Elisabetta Carlini Italiano
Portrait of Elisabetta Carlini

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 contact Publications
A crowd reaches an exit while avoiding three obstacles: a Mean Field Game equilibrium computed with a numerical scheme The thin curves are level lines of the value function u, the minimal time to reach the exit taking congestion into account. Shaded areas show where the density m of agents is higher: the crowd splits into three streams around the obstacles. Bottom right, a grid shows one semi-Lagrangian step: from the node x_i the characteristic is followed to its foot, where u is interpolated in the highlighted cell. exit xi xi + Δt α*
  • 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 ↗

  1. E. Carlini, A. Picarelli, F. J. SilvaA semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with Dirichlet boundary conditionsMathematics of Computation in press · arXiv:2503.20283
  2. 2027
    E. Carlini, A. ZorkotNewton methods for mean field games: a numerical studyJournal of Computational and Applied Mathematics 490 · arXiv:2512.12752
  3. 2026
    E. Carlini, L. SaluzziHigh order tensor-train-based schemes for high-dimensional mean field gamesSIAM Journal on Scientific Computing 48(4), A2027–A2056 · arXiv:2510.15603
  4. 2026
    E. Carlini, V. CoscettiA semi-Lagrangian scheme for first-order mean field games based on monotone operatorsIMA Journal of Numerical Analysis · arXiv:2506.10509
  5. 2025
    E. Carlini, V. Coscetti, M. PozzaError estimate for a semi-Lagrangian scheme for Hamilton–Jacobi equations on networksCommunications on Applied Mathematics and Computation · DOI · arXiv:2411.02356
  6. 2025
    E. Carlini, A. SiconolfiNumerical analysis of time-dependent Hamilton–Jacobi equations on networksNumerische Mathematik 157(6), 2055–2096 · DOI · arXiv:2310.06092
  7. 2024
    E. Carlini, R. Ferretti, M. Semplice, S. PredaA CWENO large time-step scheme for Hamilton–Jacobi equationsCommunications on Applied Mathematics and Computation · arXiv:2402.15367
  8. 2024
    E. Carlini, S. TozzaA convergent semi-Lagrangian scheme for the game ∞-LaplacianDynamic Games and Applications 15(2), 406–416
  9. 2024
    E. Calzola, E. Carlini, F. J. SilvaA high-order scheme for mean field gamesJournal of Computational and Applied Mathematics 445, 115769 · arXiv:2207.08463
  10. 2024
    E. Carlini, F. J. Silva, A. ZorkotA Lagrange–Galerkin scheme for first order mean field game systemsSIAM Journal on Numerical Analysis 62(1), 167–198
  11. 2024
    E. Carlini, S. TozzaA scheme for the game p-Laplacian and its application to image inpaintingApplied Mathematics and Computation · DOI
  12. 2023
    E. Calzola, E. Carlini, X. Dupuis, F. J. SilvaA semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditionsNumerische Mathematik 153(1), 49–84 · arXiv:2109.10228
  13. 2021
    L. Bonaventura, E. Calzola, E. Carlini, R. FerrettiSecond order fully semi-Lagrangian discretizations of advection–diffusion–reaction systemsJournal of Scientific Computing 88, 1–29 · arXiv:2002.04381
  14. 2020
    E. Carlini, A. Festa, N. ForcadelA semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations on networksSIAM Journal on Numerical Analysis 58, 3165–3196 · arXiv:1804.09429
  15. 2020
    L. Bonaventura, E. Calzola, E. Carlini, R. FerrettiA fully semi-Lagrangian method for the Navier–Stokes equations in primitive variables proceedingsLecture Notes in Computational Science and Engineering 132, 55–62 (FEF 2017)
  16. 2019
    E. Carlini, M. Falcone, R. FerrettiNumerical techniques for level set models: an image segmentation perspective book chapterIn A. El-Baz, J. S. Suri (eds.), Level Set Method in Medical Imaging Segmentation, Taylor & Francis
  17. 2018
    E. Carlini, F. J. SilvaOn the discretization of some nonlinear Fokker–Planck–Kolmogorov equations and applicationsSIAM Journal on Numerical Analysis 56(4), 2148–2177 · arXiv:1708.02042
  18. 2018
    E. Carlini, F. Camilli, C. MarchiA flame propagation model on a network with application to a blocking problemDiscrete and Continuous Dynamical Systems – Series S 11(5), 825–843 · arXiv:1411.3260
  19. 2018
    E. Carlini, F. J. SilvaA fully-discrete scheme for systems of nonlinear Fokker–Planck–Kolmogorov equations book chapterIn PDE Models for Multi-Agent Phenomena, Springer INdAM Series 28, 195–218 · arXiv:1711.10426
  20. 2017
    E. Carlini, A. Festa, F. J. Silva, M.-T. WolframA semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flowDynamic Games and Applications 7(4), 683–705 · arXiv:1601.07324
  21. 2017
    E. Carlini, R. FerrettiA semi-Lagrangian scheme with radial basis approximation for surface reconstructionComputing and Visualization in Science 18(2–3), 103–112 · arXiv:1602.04626
  22. 2017
    E. Carlini, A. Festa, F. J. SilvaThe Hughes model for pedestrian dynamics and congestion modelling proceedingsIFAC-PapersOnLine 50(1), 1655–1660 · arXiv:1611.06848
  23. 2016
    E. Carlini, F. J. SilvaA semi-Lagrangian scheme for the Fokker–Planck equation proceedingsIFAC-PapersOnLine 49(8), 272–277
  24. 2015
    E. Carlini, F. J. SilvaA semi-Lagrangian scheme for a degenerate second order mean field game systemDiscrete and Continuous Dynamical Systems – Series A 35(9), 4269–4292 · arXiv:1404.5932
  25. 2015
    E. Carlini, F. Camilli, C. MarchiA model problem for mean field games on networksDiscrete and Continuous Dynamical Systems – Series A 35(9), 4173–4192 · arXiv:1402.2467
  26. 2014
    E. Carlini, F. J. SilvaA fully discrete semi-Lagrangian scheme for a first order mean field games problemSIAM Journal on Numerical Analysis 52(1), 45–67 · arXiv:1212.4757
  27. 2013
    E. Carlini, R. FerrettiA semi-Lagrangian approximation for the AMSS model of image processingApplied Numerical Mathematics 73, 16–32
  28. 2013
    E. Carlini, M. Falcone, P. HochA generalized fast marching method on unstructured triangular meshesSIAM Journal on Numerical Analysis 51(6), 2999–3035
  29. 2013
    E. Carlini, F. J. SilvaSemi-Lagrangian schemes for mean field game models proceedings52nd IEEE Conference on Decision and Control (CDC), 3115–3120
  30. 2013
    E. Carlini, M. Falcone, A. FestaA brief survey on semi-Lagrangian schemes for image processing book chapterIn Innovations for Shape Analysis, Mathematics and Visualization, Springer, 191–218
  31. 2012
    E. Carlini, R. FerrettiA semi-Lagrangian scheme for area preserving flows proceedings21st International Conference on Pattern Recognition (ICPR), Tsukuba, 2029–2033
  32. 2011
    E. Carlini, N. Forcadel, R. MonneauA generalized fast marching method for dislocation dynamicsSIAM Journal on Numerical Analysis 49(6), 2470–2500
  33. 2010
    E. Carlini, M. Falcone, R. FerrettiConvergence of a large time-step scheme for mean curvature motionInterfaces and Free Boundaries 12(4), 409–441
  34. 2010
    A. M. Bersani, E. Carlini, P. Lanucara, M. Rorro, V. RuggieroApplication of optimal control techniques and advanced computing to the study of enzyme kineticsMathematics and Computers in Simulation 81(3), 705–716
  35. 2008
    E. Carlini, M. Falcone, N. Forcadel, R. MonneauConvergence of a generalized fast marching method for an eikonal equation with a velocity changing signSIAM Journal on Numerical Analysis 46(6), 2920–2952
  36. 2008
    E. Carlini, R. FerrettiA semi-Lagrangian approximation of min-max type for the stationary mean curvature equation proceedingsENUMATH 2007, Springer, 679–686
  37. 2008
    E. CarliniA single-pass scheme for the mean curvature motion of convex curves proceedingsENUMATH 2007, Springer, 671–678
  38. 2007
    E. Carlini, M. Falcone, R. FerrettiA semi-Lagrangian scheme for the curve shortening flow in codimension 2Journal of Computational Physics 225(2), 1388–1408
  39. 2006
    O. Alvarez, E. Carlini, R. Monneau, E. RouyConvergence of a first order scheme for a non-local eikonal equationApplied Numerical Mathematics 56(9), 1136–1146
  40. 2006
    O. Alvarez, E. Carlini, R. Monneau, E. RouyA convergent scheme for a non local Hamilton–Jacobi equation modelling dislocation dynamicsNumerische Mathematik 104(4), 413–444
  41. 2006
    E. Carlini, E. Cristiani, N. ForcadelA non-monotone fast marching scheme for a Hamilton–Jacobi equation modelling dislocation dynamics proceedingsENUMATH 2005, Springer, 724–731
  42. 2006
    E. Carlini, M. Falcone, R. FerrettiA time-adaptive semi-Lagrangian approximation to mean curvature motion proceedingsENUMATH 2005, Springer, 732–739
  43. 2005
    O. Alvarez, E. Carlini, P. Hoch, Y. Le Bouar, R. MonneauDislocation dynamics described by non-local Hamilton–Jacobi equationsMaterials Science & Engineering A 400–401, 162–165
  44. 2005
    E. Carlini, R. Ferretti, G. RussoA weighted essentially non-oscillatory, large time-step scheme for Hamilton–Jacobi equationsSIAM Journal on Scientific Computing 27(3), 1071–1091
  45. 2004
    E. Carlini, M. Falcone, R. FerrettiAn efficient algorithm for Hamilton–Jacobi equations in high dimensionComputing and Visualization in Science 7(1), 15–29

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
  1. Numerical Analysis (Istituzioni di Analisi Numerica), MSc in Mathematics for Applications, 2020–2026
  2. Programming and Computing Lab, BSc in Mathematics, 2011–2024
  3. Mathematical Methods for Engineering (MAT/08 module), Chemical Engineering, 2016–2018
  4. Computer skills: Matlab and Python, BSc in Mathematics, 2008–2023
  5. Numerical Analysis and C Programming, Master in Scientific Computing, 2008–2017
  6. Numerical methods for ordinary differential equations, BSc in Mathematics, 2009–2010
  7. Numerical Computing 1 and 2, BSc in Chemistry, 2008–2010
  8. Excellence course “Formulation and numerical approximation of some minimization problems”, with M. Ponsiglione, 2009–2010
  9. 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
  1. L. Ottavio, Un problema di tipo MFG su networks, 2025–2026
  2. E. Giansante, Riduzione di modelli per la risoluzione di equazioni di Hamilton–Jacobi, 2024–2025
  3. G. Testa, Numerical scheme for the Hughes problem using a mean-field game model, 2023–2024
  4. S. Bonaldo, Algoritmi per il calcolo di equilibri di Nash per giochi con numero finito di giocatori, 2023–2024
  5. M. Laurenti, Uno schema alle differenze finite per mean-field games, 2023–2024
  6. A. Mastrodonato, A finite difference approach for solving a mean-field game model of price formation, 2022–2023
  7. V. Coscetti, Uno schema semi-Lagrangiano per Mean Field Games deterministici basato sulle iterazioni della policy, 2022–2023
  8. L. Ferentini, Leggi di conservazione su network con schema di Godunov e applicazione al modello LWR, 2021–2022
  9. F. Puca, Approssimazione numerica del modello di Keller–Segel per la chemiotassi, 2020–2021
  10. C. Lin, Schema alle differenze finite per equazioni di Hamilton–Jacobi sui network, 2019–2020
  11. 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

I have also taken part in more than 30 national and international projects, including PRIN 2007, 2009 and 2017, the European project SADCO (2011–2014), the Programme Gaspard Monge pour l'Optimisation and KAUST grants on Mean Field Games.

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

Other conferences organized (2011–2023)
  1. Maurizio day: numerics for controls, PDEs and images, Sapienza University of Rome, 2023
  2. Nonlinear hyperbolic PDEs: numerical techniques and related models, minisymposium at WONAPDE, Concepción, Chile
  3. Numerical approximation of hyperbolic problems and applications, GNCS Project, Sapienza University of Rome, 2020
  4. Numerical methods for optimal control problems and differential games, minisymposium at SIMAI, Milan, 2016
  5. Numerical methods for PDEs: optimal control, games and image processing, Sapienza University of Rome, 2014
  6. Numerics in Image and Surface Processing, Bologna, 2014
  7. Mathematical methods in image processing, Sapienza University of Rome, 2013
  8. Recent advances on theory and applications of semi-Lagrangian methods, Sapienza University of Rome, 2011
Referee activity

SIAM Journal on Numerical Analysis, SIAM Journal on Control and Optimization, SIAM Journal on Scientific Computing, Mathematics of Computation, ESAIM: M2AN, Mathematical Methods in the Applied Sciences, Computers & Mathematics with Applications, Applied Mathematics and Computation, Dynamic Games and Applications, IEEE Transactions on Automatic Control, IEEE Conference on Decision and Control, Communications in Mathematical Sciences, IEEE Journal of Selected Topics in Signal Processing.

Committees and service
  1. Examiner and referee of PhD theses at Université Côte d'Azur and Mohammed VI Polytechnic University (2026), University of Tours (2025), University of Rouen (2025), Université de Technologie de Compiègne (2021), GSSI (2020) and Sapienza University of Rome
  2. Evaluation committees for researcher and associate professor positions in Numerical Analysis (Ferrara, Catania, Verona, Insubria)
  3. PhD committee in Mathematics, Sapienza University of Rome (2020–2022)
  4. Thesis committee (2012–2022), library committee, master's study plans committee
  5. Mathematical Olympiad project, Rome section

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

Visiting researcher at KAUST (Saudi Arabia), XLIM University of Limoges, INSA Rouen and CERMICS ENPC. Languages: Italian, English, basic French and Spanish.

Contact

Emailcarlini@mat.uniroma1.it

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