Page 533 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 533
PDE MODELS IN LIFE AND SOCIAL SCIENCES (MS-71)
Particle methods for local mean-field games
Diogo Gomes, diogo.gomes@kaust.edu.sa
King Abdullah University of Science and Technology, Saudi Arabia
We study a particle approximation for one-dimensional first-order mean-field games (MFGs)
with local interactions with planning conditions. Our problem comprises a system of a Hamilton-
Jacobi equation coupled with a transport equation. As we are dealing with the planning prob-
lem, we prescribe initial and terminal distributions for the transport equation. The particle
approximation builds on a semi-discrete variational problem. First, we address the existence
and uniqueness of the semi-discrete variational problem. Next, we show that our discretization
preserves some conserved quantities. Finally, we prove that the approximation by particle sys-
tems preserves displacement convexity. We use this last property to establish uniform estimates
for the discrete problem. All results for the discrete problem are illustrated with numerical
examples.
Asymptotic consensus in the Hegselmann-Krause model with finite speed
of information propagation
Jan Haskovec, jan.haskovec@kaust.edu.sa
King Abdullah University of Science and Technology, Saudi Arabia
We introduce a variant of the Hegselmann-Krause model of consensus formation where infor-
mation between agents propagates with a finite speed c > 0. This leads to a system of ordinary
differential equations (ODE) with state-dependent delay. Observing that the classical well-
posedness theory for ODE systems does not apply, we provide a proof of global existence and
uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in
the spatially one-dimensional setting of the model, as long as agents travel slower than c. We
also provide sufficient conditions for asymptotic consensus in the spatially multi-dimensional
setting. Finally, we discuss the mean-field limit of the model, showing that it does not facilitate
a description in terms of a Fokker-Planck equation.
Mean-field optimal control for biological pattern formation
Lisa Maria Kreusser, lmk48@cam.ac.uk
University of Cambridge, United Kingdom
In this talk I will discuss a class of interacting particle models with anisotropic repulsive-
attractive interaction forces. These models are motivated by the simulation of fingerprint databa-
ses, which are required in forensic science and biometric applications. In existing models, the
forces are isotropic and particle models lead to non-local aggregation PDEs with radially sym-
metric potentials. The central novelty in the models I consider is an anisotropy which can
describe complex patterns accurately. I will discuss the role of anisotropic interaction in the
forward models. Then, I will propose a mean-field optimal control problem for the parameter
identification of a given pattern which is an inverse problem. The cost functional is based on
the Wasserstein distance between the probability measures of the modeled and the desired pat-
terns. The first-order optimality conditions corresponding to the optimal control problem are
531
Particle methods for local mean-field games
Diogo Gomes, diogo.gomes@kaust.edu.sa
King Abdullah University of Science and Technology, Saudi Arabia
We study a particle approximation for one-dimensional first-order mean-field games (MFGs)
with local interactions with planning conditions. Our problem comprises a system of a Hamilton-
Jacobi equation coupled with a transport equation. As we are dealing with the planning prob-
lem, we prescribe initial and terminal distributions for the transport equation. The particle
approximation builds on a semi-discrete variational problem. First, we address the existence
and uniqueness of the semi-discrete variational problem. Next, we show that our discretization
preserves some conserved quantities. Finally, we prove that the approximation by particle sys-
tems preserves displacement convexity. We use this last property to establish uniform estimates
for the discrete problem. All results for the discrete problem are illustrated with numerical
examples.
Asymptotic consensus in the Hegselmann-Krause model with finite speed
of information propagation
Jan Haskovec, jan.haskovec@kaust.edu.sa
King Abdullah University of Science and Technology, Saudi Arabia
We introduce a variant of the Hegselmann-Krause model of consensus formation where infor-
mation between agents propagates with a finite speed c > 0. This leads to a system of ordinary
differential equations (ODE) with state-dependent delay. Observing that the classical well-
posedness theory for ODE systems does not apply, we provide a proof of global existence and
uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in
the spatially one-dimensional setting of the model, as long as agents travel slower than c. We
also provide sufficient conditions for asymptotic consensus in the spatially multi-dimensional
setting. Finally, we discuss the mean-field limit of the model, showing that it does not facilitate
a description in terms of a Fokker-Planck equation.
Mean-field optimal control for biological pattern formation
Lisa Maria Kreusser, lmk48@cam.ac.uk
University of Cambridge, United Kingdom
In this talk I will discuss a class of interacting particle models with anisotropic repulsive-
attractive interaction forces. These models are motivated by the simulation of fingerprint databa-
ses, which are required in forensic science and biometric applications. In existing models, the
forces are isotropic and particle models lead to non-local aggregation PDEs with radially sym-
metric potentials. The central novelty in the models I consider is an anisotropy which can
describe complex patterns accurately. I will discuss the role of anisotropic interaction in the
forward models. Then, I will propose a mean-field optimal control problem for the parameter
identification of a given pattern which is an inverse problem. The cost functional is based on
the Wasserstein distance between the probability measures of the modeled and the desired pat-
terns. The first-order optimality conditions corresponding to the optimal control problem are
531
