11,161 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
“Disguised in scarlet”. Hume and Turin in 17481
Archive Research concerning the following subjects: Hume in Turin (8 May - 29 November 1748) as secretary and aide-de-camp to General St Clair. St Clair's mission and correspondence (his letters written by Hume). Hume's account of plains and fortifications (the principle of sympathy). Hume's maladay and religion. 'Of National Characters' and Hume's reading of Montesquieu's Esprit. Hume and Lord Charlemont
Generalized Wasserstein distance and its application to transport equations with source
In this article, we generalize the Wasserstein distance to measures with
different masses. We study the properties of such distance. In particular, we
show that it metrizes weak convergence for tight sequences.
We use this generalized Wasserstein distance to study a transport equation
with source, in which both the vector field and the source depend on the
measure itself. We prove existence and uniqueness of the solution to the Cauchy
problem when the vector field and the source are Lipschitzian with respect to
the generalized Wasserstein distance
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
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