arXiv subjects

Martino Bardi

Publications and source records attributed to Martino Bardi.

At least 19 recordsLinked to original sources

arXiv2024

Long-time behaviour of deterministic Mean Field Games with non-monotone interactions

We consider deterministic Mean Field Games (MFG) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of 1st order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFG with cost functional satisfying various additional assumptions, but not the classical Lasry-Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFG some results of weak KAM theory.

math.AP↗

arXiv2024

Deep Relaxation of Controlled Stochastic Gradient Descent via Singular Perturbations

We consider a singularly perturbed system of stochastic differential equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate the Entropic Gradient Descent in the optimization of deep neural networks, via homogenisation. We embed it in a much larger class of two-scale stochastic control problems and rely on convergence results for Hamilton-Jacobi-Bellman equations with unbounded data proved recently by ourselves (ESAIM Control Optim. Calc. Var. 2023). We show that the limit of the value functions is itself the value function of an effective control problem with extended controls, and that the trajectories of the perturbed system converge in a suitable sense to the trajectories of the limiting effective control system. These rigorous results improve the understanding of the convergence of the algorithms used by Chaudhari et al., as well as of their possible extensions where some tuning parameters are modelled as dynamic controls.

math.OC↗

arXiv2024

Asymptotic properties of non-coercive Hamiltonians with drift

We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong bracket generation condition on the vector fields. We first prove the existence of a critical value of the Hamiltonian by means of the ergodic approximation. Next we prove the existence of a possibly discontinuous viscosity solution to the critical equation. We show that the long-time behaviour of solutions to the evolutive Hamilton-Jacobi equation is described in terms of the critical constant and a critical solution. As in the classical weak KAM theory we find a fixed point of the Hamilton-Jacobi-Lax-Oleinik semigroup, although possibly discontinuous. Finally we apply the existence and properties of the critical value to the periodic homogenization of stationary and evolutive H-J equations.

math.OC↗

arXiv2023

Singular perturbations in stochastic optimal control with unbounded data

We study singular perturbations of a class of two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. We construct effective Hamiltonian and initial data and prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization.

math.OC↗

arXiv2022

An Eikonal equation with vanishing Lagrangian arising in Global Optimization

We show a connection between global unconstrained optimization of a continuous function $f$ and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution $v$ of the critical Hamilton-Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then $v$ is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of $v$. Such trajectories are proved to converge to the set of minima of $f$, using tools in control theory and occupational measures. We prove also that in some cases the set of minima is reached in finite time.

math.OC↗

arXiv2022

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and zero-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

math.AP↗

arXiv2020

Liouville results for fully nonlinear equations modeled on Hörmander vector fields. I. The Heisenberg group

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.

math.AP↗

arXiv2020

Convergence of some Mean Field Games systems to aggregation and flocking models

For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modeling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the evolution of the mass density. The first model is a 2nd order MFG system with vanishing viscosity, and the limit is an aggregation equation. The result has an interpretation for models of collective animal behaviour and of crowd dynamics. The second class of problems are 1st order MFGs of acceleration and the limit is the kinetic equation associated to the Cucker-Smale model. The first problem is analyzed by PDE methods, whereas the second is studied by variational methods in the space of probability measures on trajectories.

math.OC↗

arXiv2020

Regularity of the minimum time and of viscosity solutions of degenerate eikonal equations via generalized Lie brackets

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of { set-valued} iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hoelder regularity of the minimum time function in optimal control. We then apply such result to prove Hölder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the Hölder regularity are essentially necessary, at least for smooth vector fields and target.

math.AP↗

arXiv2019

Cauchy problem and periodic homogenization for nonlocal Hamilton-Jacobi equations with coercive gradient terms

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integro-differential equations that can be of independent interest.

math.AP↗

arXiv2018

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton-Jacobi-Bellman form, such as those involving the Pucci's extremal operators over Hörmander vector fields.

math.AP↗

arXiv2018

On non-uniqueness and uniqueness of solutions in finite-horizon mean field games

This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analyzed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small T, is proved. Some results are also extended to MFGs with two populations.

math.AP↗

arXiv2017

Uniqueness of solutions in Mean Field Games with several populations and Neumann conditions

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.

math.AP↗

arXiv2016

Mean Field Games models of segregation

This paper introduces and analyses some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of Mean Field Game theory, in the stationary and in the evolutive case. Numerical methods are proposed, with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

math.AP↗

arXiv2016

Liouville properties and critical value of fully nonlinear elliptic operators

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

math.AP↗

arXiv2015

Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control

For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the associated controlled diffusion process.

math.AP↗

arXiv2014

Linear-Quadratic $N$-person and Mean-Field Games with Ergodic Cost

We consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $N$ Hamilton-Jacobi-Bellman and $N$ Kolmogorov-Fokker-Planck partial differential equations. We give necessary and sufficient conditions for the existence and uniqueness of quadratic-Gaussian solutions in terms of the solvability of suitable algebraic Riccati and Sylvester equations. Under a symmetry condition on the running costs and for nearly identical players we study the large population limit, $N$ tending to infinity, and find a unique quadratic-Gaussian solution of the pair of Mean Field Game HJB-KFP equations. Examples of explicit solutions are given, in particular for consensus problems.

math.AP↗

arXiv2014

Convergence in Multiscale Financial Models with Non-Gaussian Stochastic Volatility

We consider stochastic control systems affected by a fast mean reverting volatility $Y(t)$ driven by a pure jump Lévy process. Motivated by a large literature on financial models, we assume that $Y(t)$ evolves at a faster time scale $\frac{t}{\varepsilon}$ than the assets, and we study the asymptotics as $\varepsilon\to 0$. This is a singular perturbation problem that we study mostly by PDE methods within the theory of viscosity solutions.

math.PR↗