Structure preserving schemes for mean-field equations of collective behavior

In this paper we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang-Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic Problem

Uncertainty quantification for kinetic models in socio-economic and life sciences

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker--Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic Equations

A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit

We propose a multilevel Monte Carlo method for a particle-based asymptotic-preserving scheme for kinetic equations. Kinetic equations model transport and collision of particles in a position-velocity phase-space. With a diffusive scaling, the kinetic equation converges to an advection-diffusion equation in the limit of zero mean free path. Classical particle-based techniques suffer from a strict time-step restriction to maintain stability in this limit. Asymptotic-preserving schemes provide a solution to this time step restriction, but introduce a first-order error in the time step size. We demonstrate how the multilevel Monte Carlo method can be used as a bias reduction technique to perform accurate simulations in the diffusive regime, while leveraging the reduced simulation cost given by the asymptotic-preserving scheme. We describe how to achieve the necessary correlation between simulation paths at different levels and demonstrate the potential of the approach via numerical experiments.Comment: 20 pages, 7 figures, published in Monte Carlo and Quasi-Monte Carlo Methods 2018, correction of minor typographical error

Modelling opinion formation by means of kinetic equations

In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics

Weak and strong solutions of equations of compressible magnetohydrodynamics

International audienceThis article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling asumptions. Then, the results of existence of weak solutions, using the notion of renormalized solutions. Then, existence of strong solutions in the neighbourhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted : the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques

Women and citizenship post-trafficking : the case of Nepal

The research for this paper was funded by the Economic and Social Research Council – ESRC Res-062-23-1490: ‘Post Trafficking in Nepal: Sexuality and Citizenship in Livelihood Strategies’. Diane Richardson would like to acknowledge the support provided by the award of a Leverhulme TrustMajor Research Fellowship, ‘Transforming Citizenship: Sexuality, Gender and Citizenship Struggles’ [award MRF-2012-106].This article analyses the relationship between gender, sexuality and citizenship embedded in models of citizenship in the Global South, specifically in South Asia, and the meanings associated with having - or not having - citizenship. It does this through an examination of women's access to citizenship in Nepal in the context of the construction of the emergent nation state in the 'new' Nepal 'post-conflict'. Our analysis explores gendered and sexualized constructions of citizenship in this context through a specific focus on women who have experienced trafficking, and are beginning to organize around rights to sustainable livelihoods and actively lobby for changes in citizenship rules which discriminate against women. Building from this, in the final section we consider important implications of this analysis of post-trafficking experiences for debates about gender, sexuality and citizenship more broadly.Publisher PDFPeer reviewe

Landslide initiation and runout susceptibility modeling in the context of hill cutting and rapid urbanization: a combined approach of weights of evidence and spatial multi-criteria

Rainfall induced landslides are a common threat to the communities living on dangerous hill-slopes in Chittagong Metropolitan Area, Bangladesh. Extreme population pressure, indiscriminate hill cutting, increased precipitation events due to global warming and associated unplanned urbanization in the hills are exaggerating landslide events. The aim of this article is to prepare a scientifically accurate landslide susceptibility map by combining landslide initiation and runout maps. Land cover, slope, soil permeability, surface geology, precipitation, aspect, and distance to hill cut, road cut, drainage and stream network factor maps were selected by conditional independence test. The locations of 56 landslides were collected by field surveying. A weight of evidence (WoE) method was applied to calculate the positive (presence of landslides) and negative (absence of landslides) factor weights. A combination of analytical hierarchical process (AHP) and fuzzy membership standardization (weighs from 0 to 1) was applied for performing a spatial multi-criteria evaluation. Expert opinion guided the decision rule for AHP. The Flow-R tool that allows modeling landslide runout from the initiation sources was applied. The flow direction was calculated using the modified Holmgren’s algorithm. The AHP landslide initiation and runout susceptibility maps were used to prepare a combined landslide susceptibility map. The relative operating characteristic curve was used for model validation purpose. The accuracy of WoE, AHP, and combined susceptibility map was calculated 96%, 97%, and 98%, respectively

Numerical method for the Compton scattering operator

In this paper, we present a new discretization of the Kompaneets equation. This equation can be derived from the Quantum Boltzmann equation for cross sections localized around small changes in the energy. The two collision operators share the same properties (conservation, entropy and steady states). The numerical methods are designed in order to preserve these properties which insures the correct long time behaviour. A supplementary di#culty arises for initial data with a density larger than a critical one, associated with a Planck distribution. In this case, the equilibrium state are the sum of a smooth Planck distribution plus a Delta measure on zero energy. The scheme is able to deal with this singularity.