PDE Models for Multi-Agent Phenomena

herausgegeben von: Prof. Pierre Cardaliaguet, Prof. Alessio Porretta, Prof. Francesco Salvarani

Verlag: Springer International Publishing

Buchreihe : Springer INdAM Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This volume covers selected topics addressed and discussed during the workshop “PDE models for multi-agent phenomena,” which was held in Rome, Italy, from November 28th to December 2nd, 2016. The content mainly focuses on kinetic equations and mean field games, which provide a solid framework for the description of multi-agent phenomena. The book includes original contributions on the theoretical and numerical study of the MFG system: the uniqueness issue and finite difference methods for the MFG system, MFG with state constraints, and application of MFG to market competition. The book also presents new contributions on the analysis and numerical approximation of the Fokker-Planck-Kolmogorov equations, the isotropic Landau model, the dynamical approach to the quantization problem and the asymptotic methods for fully nonlinear elliptic equations. Chiefly intended for researchers interested in the mathematical modeling of collective phenomena, the book provides an essential overview of recent advances in the field and outlines future research directions.

Inhaltsverzeichnis

Frontmatter

Uniqueness of Solutions in Mean Field Games with Several Populations and Neumann Conditions

Abstract

We study the uniqueness of solutions to systems of PDEs arising in Mean Field Games with several populations of agents and Neumann boundary conditions. The main assumption requires the smallness of some data, e.g., the length of the time horizon. This complements the existence results for MFG models of segregation phenomena introduced by the authors and Achdou. An application to robust Mean Field Games is also given.

Martino Bardi, Marco Cirant

Finite Difference Methods for Mean Field Games Systems

Abstract

We discuss convergence results for a class of finite difference schemes approximating Mean Field Games systems either on the torus or a network. We also propose a quasi-Newton method for the computation of discrete solutions, based on a least squares formulation of the problem. Several numerical experiments are carried out including the case with two or more competing populations.

Simone Cacace, Fabio Camilli

Existence and Uniqueness for Mean Field Games with State Constraints

Abstract

In this paper, we study deterministic mean field games for agents who operate in a bounded domain. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of the solution to the associated minimization problem is no longer guaranteed. We attack the problem by interpreting equilibria as measures in a space of arcs. In such a relaxed environment the existence of solutions follows by set-valued fixed point arguments. Then, we give a uniqueness result for such equilibria under a classical monotonicity assumption.

Piermarco Cannarsa, Rossana Capuani

An Adjoint-Based Approach for a Class of Nonlinear Fokker-Planck Equations and Related Systems

Abstract

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer properties of schemes for HJ equations to FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

Adriano Festa, Diogo A. Gomes, Roberto M. Velho

Variational Mean Field Games for Market Competition

Abstract

In this paper, we explore Bertrand and Cournot Mean Field Games models for market competition with reflection boundary conditions. We prove existence, uniqueness and regularity of solutions to the system of equations, and show that this system can be written as an optimality condition of a convex minimization problem. We also provide a short proof of uniqueness to the system addressed in Graber and Bensoussan (Appl Math Optim 77:47–71, 2018), where uniqueness was only proved for small parameters 𝜖. Finally, we prove existence and uniqueness of weak solutions to the corresponding first order system at the deterministic limit.

Philip Jameson Graber, Charafeddine Mouzouni

A Review for an Isotropic Landau Model

Abstract

We consider the equation

$$\displaystyle u_t = \mathrm{div}\,(a[u]\nabla u - u\nabla a[u]),\qquad -\Delta a = u. $$

This model has attracted some attention in the recent years and several results are available in the literature. We review recent results on existence and smoothness of solutions and explain the open problems.

Maria Gualdani, Nicola Zamponi

A Gradient Flow Perspective on the Quantization Problem

Abstract

In this paper we review recent results by the author on the problem of quantization of measures. More precisely, we propose a dynamical approach, and we investigate it in dimensions 1 and 2. Moreover, we discuss a recent general result on the static problem on arbitrary Riemannian manifolds.

Mikaela Iacobelli

Asymptotic Methods in Regularity Theory for Nonlinear Elliptic Equations: A Survey

Abstract

We survey recent asymptotic methods introduced in regularity theory for fully nonlinear elliptic equations. Our presentation focuses mainly on the recession function. We detail the role of this class of techniques through examples and results. Our applications include regularity in Sobolev and Hölder spaces. In addition, we produce a density result and examine ellipticity-invariant quantities, such as the Escauriaza’s exponent.

Edgard A. Pimentel, Makson S. Santos

A Fully-Discrete Scheme for Systems of Nonlinear Fokker-Planck-Kolmogorov Equations

Abstract

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied in Carlini and Silva (SIAM J. Numer. Anal., 2018, To appear) for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games.

Elisabetta Carlini, Francisco J. Silva

Titel: PDE Models for Multi-Agent Phenomena
herausgegeben von: Prof. Pierre Cardaliaguet
Prof. Alessio Porretta
Prof. Francesco Salvarani
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-01947-1
Print ISBN: 978-3-030-01946-4
DOI: https://doi.org/10.1007/978-3-030-01947-1