I'm an assistant professor of mathematics in the Centre for Analysis, Scientific Computing and Applications at TU/e (Eindhoven University of Technology). My work deals with the numerical analysis of kinetic equations and other partial differential equations (PDEs). I'm also interested in collective dynamics, self-organisation, and the control of agent-based models.
Prior to my current post, I was a research associate at the University of Oxford, affiliated with the Oxford Centre for Nonlinear Partial Differential Equations at the Mathematical Institute. I also worked as a postdoctoral researcher at the Université de Lille with the ANEDP and Inria RAPSODI groups, under the supervision of Thomas Rey. I earned my doctorate at Imperial College London, where my advisors were José Antonio Carrillo and Pierre Degond.
My TU/e site can be found here.
Recent Publications
Aggregation-diffusion equations for collective behaviour in the sciences
Rafael Bailo
José Antonio Carrillo
David Gómez-Castro
Rafael Bailo · José Antonio Carrillo · David Gómez-Castro
In Recent Developments in Industrial and Applied Mathematics, 2026.
In Recent Developments in Industrial and Applied Mathematics, 2026.
@InCollection{BCG2026,
title={Aggregation-diffusion equations for collective behaviour in the sciences},
author={Bailo, Rafael and Carrillo, José Antonio and Gómez-Castro, David},
year={2026},
doi={10.1007/978-981-95-1446-5_9},
archivePrefix={arXiv},
arXivId={2405.16679},
eprint={2405.16679},
}This is a survey article based on the content of the plenary lecture given by José A. Carrillo at the ICIAM23 conference in Tokyo. It is devoted to produce a snapshot of the state of the art in the analysis, numerical analysis, simulation, and applications of the vast area of aggregation-diffusion equations. We also discuss the implications in mathematical biology explaining cell sorting in tissue growth as an example of this modelling framework. This modelling strategy is quite successful in other timely applications such as global optimisation, parameter estimation and machine learning.
This is a survey article based on the content of the plenary lecture given by José A. Carrillo at the ICIAM23 conference in Tokyo. It is devoted to produce a snapshot of the state of the art in the analysis, numerical analysis, simulation, and applications of the vast area of aggregation-diffusion equations. We also discuss the implications in mathematical biology explaining cell sorting in tissue growth as an example of this modelling framework. This modelling strategy is quite successful in other timely applications such as global optimisation, parameter estimation and machine learning.
A particle method for stationary transport equations
Rafael Bailo
Julie Binard
Pierre Degond
Pascal Noble
Rafael Bailo · Julie Binard · Pierre Degond · Pascal Noble
arXiv: 2511.08774, 2025.
arXiv: 2511.08774, 2025.
@Article{BBD2025,
title={A particle method for stationary transport equations},
author={Bailo, Rafael and Binard, Julie and Degond, Pierre and Noble, Pascal},
journal={Preprint arXiv: 2511.08774},
year={2025},
doi={10.48550/arXiv.2511.08774},
archivePrefix={arXiv},
arXivId={2511.08774},
eprint={2511.08774},
}We present and study a particle method for the stationary solutions of a class of transport equations. This method is inspired by non-stationary particle methods, the time variable being replaced by one spatial variable. Particles trajectories are computed using the "time-dependent" equations, and then the approximation is based on a quadrature method using the particle locations as quadrature points. We prove the convergence of the scheme under suitable regularity assumptions on the data and the solution, together with a "characteristic completeness" assumption (the characteristic curves fullfill the whole computational domain). We also provide an error estimate. The scheme is tested numerically on a two dimensional linear equation and we present a numerical study of convergence. Finally, we use this method to carry out numerical simulations of a landscape evolution model, where an erodible topography evolves under the effects of water erosion and sedimentation. The scheme is then useful to deal with wet/dry areas.
We present and study a particle method for the stationary solutions of a class of transport equations. This method is inspired by non-stationary particle methods, the time variable being replaced by one spatial variable. Particles trajectories are computed using the "time-dependent" equations, and then the approximation is based on a quadrature method using the particle locations as quadrature points. We prove the convergence of the scheme under suitable regularity assumptions on the data and the solution, together with a "characteristic completeness" assumption (the characteristic curves fullfill the whole computational domain). We also provide an error estimate. The scheme is tested numerically on a two dimensional linear equation and we present a numerical study of convergence. Finally, we use this method to carry out numerical simulations of a landscape evolution model, where an erodible topography evolves under the effects of water erosion and sedimentation. The scheme is then useful to deal with wet/dry areas.
Uncertainty quantification for the homogeneous Landau-Fokker-Planck equation via deterministic particle Galerkin methods
Rafael Bailo · José Antonio Carrillo · Andrea Medaglia · Mattia Zanella
Multiscale Modeling and Simulation, 23 (2), 2025.
Multiscale Modeling and Simulation, 23 (2), 2025.
@Article{BCM2025,
title={Uncertainty quantification for the homogeneous {L}andau-{F}okker-{P}lanck equation via deterministic particle {G}alerkin methods},
author={Bailo, Rafael and Carrillo, José Antonio and Medaglia, Andrea and Zanella, Mattia},
journal={Multiscale Model. Sim.},
year={2025},
doi={10.1137/23M1623653},
volume={23},
number={2},
archivePrefix={arXiv},
arXivId={2312.07218},
eprint={2312.07218},
}We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a sg representation of each particle. This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy production. We provide a regularity results for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases.
We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a sg representation of each particle. This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy production. We provide a regularity results for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases.