7,007 research outputs found
Stationary Mean Field Games systems defined on networks
We consider a stationary Mean Field Games system defined on a network. In
this framework, the transition conditions at the vertices play a crucial role:
the ones here considered are based on the optimal control interpretation of the
problem. We prove separately the well-posedness for each of the two equations
composing the system. Finally, we prove existence and uniqueness of the
solution of the Mean Field Games system
A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks
Three definitions of viscosity solutions for Hamilton-Jacobi equations on
networks recently appeared in literature ([1,4,6]). Being motivated by various
applications, they appear to be considerably different. Aim of this note is to
establish their equivalence
Continuous dependence estimates and homogenization of quasi-monotone systems of fully nonlinear second order parabolic equations
Aim of this paper is to extend the continuous dependence estimates proved in
\cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic
equations. As by-product of these estimates, we get an H\"older estimate for
bounded solutions of systems and a rate of convergence estimate for the
vanishing viscosity approximation. In the second part of the paper we employ
similar techniques to study the periodic homogenization of quasi-monotone
systems of fully nonlinear second-order uniformly parabolic equations. Finally,
some examples are discussed
Online Reciprocal Recommendation with Theoretical Performance Guarantees
A reciprocal recommendation problem is one where the goal of learning is not
just to predict a user's preference towards a passive item (e.g., a book), but
to recommend the targeted user on one side another user from the other side
such that a mutual interest between the two exists. The problem thus is sharply
different from the more traditional items-to-users recommendation, since a good
match requires meeting the preferences of both users. We initiate a rigorous
theoretical investigation of the reciprocal recommendation task in a specific
framework of sequential learning. We point out general limitations, formulate
reasonable assumptions enabling effective learning and, under these
assumptions, we design and analyze a computationally efficient algorithm that
uncovers mutual likes at a pace comparable to those achieved by a clearvoyant
algorithm knowing all user preferences in advance. Finally, we validate our
algorithm against synthetic and real-world datasets, showing improved empirical
performance over simple baselines
A model problem for Mean Field Games on networks
In [14], Gueant, Lasry and Lions considered the model problem ``What time
does meeting start?'' as a prototype for a general class of optimization
problems with a continuum of players, called Mean Field Games problems. In this
paper we consider a similar model, but with the dynamics of the agents defined
on a network. We discuss appropriate transition conditions at the vertices
which give a well posed problem and we present some numerical results
Eikonal equations on the Sierpinski gasket
We study the eikonal equation on the Sierpinski gasket in the spirit of the
construction of the Laplacian in Kigami [8]: we consider graph eikonal
equations on the prefractals and we show that the solutions of these problems
converge to a function defined on the fractal set. We characterize this limit
function as the unique metric viscosity solution to the eikonal equation on the
Sierpinski gasket according to the definition introduced in [3]
A numerical method for Mean Field Games on networks
We propose a numerical method for stationary Mean Field Games defined on a
network. In this framework a correct approximation of the transition conditions
at the vertices plays a crucial role. We prove existence, uniqueness and
convergence of the scheme and we also propose a least squares method for the
solution of the discrete system. Numerical experiments are carried out
The vanishing viscosity limit for Hamilton-Jacobi equations on Networks
For a Hamilton-Jacobi equation defined on a network, we introduce its
vanishing viscosity approximation. The elliptic equation is given on the edges
and coupled with Kirchhoff-type conditions at the transition vertices. We prove
that there exists exactly one solution of this elliptic approximation and
mainly that, as the viscosity vanishes, it converges to the unique solution of
the original problem
Eikonal equations on ramified spaces
We generalize the results in [16] to higher dimensional ramified spaces. For
this purpose we introduce ramified manifolds and, as special cases, locally
elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a
theory of viscosity solutions for Hamilton-Jacobi equations, providing
existence and uniqueness results
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