Author
Diogo A. Gomes
Other affiliations: Federal University of Ceará, Instituto Superior Técnico, University of Lisbon ...read more
Bio: Diogo A. Gomes is an academic researcher from King Abdullah University of Science and Technology. The author has contributed to research in topics: Uniqueness & Hamilton–Jacobi equation. The author has an hindex of 32, co-authored 177 publications receiving 3266 citations. Previous affiliations of Diogo A. Gomes include Federal University of Ceará & Instituto Superior Técnico.
Papers published on a yearly basis
Papers
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TL;DR: A brief survey of mean-field models as well as recent results and techniques is presented, and a definition of relaxed solution for mean- field games that allows to establish uniqueness under minimal regularity hypothesis is proposed.
Abstract: The mean-field framework was developed to study systems with an infinite number of rational agents in competition, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by Lasry and Lions, and independently around the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhame. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this paper we present a brief survey of mean-field models as well as recent results and techniques. In the first part of this paper, we study reduced mean-field games, that is, mean-field games, which are written as a system of a Hamilton–Jacobi equation and a transport or Fokker–Planck equation. We start by the derivation of the models and by describing some of the existence results available in the literature. Then we discuss the uniqueness of a solution and propose a definition of relaxed solution for mean-field games that allows to establish uniqueness under minimal regularity hypothesis. A special class of mean-field games that we discuss in some detail is equivalent to the Euler–Lagrange equation of suitable functionals. We present in detail various additional examples, including extensions to population dynamics models. This section ends with a brief overview of the random variables point of view as well as some applications to extended mean-field games models. These extended models arise in problems where the costs incurred by the agents depend not only on the distribution of the other agents, but also on their actions. The second part of the paper concerns mean-field games in master form. These mean-field games can be modeled as a partial differential equation in an infinite dimensional space. We discuss both deterministic models as well as problems where the agents are correlated. We end the paper with a mean-field model for price impact.
324 citations
TL;DR: In this article, the authors report on some recent results for mean field models in discrete time with a finite number of states, and address existence, uniqueness and exponential convergence, and check to equilibrium results.
210 citations
TL;DR: In this article, the authors apply partial differential equation (PDE) methods to understand the structure of certain Hamiltonian flows, based upon the ideas of Mather, Moser, Fathi, E and others.
Abstract: This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE (partial differential equation) methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell” or “corrector” PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study of periodic homogenization for Hamilton-Jacobi equations, formally induces a canonical change of variables, in terms of which the dynamics are trivial. We investigate to what extent this observation can be made rigorous in the case that the Hamiltonian is strictly convex in the momenta, given that the relevant PDE does not usually in fact admit a smooth solution.
133 citations
TL;DR: Gomes et al. as mentioned in this paper considered extended stationary mean-field games, which depend on the velocity field of the players, and established the existence of smooth solutions under fairly general conditions by applying the continuity method.
Abstract: In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in Gomes et al. (2012). In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in Lasry and Lions (2006), Gomes and Sanchez-Morgado (2011) or Gomes et al. (2012) this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved.
126 citations
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01 Jan 2006
TL;DR: Probability distributions of linear models for regression and classification are given in this article, along with a discussion of combining models and combining models in the context of machine learning and classification.
Abstract: Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.
10,141 citations
Book•
02 Jan 2013
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.
5,524 citations
TL;DR: The Essay concludes that practitioners theorize, and theorists practice, use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ.
Abstract: Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.
2,077 citations
Posted Content•
TL;DR: This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry, based upon the articles of the three authors and upon unpublished materials they developed.
Abstract: This text is inspired from a "Cours Bachelier" held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials developed by the authors. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class. The content of this text is therefore far more important than the actual "Cours Bachelier" conferences, though the guiding principle is the same and consists in a progressive introduction of the concepts, methodologies and mathematical tools of mean field games theory.
487 citations
01 Jan 2011
TL;DR: The Course Bachelier 2009 as discussed by the authors was inspired from a course inspired by the work of Jean-Michel Lasry, and the course was based upon the articles of the three authors and upon unpublished materials they developed.
Abstract: This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.
479 citations