arXiv subjects

Fabio Camilli

Publications and source records attributed to Fabio Camilli.

At least 19 recordsLinked to original sources

arXiv2026

Markov Chain Approximation of Sticky Diffusions and Hamilton-Jacobi-Bellman Equations on Networks

We propose a discrete Markov-chain approximation of diffusion processes on networks with both Kirchhoff and sticky vertex conditions. Stickiness is modeled by a probabilistic residence mechanism at the vertex, while the motion along the edges follows an Euler-Maruyama-type update at the diffusive scale. We prove that the associated time-interpolated chain converges in distribution to the limiting diffusion in the Skorokhod space using the Ethier-Kurtz framework. Based on this construction, we derive a fully discrete semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations on networks and establish its convergence using viscosity solution techniques.

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arXiv2026

Kolmogorov $\varepsilon$-entropy of numerical solutions for scalar conservation laws with convex flux

Building on the information-theoretic perspective of P.~D.~Lax [\textit{Proc.\ Sympos., Math.\ Res.\ Center, Univ.\ Wisconsin}, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation laws with a uniformly convex flux, expressed in terms of Kolmogorov $\varepsilon$-entropy. We prove that, under specific grid constraints, conservative, monotone finite-difference schemes satisfying a discrete one-sided Lipschitz condition (OSLC) preserve the $1/\varepsilon$ Kolmogorov entropy scaling of the corresponding exact entropy solution set, matching the bounds obtained by De~Lellis and Golse [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{58} (2005)] and by Ancona, Glass, and Nguyen [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{65} (2012)]. Specifically, the upper bound follows from the discrete OSLC, while the lower bound relies on a uniform approximation argument on a bounded-variation precursor class. Our results show that prototypical first-order methods are high-resolution in Lax's sense. Finally, we abstract the lower bound mechanism into a general transfer principle, discuss implications for information recovery via post-processing, and indicate directions for future work.

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arXiv2026

Semi-Discrete Approximation of Aubry and Mather sets

We study the semi-discrete approximation of Aubry and Mather sets for Tonelli Lagrangians on the flat torus. Starting from the discrete Lax--Oleinik equation, we introduce natural discrete analogues of these sets and analyze their convergence, as the time step tends to zero, in the sense of Kuratowski. Our results show that the semi-discrete variational framework captures not only the ergodic constant, but also the minimizing invariant geometry of the continuous dynamics. In full generality, we prove upper Kuratowski limit inclusions for both the Aubry and Mather sets. For the Aubry set, we establish full convergence under a hyperbolicity assumption on the continuous Aubry set. For the Mather set, we prove full convergence under a genericity assumption ensuring that the Lagrangian admits finitely many ergodic Mather measures. This provides a first rigorous step toward a structure-preserving approximation theory for Aubry and Mather sets in the Tonelli setting, and clarifies how discrete variational models recover the central geometric objects of weak KAM and Aubry--Mather theory.

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arXiv2026

A Mean Field Games Perspective on Evolutionary Clustering

We propose a control-theoretic framework for evolutionary clustering based on quasi-stationary Mean Field Games. Each cluster is represented by a probability density whose evolution is governed by a Fokker--Planck equation, while the associated velocity field is determined through a stationary Hamilton--Jacobi equation. The general formulation does not prescribe a finite-dimensional statistical shape for the component densities, although the number of components is fixed. In the Gaussian specialization, we show that suitable affine dynamics reproduce the mean and covariance trajectories generated by the classical Expectation--Maximization procedure. To improve temporal coherence in the presence of noise and temporary cluster overlaps, we introduce causal and non-causal time-averaged log-likelihood objectives. We also develop a fully density-based numerical implementation for non-Gaussian components. The proposed formulations are assessed on synthetic and real time-dependent datasets and compared with independent snapshot Expectation--Maximization and with the same method applied to temporally smoothed observations. In the two-dimensional benchmark, an established evolutionary $k$-means method is also included as an external dynamic-clustering baseline.

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arXiv2025

$L^p$ Estimates for Numerical Approximation of Convex Hamilton-Jacobi Equations

We establish $L^p$ error estimates for monotone numerical schemes approximating convex Hamilton-Jacobi equations on the $d$-dimensional torus. Using the adjoint method and semiconcavity estimates, we first prove an $L^1$ error bound of order one for classical monotone schemes of Crandall-Lions type and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian and semiconcavity assumptions on the initial datum. By interpolation with the classical $L^\infty$ estimate, we obtain $L^p$ estimates for every $1\le p<+\infty$.

math.AP↗

arXiv2025

A semi-Lagrangian method for solving state constraint Mean Field Games in Macroeconomics

We study continuous-time heterogeneous agent models cast as Mean Field Games, in the Aiyagari-Bewley-Huggett framework. The model couples a Hamilton-Jacobi-Bellman equation for individual optimization with a Fokker-Planck-Kolmogorov equation for the wealth distribution. We establish a comparison principle for constrained viscosity solutions of the HJB equation and propose a semi-Lagrangian (SL) scheme for its numerical solution, proving convergence via the Barles-Souganidis method. A policy iteration algorithm handles state constraints, and a dual SL scheme is used for the FPK equation. Numerical methods are presented in a fully discrete, implementable form.

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arXiv2025

Approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields

The aim of this work is to investigate semi-Lagrangian approximation schemes on unstructured grids for viscous transport and conservative equations with measurable coefficients that satisfy a one-sided Lipschitz condition. To establish the convergence of the schemes, we exploit the characterization of the solution for these equations expressed in terms of measurable time-dependent viscosity solution and, respectively, duality solution. We supplement our theoretical analysis with various numerical examples to illustrate the features of the schemes.

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arXiv2025

Li-Yau inequality and related properties on metric star graphs

We prove a Li-Yau gradient estimate for positive solutions to the heat equation defined on a metric star graph $\mG$ given by the heat kernel formula. As consequence, we derive a Harnack estimate and a Liouville property for bounded harmonic functions. The argument exploits an explicit representation formula for the heat kernel on $\mG$.

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arXiv2024

A note on first order quasi-stationary Mean Field Games

Quasi-stationary Mean Field Games models consider agents who base their strategies on current information without forecasting future states. In this paper we address the first-order quasi-stationary Mean Field Games system, which involves an ergodic Hamilton-Jacobi equation and an evolutive continuity equation. Our approach relies on weak KAM theory. We introduce assumptions on the Hamiltonian and coupling cost to ensure continuity of the Peierls barrier and the Aubry set over time. These assumptions, though restrictive, cover interesting cases such as perturbed mechanical Hamiltonians.

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arXiv2024

A network model for urban planning

We study a mathematical model to describe the evolution of a city, which is determined by the interaction of two large populations of agents, workers and firms. The map of the city is described by a network with the edges representing at the same time residential areas and communication routes. The two populations compete for space while interacting through the labour market. The resulting model is described by a two population Mean-Field Game system coupled with an Optimal Transport problem.We prove existence and uniqueness of the solution and we provide several numerical simulations.

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arXiv2024

Stationary Mean Field Games on networks with sticky transition conditions

We study stochastic Mean Field Games on networks with sticky transition conditions. In this setting, the diffusion process governing the agent's dynamics can spend finite time both in the interior of the edges and at the vertices. The corresponding generator is subject to limitations concerning second-order derivatives and the invariant measure breaks down into a combination of an absolutely continuous measure within the edges and a sum of Dirac measures positioned at the vertices. Additionally, the value function, solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized Kirchhoff conditions at the vertices.

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arXiv2024

Learning equilibria in Cournot mean field games of controls

We consider Cournot mean field games of controls, a model originally developed for the production of an exhaustible resource by a continuum of producers. We prove uniqueness of the solution under general assumptions on the price function. Then, we prove convergence of a learning algorithm which gives existence of a solution to the mean field games system. The learning algorithm is implemented with a suitable finite difference discretization to get a numerical method to the solution. We supplement our theoretical analysis with several numerical examples and illustrate the impacts of model parameters.

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arXiv2023

On the quadratic convergence of Newton's method for Mean Field Games with non-separable Hamiltonian

We analyze asymptotic convergence properties of Newton's method for a class of evolutive Mean Field Games systems with non-separable Hamiltonian arising in mean field type models with congestion. We prove the well posedness of the Mean Field Game system with non-separable Hamiltonian and of the linear system giving the Newton iterations. Then, by forward induction and assuming that the initial guess is sufficiently close to the solution of problem, we show a quadratic rate of convergence for the approximation of the Mean Field Game system by Newton's method. We also consider the case of a nonlocal coupling, but with separable Hamiltonian, and we show a similar rate of convergence.

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arXiv2023

Quantitative and qualitative properties for Hamilton-Jacobi PDEs via the nonlinear adjoint method

We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from $L^p$-rates of convergence for the vanishing viscosity approximation to regularizing effects for the Cauchy problem in the whole Euclidean space and Liouville-type theorems. Our approach is based on duality techniques \`a la Evans and a careful study of advection-diffusion equations. The optimality of the results is discussed by several examples.

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arXiv2023

A continuous dependence estimate for viscous Hamilton-Jacobi equations on networks with applications

We study continuous dependence estimates for viscous Hamilton- Jacobi equations defined on a network Gamma. 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 coefficients. 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 Gamma; 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.

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arXiv2022

Approximation of the value function for optimal control problems on stratified domains

In optimal control problems defined on stratified domains, the dynamics and the running cost may have discontinuities on a finite union of submanifolds of RN. In [8, 5], the corresponding value function is characterized as the unique viscosity solution of a discontinuous Hamilton-Jacobi equation satisfying additional viscosity conditions on the submanifolds. In this paper, we consider a semi-Lagrangian approximation scheme for the previous problem. Relying on a classical stability argument in viscosity solution theory, we prove the convergence of the scheme to the value function. We also present HJSD, a free software we developed for the numerical solution of control problems on stratified domains in two and three dimensions, showing, in various examples, the particular phenomena that can arise with respect to the classical continuous framework.

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arXiv2022

On quasi-stationary Mean Field Games of Controls

In 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, which differs from the standard case in the strategy-choice mechanism of the agent: it 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.

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arXiv2021

Rates of convergence for the policy iteration method for Mean Field Games systems

Convergence 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.

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