arXiv subjects

Elisabetta Carlini

Publications and source records attributed to Elisabetta Carlini.

18 recordsLinked to original sources

arXiv2026

A semi-Lagrangian scheme for First-Order Mean Field Games based on monotone operators

We construct a semi-Lagrangian scheme for first-order, time-dependent, and non-local Mean Field Games. The convergence of the scheme to a weak solution of the system is analyzed by exploiting a key monotonicity property. To solve the resulting discrete problem, we implement a Learning Value Algorithm, prove its convergence, and propose an acceleration strategy based on a Policy iteration method. Finally, we present numerical experiments that validate the effectiveness of the proposed schemes and show that the accelerated version significantly improves performance.

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arXiv2026

High order Tensor-Train-Based Schemes for High-Dimensional Mean Field Games

We introduce a fully discrete scheme to solve a class of high-dimensional Mean Field Games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By reformulating the classical Hamilton-Jacobi-Bellman and Fokker-Planck equations as a sequence of advection-diffusion-reaction subproblems within a smoothed policy iteration, we construct both first and second order in time SL schemes. The TT format and appropriate quadrature rules reduce storage and computational cost from exponential to polynomial in the dimension. Numerical experiments demonstrate that our TT-accelerated SL methods achieve their theoretical convergence rates, exhibit modest growth in memory usage and runtime with dimension, and significantly outperform grid-based SL in accuracy per CPU second.

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arXiv2026

Newton Methods for Mean Field Games: A Numerical Study

We address the numerical solution of second-order Mean Field Game problems through Newton iterations in infinite dimensions, introduced in [14], where quadratic convergence of the method was rigorously established. Building upon this theoretical framework, we develop new numerical discretization techniques, including both a finite difference and a semi-Lagrangian scheme, that enable an effective computational implementation of the infinite-dimensional iterations. The proposed methods are tested on several benchmark problems, and the resulting numerical experiments demonstrate their robustness, accuracy, and efficiency. A comparative analysis between the two schemes and existing approaches from the literature is also presented, highlighting the potential of Newton-based solvers for MFG systems.

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arXiv2025

A Semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with Dirichlet boundary conditions

We study the numerical approximation of time-dependent, possibly degenerate, second-order Hamilton-Jacobi-Bellman equations in bounded domains with nonhomogeneous Dirichlet boundary conditions. It is well known that convergence towards the exact solution of the equation, considered here in the viscosity sense, holds if the scheme is monotone, consistent, and stable. While standard finite difference schemes are, in general, not monotone, the so-called semi-Lagrangian schemes are monotone by construction. On the other hand, these schemes make use of a wide stencil and, when the equation is set in a bounded domain, this typically causes an overstepping of the boundary and hence the loss of consistency. We propose here a semi-Lagrangian scheme defined on an unstructured mesh, with a suitable treatment at grid points near the boundary to preserve consistency, and show its convergence for problems where the viscosity solution can even be discontinuous. We illustrate the numerical convergence in several tests, including degenerate and first-order equations.

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arXiv2024

Error Estimate for a Semi-Lagrangian Scheme for Hamilton-Jacobi Equations on Networks

We examine the numerical approximation of time-dependent Hamilton-Jacobi equations on networks, providing a convergence error estimate for the semi-Lagrangian scheme introduced in (Carlini and Siconolfi, 2023), where convergence was proven without an error estimate. We derive a convergence error estimate of order one-half. This is achieved showing the equivalence between two definitions of solutions to this problem proposed in (Imbert and Monneau, 2017) and (Siconolfi, 2022), a result of independent interest, and applying a general convergence result from (Carlini, Festa and Forcadel, 2020).

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arXiv2023

A high-order scheme for mean field games

In this paper we propose a high-order numerical scheme for time-dependent mean field games systems. The scheme, which is built by combining Lagrange-Galerkin and semi-Lagrangian techniques, is consistent and stable for large time steps compared with the space steps. We provide a convergence analysis for the exactly integrated Lagrange-Galerkin scheme applied to the Fokker-Planck equation, and we propose an implementable version with inexact integration. Finally, we validate the convergence rate of the proposed scheme through the numerical approximation of two mean field games systems.

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arXiv2023

Numerical Analysis of time-dependent Hamilton-Jacobi Equations on Networks

A new algorithm for time dependent Hamilton Jacobi equations on networks, based on semi Lagrangian scheme, is proposed. It is based on the definition of viscosity solution for this kind of problems recently given in. A thorough convergence analysis, not requiring weak semilimits, is provided. In particular, the check of the supersolution property at the vertices is performed through a dynamical technique which seems new. The scheme is efficient, explicit, allows long time steps, and is suitable to be implemented in a parallel algorithm. We present some numerical tests, showing the advantage in terms of computational cost over the one proposed in [7]

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arXiv2021

A semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with oblique boundary conditions

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton-Jacobi-Bellman (HJB) equations on a bounded domain with oblique boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes. We present some numerical results that confirm the numerical convergence of the scheme.

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arXiv2020

Second order fully semi-Lagrangian discretizations of advection-diffusion-reaction systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations

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arXiv2018

A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games.

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arXiv2018

A semi-Lagrangian scheme for Hamilton-Jacobi equations on networks with application to traffic flow models

We present a semi-Lagrangian scheme for the approximation of a class of Hamilton-Jacobi-Bellman equations on networks. The scheme is explicit and stable under some technical conditions. We prove a convergence theorem and some error estimates. Additionally, the theoretical results are validated by numerical tests. Finally, we apply the scheme to simulate traffic flows modeling problems.

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arXiv2018

On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications

In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity of the solution, conserves the mass and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. The main assumptions to obtain a convergence result are that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, we obtain a new proof of existence of solutions for such equations. We apply our results to several examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.

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arXiv2017

A flame propagation model on a network with application to a blocking problem

We consider the Cauchy problem \[\partial_t u+H(x,Du)=0 \quad (x,t)\inΓ\times (0,T),\quad u(x,0)=u_0(x) \quad x\inΓ\] where $Γ$ is a network and $H$ is a convex and positive homogeneous Hamiltonian which may change from edge to edge. In the former part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.

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arXiv2016

The Hughes model for pedestrian dynamics and congestion modelling

In this paper we present a numerical study of some variations of the Hughes model for pedestrian flow under different types of congestion effects. The general model consists of a coupled non-linear PDE system involving an eikonal equation and a first order conservation law, and it intends to approximate the flow of a large pedestrian group aiming to reach a target as fast as possible, while taking into account the congestion of the crowd. We propose an efficient semi-Lagrangian scheme (SL) to approximate the solution of the PDE system and we investigate the macroscopic effects of different penalization functions modelling the congestion phenomena.

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arXiv2016

A Semi-Lagrangian Scheme with Radial Basis Approximation for Surface Reconstruction

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces from sparse, possibly noisy data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.

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arXiv2016

A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow

In this paper we present a Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow. Hughes originally proposed a coupled nonlinear PDE system describing the evolution of a large pedestrian group trying to exit a domain as fast as possible. The original model corresponds to a system of a conservation law for the pedestrian density and an Eikonal equation to determine the weighted distance to the exit. We consider this model in presence of small diffusion and discuss the numerical analysis of the proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small diffusion on the exit time with various numerical experiments.

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arXiv2014

A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system

In this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the state dimension is equals to one, we prove a convergence result. Some numerical simulations are provided, evidencing the convergence of the approximation and also the difference between the numerical results for the degenerate and non-degenerate cases.

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arXiv2014

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.

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