277 research outputs found
Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes
Motivated by pedestrian modelling, we study evolution of measures in the
Wasserstein space. In particular, we consider the Cauchy problem for a
transport equation, where the velocity field depends on the measure itself.
We deal with numerical schemes for this problem and prove convergence of a
Lagrangian scheme to the solution, when the discretization parameters approach
zero. We also prove convergence of an Eulerian scheme, under more strict
hypotheses. Both schemes are discretizations of the push-forward formula
defined by the transport equation. As a by-product, we obtain existence and
uniqueness of the solution.
All the results of convergence are proved with respect to the Wasserstein
distance. We also show that spaces are not natural for such equations,
since we lose uniqueness of the solution
Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints
This paper is concerned with mathematical modeling of intelligent systems,
such as human crowds and animal groups. In particular, the focus is on the
emergence of different self-organized patterns from non-locality and anisotropy
of the interactions among individuals. A mathematical technique by
time-evolving measures is introduced to deal with both macroscopic and
microscopic scales within a unified modeling framework. Then self-organization
issues are investigated and numerically reproduced at the proper scale,
according to the kind of agents under consideration.Comment: 24 pages, 13 figure
Multiscale modeling of granular flows with application to crowd dynamics
In this paper a new multiscale modeling technique is proposed. It relies on a
recently introduced measure-theoretic approach, which allows to manage the
microscopic and the macroscopic scale under a unique framework. In the
resulting coupled model the two scales coexist and share information. This
allows to perform numerical simulations in which the trajectories and the
density of the particles affect each other. Crowd dynamics is the motivating
application throughout the paper.Comment: 30 pages, 9 figure
Mean-Field Sparse Optimal Control
We introduce the rigorous limit process connecting finite dimensional sparse
optimal control problems with ODE constraints, modeling parsimonious
interventions on the dynamics of a moving population divided into leaders and
followers, to an infinite dimensional optimal control problem with a constraint
given by a system of ODE for the leaders coupled with a PDE of Vlasov-type,
governing the dynamics of the probability distribution of the followers. In the
classical mean-field theory one studies the behavior of a large number of small
individuals freely interacting with each other, by simplifying the effect of
all the other individuals on any given individual by a single averaged effect.
In this paper we address instead the situation where the leaders are actually
influenced also by an external policy maker, and we propagate its effect for
the number of followers going to infinity. The technical derivation of the
sparse mean-field optimal control is realized by the simultaneous development
of the mean-field limit of the equations governing the followers dynamics
together with the -limit of the finite dimensional sparse optimal
control problems.Comment: arXiv admin note: text overlap with arXiv:1306.591
How can macroscopic models reveal self-organization in traffic flow?
In this paper we propose a new modeling technique for vehicular traffic flow,
designed for capturing at a macroscopic level some effects, due to the
microscopic granularity of the flow of cars, which would be lost with a purely
continuous approach. The starting point is a multiscale method for pedestrian
modeling, recently introduced in Cristiani et al., Multiscale Model. Simul.,
2011, in which measure-theoretic tools are used to manage the microscopic and
the macroscopic scales under a unique framework. In the resulting coupled model
the two scales coexist and share information, in the sense that the same system
is simultaneously described from both a discrete (microscopic) and a continuous
(macroscopic) perspective. This way it is possible to perform numerical
simulations in which the single trajectories and the average density of the
moving agents affect each other. Such a method is here revisited in order to
deal with multi-population traffic flow on networks. For illustrative purposes,
we focus on the simple case of the intersection of two roads. By exploiting one
of the main features of the multiscale method, namely its
dimension-independence, we treat one-dimensional roads and two-dimensional
junctions in a natural way, without referring to classical network theory.
Furthermore, thanks to the coupling between the microscopic and the macroscopic
scales, we model the continuous flow of cars without losing the right amount of
granularity, which characterizes the real physical system and triggers
self-organization effects, such as, for example, the oscillatory patterns
visible at jammed uncontrolled crossroads.Comment: 7 pages, 7 figure
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