Publications

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The collisional particle-in-cell method for the Vlasov-Maxwell-Landau equations

Rafael Bailo · José Antonio Carrillo · Jingwei Hu

Journal of Plasma Physics, 2024 (to appear).

Journal of Plasma Physics, 2024 (to appear).

@Article{BCH2024,
	title={The collisional particle-in-cell method for the {V}lasov-{M}axwell-{L}andau equations},
	author={Bailo, Rafael and Carrillo, José Antonio and Hu, Jingwei},
	journal={J. Plasma Phys. (to appear)},
	year={2024},
	doi={10.48550/arXiv.2401.01689},
	archivePrefix={arXiv},
	arXivId={2401.01689},
	eprint={2401.01689},
}

We introduce an extension of the particle-in-cell (PIC) method that captures the Landau collisional effects in the Vlasov-Maxwell-Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum, and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport-collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability, and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.

We introduce an extension of the particle-in-cell (PIC) method that captures the Landau collisional effects in the Vlasov-Maxwell-Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum, and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport-collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability, and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.

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CBX: Python and Julia packages for consensus-based interacting particle methods

Journal of Open Source Software, 9 (98), 2024.

Journal of Open Source Software, 9 (98), 2024.

@Article{BBG2024,
	title={{CBX}: {P}ython and {J}ulia packages for
    consensus-based interacting particle methods},
	author={Bailo, Rafael and Barbaro, Alethea and Gomes, Susana N. and Riedl, Konstantin and Roith, Tim and Totzeck, Claudia and Vaes, Urbain},
	journal={Journal of Open Source Software},
	year={2024},
	doi={10.21105/joss.06611},
	volume={9},
	number={98},
	archivePrefix={arXiv},
	arXivId={2403.14470},
	eprint={2403.14470},
}

We introduce CBXPy and ConsensusBasedX.jl, Python and Julia implementations of consensus-based interacting particle systems (CBX), which generalise consensus-based optimisation methods (CBO) for global, derivative-free optimisation. The raison d'être of our libraries is twofold: on the one hand, to offer high-performance implementations of CBX methods that the community can use directly, while on the other, providing a general interface that can accommodate and be extended to further variations of the CBX family. Python and Julia were selected as the leading high-level languages in terms of usage and performance, as well as their popularity among the scientific computing community. Both libraries have been developed with a common ethos, ensuring a similar API and core functionality, while leveraging the strengths of each language and writing idiomatic code.

We introduce CBXPy and ConsensusBasedX.jl, Python and Julia implementations of consensus-based interacting particle systems (CBX), which generalise consensus-based optimisation methods (CBO) for global, derivative-free optimisation. The raison d'être of our libraries is twofold: on the one hand, to offer high-performance implementations of CBX methods that the community can use directly, while on the other, providing a general interface that can accommodate and be extended to further variations of the CBX family. Python and Julia were selected as the leading high-level languages in terms of usage and performance, as well as their popularity among the scientific computing community. Both libraries have been developed with a common ethos, ensuring a similar API and core functionality, while leveraging the strengths of each language and writing idiomatic code.

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Aggregation-diffusion equations for collective behaviour in the sciences

arXiv: 2405.16679, 2024.

arXiv: 2405.16679, 2024.

@Article{BCG2024,
	title={Aggregation-diffusion equations for collective behaviour in the sciences},
	author={Bailo, Rafael and Carrillo, José Antonio and Gómez-Castro, David},
	journal={Preprint arXiv: 2405.16679},
	year={2024},
	doi={10.48550/arXiv.2405.16679},
	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.

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A finite-volume scheme for fractional diffusion on bounded domains

European Journal of Applied Mathematics, 2024 (to appear).

European Journal of Applied Mathematics, 2024 (to appear).

@Article{BCF2024,
	title={A finite-volume scheme for fractional diffusion on bounded domains},
	author={Bailo, Rafael and Carrillo, José Antonio and Fronzoni, Stefano and Gómez-Castro, David},
	journal={European J. Appl. Math. (to appear)},
	year={2024},
	doi={10.48550/arXiv.2309.08283},
	archivePrefix={arXiv},
	arXivId={2309.08283},
	eprint={2309.08283},
}

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the L'evy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the L'evy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

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Pedestrian models with congestion effects

Mathematical Models and Methods in Applied Sciences, 34 (6), 2024.

Mathematical Models and Methods in Applied Sciences, 34 (6), 2024.

@Article{ABD2024,
	title={Pedestrian models with congestion effects},
	author={Aceves-Sánchez, Pedro and Bailo, Rafael and Degond, Pierre and Mercier, Zoé},
	journal={Math. Models Methods Appl. Sci.},
	year={2024},
	doi={10.1142/S0218202524400050},
	volume={34},
	number={6},
	archivePrefix={arXiv},
	arXivId={2401.08630},
	halId={hal-04334055},
	eprint={2401.08630},
}

We study the validity of the dissipative Aw-Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while inducing a steering behaviour. Furthermore, we introduce a semi-implicit, structure-preserving, and asymptotic-preserving numerical scheme which can handle the numerical solution of the model efficiently. We perform the first numerical simulations of the dissipative Aw-Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical experiments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.

We study the validity of the dissipative Aw-Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while inducing a steering behaviour. Furthermore, we introduce a semi-implicit, structure-preserving, and asymptotic-preserving numerical scheme which can handle the numerical solution of the model efficiently. We perform the first numerical simulations of the dissipative Aw-Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical experiments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.

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Uncertainty quantification for the homogeneous Landau-Fokker-Planck equation via deterministic particle Galerkin methods

arXiv: 2312.07218, 2023.

arXiv: 2312.07218, 2023.

@Article{BCM2023,
	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={Preprint arXiv: 2312.07218},
	year={2023},
	doi={10.48550/arXiv.2312.07218},
	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.

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Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation

Communications in Computational Physics, 34 (3), 2023.

Communications in Computational Physics, 34 (3), 2023.

@Article{BCK2023,
	title={Unconditional bound-preserving and energy-dissipating finite-volume schemes for the {C}ahn-{H}illiard equation},
	author={Bailo, Rafael and Carrillo, José Antonio and Kalliadasis, Serafim and Perez, Sergio P.},
	journal={Commun. Comput. Phys.},
	year={2023},
	doi={10.4208/cicp.OA-2023-0049},
	volume={34},
	number={3},
	archivePrefix={arXiv},
	arXivId={2105.05351},
	eprint={2105.05351},
}

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

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Bound-preserving finite-volume schemes for systems of continuity equations with saturation

Rafael Bailo · José Antonio Carrillo · Jingwei Hu

SIAM Journal on Applied Mathematics, 83 (3), 2023.

SIAM Journal on Applied Mathematics, 83 (3), 2023.

@Article{BCH2023,
	title={Bound-preserving finite-volume schemes for systems of continuity equations with saturation},
	author={Bailo, Rafael and Carrillo, José Antonio and Hu, Jingwei},
	journal={SIAM J. Appl. Math.},
	year={2023},
	doi={10.1137/22M1488703},
	volume={83},
	number={3},
	archivePrefix={arXiv},
	arXivId={2110.08186},
	eprint={2110.08186},
}

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

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Projective and telescopic projective integration for non-linear kinetic mixtures

Rafael Bailo
Thomas Rey

Rafael Bailo · Thomas Rey

Journal of Computational Physics, 458, 2022.

Journal of Computational Physics, 458, 2022.

@Article{BR2022,
	title={Projective and telescopic projective integration for non-linear kinetic mixtures},
	author={Bailo, Rafael and Rey, Thomas},
	journal={J. Comput. Phys.},
	year={2022},
	doi={10.1016/j.jcp.2022.111082},
	volume={458},
	archivePrefix={arXiv},
	arXivId={2106.08811},
	eprint={2106.08811},
}

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.

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Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations

Mathematical Models and Methods in Applied Sciences, 30 (13), 2020.

Mathematical Models and Methods in Applied Sciences, 30 (13), 2020.

@Article{BCM2020,
	title={Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations},
	author={Bailo, Rafael and Carrillo, José Antonio and Murakawa, Hideki and Schmidtchen, Markus},
	journal={Math. Models Methods Appl. Sci.},
	year={2020},
	doi={10.1142/S0218202520500487},
	volume={30},
	number={13},
	archivePrefix={arXiv},
	arXivId={2002.10821},
	eprint={2002.10821},
}

We study an implicit finite-volume scheme for non-linear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced by Bailo, Carrillo, and Hu (2020). Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

We study an implicit finite-volume scheme for non-linear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced by Bailo, Carrillo, and Hu (2020). Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

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Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure

Rafael Bailo · José Antonio Carrillo · Jingwei Hu

Communications in Mathematical Sciences, 18 (5), 2020.

Communications in Mathematical Sciences, 18 (5), 2020.

@Article{BCH2020,
	title={Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure},
	author={Bailo, Rafael and Carrillo, José Antonio and Hu, Jingwei},
	journal={Commun. Math. Sci.},
	year={2020},
	doi={10.4310/CMS.2020.v18.n5.a5},
	volume={18},
	number={5},
	archivePrefix={arXiv},
	arXivId={1811.11502},
	eprint={1811.11502},
}

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker–Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the efficient construction of these schemes in higher dimensions while preserving their structural properties. Numerical experiments validate the schemes and show their ability to handle complicated phenomena typical in aggregation-diffusion equations, such as free boundaries, metastability, merging and phase transitions.

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker–Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the efficient construction of these schemes in higher dimensions while preserving their structural properties. Numerical experiments validate the schemes and show their ability to handle complicated phenomena typical in aggregation-diffusion equations, such as free boundaries, metastability, merging and phase transitions.

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Optimal consensus control of the Cucker-Smale model

IFAC-PapersOnLine, 51 (13), 2018.

IFAC-PapersOnLine, 51 (13), 2018.

@Article{BBC2018,
	title={Optimal consensus control of the {C}ucker-{S}male model},
	author={Bailo, Rafael and Bongini, Mattia and Carrillo, José Antonio and Kalise, Dante},
	journal={IFAC-PapersOnLine},
	year={2018},
	doi={10.1016/j.ifacol.2018.07.245},
	volume={51},
	number={13},
	archivePrefix={arXiv},
	arXivId={1802.01529},
	eprint={1802.01529},
}

We study the numerical realisation of optimal consensus control laws for agent-based models. For a nonlinear multi-agent system of Cucker-Smale type, consensus control is cast as a dynamic optimisation problem for which we derive first-order necessary optimality conditions. In the case of a smooth penalisation of the control energy, the optimality system is numerically approximated via a gradient-descent method. For sparsity promoting, non-smooth l1-norm control penalisations, the optimal controllers are realised by means of heuristic methods. For an increasing number of agents, we discuss the approximation of the consensus control problem by following a mean-field modelling approach.

We study the numerical realisation of optimal consensus control laws for agent-based models. For a nonlinear multi-agent system of Cucker-Smale type, consensus control is cast as a dynamic optimisation problem for which we derive first-order necessary optimality conditions. In the case of a smooth penalisation of the control energy, the optimality system is numerically approximated via a gradient-descent method. For sparsity promoting, non-smooth l1-norm control penalisations, the optimal controllers are realised by means of heuristic methods. For an increasing number of agents, we discuss the approximation of the consensus control problem by following a mean-field modelling approach.

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Pedestrian models based on rational behaviour

In Crowd Dynamics, Volume 1, 2018.

In Crowd Dynamics, Volume 1, 2018.

@InCollection{BCD2018,
	title={Pedestrian models based on rational behaviour},
	author={Bailo, Rafael and Carrillo, José Antonio and Degond, Pierre},
	year={2018},
	doi={10.1007/978-3-030-05129-7_9},
	archivePrefix={arXiv},
	arXivId={1808.07426},
	eprint={1808.07426},
}

Following the paradigm set by attraction-repulsion-alignment schemes, a myriad of individual based models have been proposed to calculate the evolution of abstract agents. While the emergent features of many agent systems have been described astonishingly well with force-based models, this is not the case for pedestrians. Many of the classical schemes have failed to capture the fine detail of crowd dynamics, and it is unlikely that a purely mechanical model will succeed. As a response to the mechanistic literature, we will consider a model for pedestrian dynamics that attempts to reproduce the rational behaviour of individual agents through the means of anticipation. Each pedestrian undergoes a two-step time evolution based on a perception stage and a decision stage. We will discuss the validity of this game theoretical based model in regimes with varying degrees of congestion, ultimately presenting a correction to the mechanistic model in order to achieve realistic high-density dynamics.

Following the paradigm set by attraction-repulsion-alignment schemes, a myriad of individual based models have been proposed to calculate the evolution of abstract agents. While the emergent features of many agent systems have been described astonishingly well with force-based models, this is not the case for pedestrians. Many of the classical schemes have failed to capture the fine detail of crowd dynamics, and it is unlikely that a purely mechanical model will succeed. As a response to the mechanistic literature, we will consider a model for pedestrian dynamics that attempts to reproduce the rational behaviour of individual agents through the means of anticipation. Each pedestrian undergoes a two-step time evolution based on a perception stage and a decision stage. We will discuss the validity of this game theoretical based model in regimes with varying degrees of congestion, ultimately presenting a correction to the mechanistic model in order to achieve realistic high-density dynamics.