have been a powerful tool to translate the microscopic interactions to the corresponding macroscopic partial differential equations.The aim of the project 'Computational Mean Field Games and Applications' was to develop numerical methods for mean field games that allow for accurate and efficient computer simulations. The modeling was driven by various applications like pedestrian motion, animal herding or synthetic and biological channels. An important aspect was the consistent modeling and the analytic behavior of the derived mathematical equations. We were able to obtain a better mathematical understanding of several models, which have been proposed in the literature. This was accomplished by fruitful interactions between analyzing the mathematical equations and simulating their behavior on the computer.Furthermore we revealed novel connections between different modeling approaches, which linked different areas of mathematical research.The project also contributed to the development of the software package MsSimPore for experimentalists, which work in the lab on ion channels and nanopores. MsSimPore provides a graphical user interface to simulate the ow of charged particle through narrow geometries like nanopores or ion channels. It can be used to predict the transportation properties for various types of pores, which is useful for the more efficient development of nanopore diodes or sensors.