Elisabetta Carlini

Associate Professor of Numerical Analysis
Department of Mathematics, Sapienza University of Rome

I work on numerical methods for nonlinear partial differential equations: semi-Lagrangian and finite difference schemes for Hamilton–Jacobi–Bellman equations, Mean Field Games, Fokker–Planck equations, problems on networks, and applications to image processing and crowd dynamics.

Email: carlini@mat.uniroma1.it
Interactive version: elisabettacarlini-collab.github.io

━ Value function u: Hamilton–Jacobi equation, solved backward in time
■ Agent density m: Fokker–Planck equation, forward in time
┄ Semi-Lagrangian step: follow the characteristic from the node and interpolate at its foot

Research

32 journal articles · 565 Scopus citations · Scopus h-index 12 · Scopus profile ↗

Hamilton–Jacobi–Bellman equations

Semi-Lagrangian and finite difference schemes for first- and second-order HJB equations. High-order schemes coupled with WENO and CWENO interpolation, algorithms in high dimension, generalized Fast Marching methods for fronts with sign-changing and non-local velocities (dislocation dynamics), and the treatment of nonlinear Neumann and Dirichlet boundary conditions.

Mean Field Games

Discretization of the coupled Hamilton–Jacobi / Fokker–Planck system describing equilibria of games with a very large number of agents. Semi-Lagrangian and Lagrange–Galerkin schemes with convergence results, high-order schemes, Newton-type methods, tensor-train schemes for high-dimensional problems, and applications to the Hughes model for crowd evacuation.

Equations on networks

Numerical approximation of Hamilton–Jacobi equations and MFG models on graphs, including the discretization of vertex conditions, convergence analysis and error estimates. Applications to traffic models and front propagation.

Mean curvature motion and image processing

Semi-Lagrangian schemes for mean curvature motion, the AMSS model and area-preserving flows; surface reconstruction from sparse data with radial basis functions; schemes for the game p-Laplacian and ∞-Laplacian with applications to image inpainting.

Member of GNCS (National Group for Scientific Computing, INdAM). Scopus metrics updated October 2026; current values are on the Scopus profile.

Publications

Preprints: all my papers on arXiv ↗

In press (1)

E. Carlini, A. Picarelli, F. J. Silva, A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with Dirichlet boundary conditions, Mathematics of Computation · arXiv:2503.20283

Journal articles (32)

E. Carlini, A. Zorkot, Newton methods for mean field games: a numerical study, Journal of Computational and Applied Mathematics 490, 2027 · arXiv:2512.12752
E. Carlini, L. Saluzzi, High order tensor-train-based schemes for high-dimensional mean field games, SIAM Journal on Scientific Computing 48(4), A2027–A2056, 2026 · arXiv:2510.15603
E. Carlini, V. Coscetti, A semi-Lagrangian scheme for first-order mean field games based on monotone operators, IMA Journal of Numerical Analysis, 2026 · arXiv:2506.10509
E. Carlini, V. Coscetti, M. Pozza, Error estimate for a semi-Lagrangian scheme for Hamilton–Jacobi equations on networks, Communications on Applied Mathematics and Computation, 2025 · DOI · arXiv:2411.02356
E. Carlini, A. Siconolfi, Numerical analysis of time-dependent Hamilton–Jacobi equations on networks, Numerische Mathematik 157(6), 2055–2096, 2025 · DOI · arXiv:2310.06092
E. Carlini, R. Ferretti, M. Semplice, S. Preda, A CWENO large time-step scheme for Hamilton–Jacobi equations, Communications on Applied Mathematics and Computation, 2024 · arXiv:2402.15367
E. Carlini, S. Tozza, A convergent semi-Lagrangian scheme for the game ∞-Laplacian, Dynamic Games and Applications 15(2), 406–416, 2024
E. Calzola, E. Carlini, F. J. Silva, A high-order scheme for mean field games, Journal of Computational and Applied Mathematics 445, 115769, 2024 · arXiv:2207.08463
E. Carlini, F. J. Silva, A. Zorkot, A Lagrange–Galerkin scheme for first order mean field game systems, SIAM Journal on Numerical Analysis 62(1), 167–198, 2024
E. Carlini, S. Tozza, A scheme for the game p-Laplacian and its application to image inpainting, Applied Mathematics and Computation, 2024 · DOI
E. Calzola, E. Carlini, X. Dupuis, F. J. Silva, A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions, Numerische Mathematik 153(1), 49–84, 2023 · arXiv:2109.10228
L. Bonaventura, E. Calzola, E. Carlini, R. Ferretti, Second order fully semi-Lagrangian discretizations of advection–diffusion–reaction systems, Journal of Scientific Computing 88, 1–29, 2021 · arXiv:2002.04381
E. Carlini, A. Festa, N. Forcadel, A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations on networks, SIAM Journal on Numerical Analysis 58, 3165–3196, 2020 · arXiv:1804.09429
E. Carlini, F. J. Silva, On the discretization of some nonlinear Fokker–Planck–Kolmogorov equations and applications, SIAM Journal on Numerical Analysis 56(4), 2148–2177, 2018 · arXiv:1708.02042
E. Carlini, F. Camilli, C. Marchi, A flame propagation model on a network with application to a blocking problem, Discrete and Continuous Dynamical Systems – Series S 11(5), 825–843, 2018 · arXiv:1411.3260
E. Carlini, A. Festa, F. J. Silva, M.-T. Wolfram, A semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dynamic Games and Applications 7(4), 683–705, 2017 · arXiv:1601.07324
E. Carlini, R. Ferretti, A semi-Lagrangian scheme with radial basis approximation for surface reconstruction, Computing and Visualization in Science 18(2–3), 103–112, 2017 · arXiv:1602.04626
E. Carlini, F. J. Silva, A semi-Lagrangian scheme for a degenerate second order mean field game system, Discrete and Continuous Dynamical Systems – Series A 35(9), 4269–4292, 2015 · arXiv:1404.5932
E. Carlini, F. Camilli, C. Marchi, A model problem for mean field games on networks, Discrete and Continuous Dynamical Systems – Series A 35(9), 4173–4192, 2015 · arXiv:1402.2467
E. Carlini, F. J. Silva, A fully discrete semi-Lagrangian scheme for a first order mean field games problem, SIAM Journal on Numerical Analysis 52(1), 45–67, 2014 · arXiv:1212.4757
E. Carlini, R. Ferretti, A semi-Lagrangian approximation for the AMSS model of image processing, Applied Numerical Mathematics 73, 16–32, 2013
E. Carlini, M. Falcone, P. Hoch, A generalized fast marching method on unstructured triangular meshes, SIAM Journal on Numerical Analysis 51(6), 2999–3035, 2013
E. Carlini, N. Forcadel, R. Monneau, A generalized fast marching method for dislocation dynamics, SIAM Journal on Numerical Analysis 49(6), 2470–2500, 2011
E. Carlini, M. Falcone, R. Ferretti, Convergence of a large time-step scheme for mean curvature motion, Interfaces and Free Boundaries 12(4), 409–441, 2010
A. M. Bersani, E. Carlini, P. Lanucara, M. Rorro, V. Ruggiero, Application of optimal control techniques and advanced computing to the study of enzyme kinetics, Mathematics and Computers in Simulation 81(3), 705–716, 2010
E. Carlini, M. Falcone, N. Forcadel, R. Monneau, Convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign, SIAM Journal on Numerical Analysis 46(6), 2920–2952, 2008
E. Carlini, M. Falcone, R. Ferretti, A semi-Lagrangian scheme for the curve shortening flow in codimension 2, Journal of Computational Physics 225(2), 1388–1408, 2007
O. Alvarez, E. Carlini, R. Monneau, E. Rouy, Convergence of a first order scheme for a non-local eikonal equation, Applied Numerical Mathematics 56(9), 1136–1146, 2006
O. Alvarez, E. Carlini, R. Monneau, E. Rouy, A convergent scheme for a non local Hamilton–Jacobi equation modelling dislocation dynamics, Numerische Mathematik 104(4), 413–444, 2006
O. Alvarez, E. Carlini, P. Hoch, Y. Le Bouar, R. Monneau, Dislocation dynamics described by non-local Hamilton–Jacobi equations, Materials Science & Engineering A 400–401, 162–165, 2005
E. Carlini, R. Ferretti, G. Russo, A weighted essentially non-oscillatory, large time-step scheme for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 27(3), 1071–1091, 2005
E. Carlini, M. Falcone, R. Ferretti, An efficient algorithm for Hamilton–Jacobi equations in high dimension, Computing and Visualization in Science 7(1), 15–29, 2004

Proceedings (9)

L. Bonaventura, E. Calzola, E. Carlini, R. Ferretti, A fully semi-Lagrangian method for the Navier–Stokes equations in primitive variables, Lecture Notes in Computational Science and Engineering 132, 55–62 (FEF 2017), 2020
E. Carlini, A. Festa, F. J. Silva, The Hughes model for pedestrian dynamics and congestion modelling, IFAC-PapersOnLine 50(1), 1655–1660, 2017 · arXiv:1611.06848
E. Carlini, F. J. Silva, A semi-Lagrangian scheme for the Fokker–Planck equation, IFAC-PapersOnLine 49(8), 272–277, 2016
E. Carlini, F. J. Silva, Semi-Lagrangian schemes for mean field game models, 52nd IEEE Conference on Decision and Control (CDC), 3115–3120, 2013
E. Carlini, R. Ferretti, A semi-Lagrangian scheme for area preserving flows, 21st International Conference on Pattern Recognition (ICPR), Tsukuba, 2029–2033, 2012
E. Carlini, R. Ferretti, A semi-Lagrangian approximation of min-max type for the stationary mean curvature equation, ENUMATH 2007, Springer, 679–686, 2008
E. Carlini, A single-pass scheme for the mean curvature motion of convex curves, ENUMATH 2007, Springer, 671–678, 2008
E. Carlini, E. Cristiani, N. Forcadel, A non-monotone fast marching scheme for a Hamilton–Jacobi equation modelling dislocation dynamics, ENUMATH 2005, Springer, 724–731, 2006
E. Carlini, M. Falcone, R. Ferretti, A time-adaptive semi-Lagrangian approximation to mean curvature motion, ENUMATH 2005, Springer, 732–739, 2006

Book chapters (3)

E. Carlini, M. Falcone, R. Ferretti, Numerical techniques for level set models: an image segmentation perspective, In A. El-Baz, J. S. Suri (eds.), Level Set Method in Medical Imaging Segmentation, Taylor & Francis, 2019
E. Carlini, F. J. Silva, A fully-discrete scheme for systems of nonlinear Fokker–Planck–Kolmogorov equations, In PDE Models for Multi-Agent Phenomena, Springer INdAM Series 28, 195–218, 2018 · arXiv:1711.10426
E. Carlini, M. Falcone, A. Festa, A brief survey on semi-Lagrangian schemes for image processing, In Innovations for Shape Analysis, Mathematics and Visualization, Springer, 191–218, 2013

Teaching

Courses in academic year 2026–2027

MSc · Numerical Methods for Nonlinear Partial Differential Equations — MSc in Mathematics for Applications · with T. Taddei · since 2026
MSc · Numerical Methods for Partial Differential Equations — MSc in Mathematics for Applications · since 2011
BSc · Mathematics with Elements of Computer Science — BSc in Pharmacy · since 2026

PhD courses and schools

2025 · Semi-Lagrangian schemes for Hamilton–Jacobi equations — CIMPA School “Transport optimal, EDP et optimisation”, Essaouira, Morocco
2024 · Approximation of Mean Field Games and related problems — PhD course, Sapienza University of Rome, with F. J. Silva (University of Limoges)
2022 · Approximation of Mean Field Games and related problems — PhD course, KAUST, Saudi Arabia
2015 · Numerical Methods for Partial Differential Equations — School on Numerical Methods, IMCA, Lima, Peru

Previous teaching

Numerical Analysis (Istituzioni di Analisi Numerica), MSc in Mathematics for Applications, 2020–2026
Programming and Computing Lab, BSc in Mathematics, 2011–2024
Mathematical Methods for Engineering (MAT/08 module), Chemical Engineering, 2016–2018
Computer skills: Matlab and Python, BSc in Mathematics, 2008–2023
Numerical Analysis and C Programming, Master in Scientific Computing, 2008–2017
Numerical methods for ordinary differential equations, BSc in Mathematics, 2009–2010
Numerical Computing 1 and 2, BSc in Chemistry, 2008–2010
Excellence course “Formulation and numerical approximation of some minimization problems”, with M. Ponsiglione, 2009–2010
Teaching assistant for Numerical Analysis and Programming and Computing Lab, Sapienza University of Rome and Roma Tre University, 2001–2011

Supervision

PhD students

2023–2026 · Valentina Coscetti · Semi-Lagrangian schemes for first order mean field games and for Hamilton–Jacobi equations on networks — PhD in Mathematics, Sapienza University of Rome, cycle XXXIX
2021–2023 · Ahmad Zorkot · Approximation to Mean Field Games — PhD in Applied Mathematics, XLIM - DMI UMR CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges · co-supervised with F. J. Silva
2018–2021 · Elisa Calzola · Semi-Lagrangian schemes for parabolic equations: second order accuracy and boundary conditions — PhD in Mathematics, Sapienza University of Rome, cycle XXXIV · joint PhD with F. J. Silva (University of Limoges)

Theses

I have supervised 22 master's theses in Mathematics for Applications, 18 bachelor's theses, 2 second-level Master's theses in Scientific Computing and 2 international internships (INSA Rouen).

Recent master's theses

L. Ottavio, Un problema di tipo MFG su networks, 2025–2026
E. Giansante, Riduzione di modelli per la risoluzione di equazioni di Hamilton–Jacobi, 2024–2025
G. Testa, Numerical scheme for the Hughes problem using a mean-field game model, 2023–2024
S. Bonaldo, Algoritmi per il calcolo di equilibri di Nash per giochi con numero finito di giocatori, 2023–2024
M. Laurenti, Uno schema alle differenze finite per mean-field games, 2023–2024
A. Mastrodonato, A finite difference approach for solving a mean-field game model of price formation, 2022–2023
V. Coscetti, Uno schema semi-Lagrangiano per Mean Field Games deterministici basato sulle iterazioni della policy, 2022–2023
L. Ferentini, Leggi di conservazione su network con schema di Godunov e applicazione al modello LWR, 2021–2022
F. Puca, Approssimazione numerica del modello di Keller–Segel per la chemiotassi, 2020–2021
C. Lin, Schema alle differenze finite per equazioni di Hamilton–Jacobi sui network, 2019–2020
M. Abrugia, Approssimazione numerica di equazioni di Hamilton–Jacobi sui network, 2018–2019

Thesis proposals

Students interested in a thesis on numerical methods for Hamilton–Jacobi equations, Mean Field Games or problems on networks are welcome to write to me to arrange a meeting.

Projects

As principal investigator

2025– · Mean-Field Games: models, theory, and computational aspects — KAUST Competitive Research Grant, local PI (PI: D. Gomes)
2022 · Optimal control problems: analysis, approximation and applications — PRIN 2022, local unit PI (national PI: G. Colombo)
2022 · Numerical approximation of differential models and applications — University project, Sapienza University of Rome
2019 · Numerical approximation of hyperbolic problems and applications — GNCS-INdAM Project
2018 · International inter-university agreement with IMCA, Lima — Sapienza University of Rome
2013 · Analysis and approximation of nonlinear differential problems and applications — University project, Sapienza University of Rome
2011 · Theoretical and applied developments of semi-Lagrangian methods — GNCS-INdAM Project
2009 · Front evolution: models, algorithms and applications — Ateneo Federato AST Project

Scientific activities

Recent invited talks

2026 · High order tensor-train-based schemes for high-dimensional mean field games — Invited speaker, workshop Numerical methods for Hamilton–Jacobi and mean-field PDEs, ICMS, Bayes Centre, Edinburgh · September 2026
2026 · Error estimate for a semi-Lagrangian scheme for a time-dependent Hamilton–Jacobi equation on networks and High order tensor-train-based schemes for high-dimensional mean field games — Two talks in special sessions, AIMS Conference, Athens · July 2026
2026 · Algorithm for deterministic mean-field games based on monotone operators — Invited speaker, workshop “4ème rencontre ANR Control On Stratified Structures”, Tours · March 2026
2025 · Numerical approximations of the Newton algorithm for mean field games — Minisymposium, 30th Biennial Numerical Analysis Conference, Glasgow · June 2025
2025 · Numerical approximations of the Newton algorithm for mean field games — Invited speaker, workshop “Mean Field Games, Mean Field Control and Applications to Economics and Finance”, University of Pavia · April 2025
2025 · Algorithm for deterministic mean-field games — Invited speaker, Durham Symposium on Mean Field Games, Durham University · March 2025
2024 · Algorithm for deterministic mean-field games — Special session “Recent advances in mean field games for crowd dynamics”, 14th AIMS International Conference, Abu Dhabi · December 2024
2024 · Newton methods for mean field games — Invited speaker, workshop “Numerical methods for optimal transport problems, mean field games, and multi-agent dynamics”, Valparaíso, Chile · January 2024
2023 · A high-order Lagrange–Galerkin scheme for a class of Fokker–Planck equations and applications to mean field games — Session “Metodi numerici per le equazioni alle derivate parziali”, UMI 2023 Congress, Pisa · September 2023
2023 · Lagrange–Galerkin schemes for mean field games — Invited speaker, workshop “Numerical Methods for Mean Field Games and related PDE”, University College London · July 2023
2023 · Lagrange–Galerkin schemes for mean field games — Invited speaker, workshop “Numerical aspects of hyperbolic balance laws and related problems”, Cortona · June 2023
2023 · A high-order scheme for mean field games — Invited speaker, 13th ISDG Workshop, Naples · June 2023
2023 · A high-order Lagrange–Galerkin scheme for mean field games — Invited speaker, workshop “Advances in nonlinear elliptic and parabolic PDEs”, AlUla, Saudi Arabia · May 2023
2023 · A numerical scheme for evolutive Hamilton–Jacobi equations on networks — Invited speaker, final workshop of PRIN 2017, Sapienza University of Rome · February 2023
2023 · A numerical scheme for evolutive Hamilton–Jacobi equations on networks — Session “New developments in Mean Field Games and Hamilton–Jacobi equations”, LACIAM 2023, Rio de Janeiro · February 2023
2023 · A numerical scheme for evolutive Hamilton–Jacobi equations on networks — Invited speaker, workshop “First order problems on networks and applications”, Sapienza University of Rome · January 2023

Since 2002, more than 40 invited talks at international conferences, including ICIAM, ENUMATH, IEEE CDC, ICCOPT, MTNS, IPAM-UCLA, Banff, CIRM and RICAM.

Conference organization

2026 · Hamilton–Jacobi Equations and Mean Field Games: From Modeling to Numerics via Analysis — Palazzo del Castelletto, Aula Dini, Pisa · 1–2 October 2026 · co-organized with Fabio Camilli, Annalisa Cesaroni and Claudio Marchi
2025 · Optimal Control and Agent Systems — Sapienza University of Rome
2024 · Maurizio day II: numerics for controls, PDEs and images — Sapienza University of Rome, celebrating Maurizio Falcone's 70th birthday
2024 · Machine Learning and Optimal Control — Summer school, Gaeta
2023 · Nonlinear PDEs: theory, numerics and applications, in memory of Maurizio Falcone — Sapienza University of Rome
2023 · First order problems on networks and applications — Sapienza University of Rome
2022 · Theory and numerics of Mean Field Games and Hamilton–Jacobi equations — Sapienza University of Rome

Other conferences organized (2011-2023)

Maurizio day: numerics for controls, PDEs and images, Sapienza University of Rome, 2023
Nonlinear hyperbolic PDEs: numerical techniques and related models, minisymposium at WONAPDE, Concepción, Chile
Numerical approximation of hyperbolic problems and applications, GNCS Project, Sapienza University of Rome, 2020
Numerical methods for optimal control problems and differential games, minisymposium at SIMAI, Milan, 2016
Numerical methods for PDEs: optimal control, games and image processing, Sapienza University of Rome, 2014
Numerics in Image and Surface Processing, Bologna, 2014
Mathematical methods in image processing, Sapienza University of Rome, 2013
Recent advances on theory and applications of semi-Lagrangian methods, Sapienza University of Rome, 2011

Curriculum vitae

2026 · National Scientific Qualification as Full Professor — Field 01/A5, Numerical Analysis
2018–present · Associate Professor — Department of Mathematics, Sapienza University of Rome
2008–2018 · Researcher — Department of Mathematics, Sapienza University of Rome
2005–2007 · Research fellow — Istituto Nazionale di Alta Matematica “Francesco Severi”
2004 · Postdoctoral researcher — CERMICS, École Nationale des Ponts et Chaussées, France
2004 · PhD in Mathematical Models and Methods for Technology and Society — Sapienza University of Rome · thesis: Semi-Lagrangian schemes for first and second order Hamilton–Jacobi equations
2002 · Mathematics in Nanoscale Science and Engineering program — IPAM, University of California, Los Angeles
2000 · Degree in Mathematics — Roma Tre University

Contact

Email
carlini@mat.uniroma1.it

Address
Department of Mathematics
Sapienza University of Rome
Piazzale Aldo Moro 5, 00185 Rome, Italy

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