1,721,031 research outputs found
A quadratic mean field games model for the langevin equation
Full text linkWe consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic Fokker–Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system
Approximation scheme for the maximal solution of the shape-from-shading model
Full text linkThe shape-from-shading model leads to a first order Hamilton-Jacobi equation coupled with a boundary condition, f.e. of Dirichlet type. The analytical characterization of the solution presents some difficulties since this is an eikonal type equation which has several weak solutions (in the viscosity sense). The lack of uniqueness is also a big trouble when we try to compute a solution. In order to avoid those difficulties the problem is usually solved adding some additional informations such as the height at points where the brightness has a maximum, or the complete knowledge of a level curve. Here we use recent results in the theory of viscosity solutions to characterize the maximal solution without extra informations besides the equation and we construct an algorithm which converges to that solution. Some examples show the accuracy of the algorithm
A continuous dependence estimate for viscous Hamilton–Jacobi equations on networks with applications
Full text linkWe study continuous dependence estimates for viscous Hamilton–Jacobi equations defined on a network Γ . Given two Hamilton–Jacobi equations, we prove an estimate of the C2 -norm of the difference between the corresponding solutions in terms of the distance among the Hamiltonians. We also provide two applications of the previous estimate: the first one is an existence and uniqueness result for a quasi-stationary Mean Field Games defined on the network Γ ; the second one is an estimate of the rate of convergence for homogenization of Hamilton–Jacobi equations defined on a periodic network, when the size of the cells vanishes and the limit problem is defined in the whole Euclidean space
A System of Hamilton-Jacobi Equations Characterizing Geodesic Centroidal Tessellations
No full textWe introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e., tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and, therefore, the associated tessellations. We propose various numerical examples to illustrate the features of the technique
Rates of convergence for the policy iteration method for Mean Field Games systems
Full text linkConvergence of the policy iteration method for discrete and continuous optimal control problems holds under general assumptions. Moreover, in some circumstances, it is also possible to show a quadratic rate of convergence for the algorithm. For Mean Field Games, convergence of the policy iteration method has been recently proved in [9]. Here, we provide an estimate of its rate of convergence
A System of Hamilton-Jacobi Equations Characterizing Geodesic Centroidal Tessellations
Full text linkWe introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e., tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and, therefore, the associated tessellations. We propose various numerical examples to illustrate the features of the technique
Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative
Full text linkWe study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton–Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem
Rates of convergence in periodic homogenization offully nonlinear uniformly elliptic PDEs
No full textWe consider periodic homogenization of the fully nonlinear uniformly elliptic equation u(epsilon) + H(x, x/epsilon, Du(epsilon), D(2)u(epsilon)) = 0. We give an estimate of the rate of convergence of u(epsilon) to the solution u of the homogenized problem u + H(x, Du, D(2)u) = 0. Moreover we describe a numerical scheme for the approximation of the effective nonlinearity H and we estimate the corresponding rate of convergence
On Quasi-Stationary Mean Field Games of Controls
Full text linkIn Mean Field Games of Controls, the dynamics of the single agent is influenced not only by the distribution of the agents, as in the classical theory, but also by the distribution of their optimal strategies. In this paper, we study quasi-stationary Mean Field Games of Controls, in which the strategy-choice mechanism of the agent is different from the classical case: the generic agent cannot predict the evolution of the population, but chooses its strategy only on the basis of the information available at the given instant of time, without anticipating. We prove existence and uniqueness for the solution of the corresponding quasi-stationary Mean Field Games system under different sets of hypotheses and we provide some examples of models which fall within these hypotheses
A note on Kazdan-Warner equation on networks
Full text linkWe investigate the Kazdan-Warner equation on a network. In this case, the differential equation is defined on each edge, while appropriate transition conditions of Kirchhoff type are prescribed at the vertices. We show that the whole Kazdan-Warner theory, both for the noncritical and the critical case, extends to the present setting
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