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2017 | Book

Continuum Modeling of Biological Network Formation Read chapter Read first chapter

Active Particles, Volume 1

Advances in Theory, Models, and Applications

Editors: Nicola Bellomo, Pierre Degond, Eitan Tadmor

Publisher: Springer International Publishing

Book Series : Modeling and Simulation in Science, Engineering and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume collects ten surveys on the modeling, simulation, and applications of active particles using methods ranging from mathematical kinetic theory to nonequilibrium statistical mechanics. The contributing authors are leading experts working in this challenging field, and each of their chapters provides a review of the most recent results in their areas and looks ahead to future research directions. The approaches to studying active matter are presented here from many different perspectives, such as individual-based models, evolutionary games, Brownian motion, and continuum theories, as well as various combinations of these. Applications covered include biological network formation and network theory; opinion formation and social systems; control theory of sparse systems; theory and applications of mean field games; population learning; dynamics of flocking systems; vehicular traffic flow; and stochastic particles and mean field approximation. Mathematicians and other members of the scientific community interested in active matter and its many applications will find this volume to be a timely, authoritative, and valuable resource.

Table of Contents

Frontmatter
Continuum Modeling of Biological Network Formation
Abstract
We present an overview of recent analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transportation networks. The model describes the pressure field using a Darcy type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro- and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.
Giacomo Albi, Martin Burger, Jan Haskovec, Peter Markowich, Matthias Schlottbom
Recent Advances in Opinion Modeling: Control and Social Influence
Abstract
We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents’ role in the opinion exchange process.
Giacomo Albi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella
Interaction Network, State Space, and Control in Social Dynamics
Abstract
In the present chapter, we study the emergence of global patterns in large groups in first- and second-order multiagent systems, focusing on two ingredients that influence the dynamics: the interaction network and the state space. The state space determines the types of equilibrium that can be reached by the system. Meanwhile, convergence to specific equilibria depends on the connectivity of the interaction network and on the interaction potential. When the system does not satisfy the necessary conditions for convergence to the desired equilibrium, control can be exerted, both on finite-dimensional systems and on their mean-field limit.
Aylin Aydoğdu, Marco Caponigro, Sean McQuade, Benedetto Piccoli, Nastassia Pouradier Duteil, Francesco Rossi, Emmanuel Trélat
Variational Mean Field Games
Abstract
This paper is a brief presentation of those mean field games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e., with diffusion) is also presented in both the stationary and dynamic cases. The variational problems relevant to MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on augmented Lagrangian algorithms, and some new simulations are shown at the end of the paper.
Jean-David Benamou, Guillaume Carlier, Filippo Santambrogio
Sparse Control of Multiagent Systems
Abstract
In recent years, numerous studies have focused on the mathematical modeling of social dynamics, with self-organization, i.e., the autonomous pattern formation, as the main driving concept. Usually, first- or second-order models are employed to reproduce, at least qualitatively, certain global patterns (such as bird flocking, milling schools of fish, or queue formations in pedestrian flows, just to mention a few). It is, however, common experience that self-organization does not always spontaneously occur in a society. In this review chapter, we aim to describe the limitations of decentralized controls in restoring certain desired configurations and to address the question of whether it is possible to externally and parsimoniously influence the dynamics to reach a given outcome. More specifically, we address the issue of finding the sparsest control strategy for finite agent-based models in order to lead the dynamics optimally toward a desired pattern.
Mattia Bongini, Massimo Fornasier
A Kinetic Theory Approach to the Modeling of Complex Living Systems
Abstract
In this chapter, a mathematical structure is derived to provide a general framework toward the modeling of space-homogeneous living systems, according to the kinetic theory for active particles. This structure can be adapted to study a variety of processes such as collective learning and social dynamics. Simple models regarding learning in a classroom and the dynamics of the criminality are presented to illustrate how the general modeling strategy operates in well-defined applications. Future research directions using the proposed approach are discussed.
Diletta Burini, Livio Gibelli, Nisrine Outada
A Review on Attractive–Repulsive Hydrodynamics for Consensus in Collective Behavior
Abstract
This survey summarizes and illustrates the main qualitative properties of hydrodynamics models for collective behavior. These models include a velocity consensus term together with attractive–repulsive potentials leading to non-trivial flock profiles. The connection between the underlying particle systems and the swarming hydrodynamic equations is performed through kinetic theory modeling arguments. We focus on Lagrangian schemes for the hydrodynamic systems showing the different qualitative behaviors of the systems and its capability of keeping properties of the original particle models. We illustrate the known results concerning large-time profiles and blowup in finite time of the hydrodynamic systems to validate the numerical scheme. We finally explore the unknown situations making use of the numerical scheme showcasing a number of conjectures based on the numerical results.
José A. Carrillo, Young-Pil Choi, Sergio P. Perez
Emergent Dynamics of the Cucker–Smale Flocking Model and Its Variants
Abstract
In this chapter, we present the Cucker–Smale-type flocking models and discuss their mathematical structures and flocking theorems in terms of coupling strength, interaction topologies, and initial data. In 2007, two mathematicians Felipe Cucker and Steve Smale introduced a second-order particle model which resembles Newton’s equations in N-body system and present how their simple model can exhibit emergent flocking behavior under sufficient conditions expressed only in terms of parameters and initial data. After Cucker–Smale’s seminal works in [31, 32], their model has received lots of attention from applied math and control engineering communities. We discuss the state of the art for the flocking theorems to Cucker–Smale-type flocking models.
Young-Pil Choi, Seung-Yeal Ha, Zhuchun Li
Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows
Abstract
We review the recent results and present new ones on a deterministic follow-the-leader particle approximation of first-and second-order models for traffic flow and pedestrian movements. We start by constructing the particle scheme for the first-order Lighthill–Whitham–Richards (LWR) model for traffic flow. The approximation is performed by a set of ODEs following the position of discretized vehicles seen as moving particles. The convergence of the scheme in the many particle limit toward the unique entropy solution of the LWR equation is proven in the case of the Cauchy problem on the real line. We then extend our approach to the initial–boundary value problem (IBVP) with time-varying Dirichlet data on a bounded interval. In this case, we prove that our scheme is convergent strongly in \(\mathbf {L^{1}}\) up to a subsequence. We then review extensions of this approach to the Hughes model for pedestrian movements and to the second-order Aw–Rascle–Zhang (ARZ) model for vehicular traffic. Finally, we complement our results with numerical simulations. In particular, the simulations performed on the IBVP and the ARZ model suggest the consistency of the corresponding schemes, which is easy to prove rigorously in some simple cases.
M. Di Francesco, S. Fagioli, M. D. Rosini, G. Russo
Mean Field Limit for Stochastic Particle Systems
Abstract
We review some classical and more recent results for the derivation of mean field equations from systems of many particles, focusing on the stochastic case where a large system of SDEs leads to a McKean–Vlasov PDE as the number N of particles goes to infinity. Classical mean field limit results require that the interaction kernel be essentially Lipschitz. To handle more singular interaction kernels is a long-standing and challenging question but which has had some recent successes.
Pierre-Emmanuel Jabin, Zhenfu Wang
Metadata
Title
Active Particles, Volume 1
Editors
Nicola Bellomo
Pierre Degond
Eitan Tadmor
Copyright Year
2017
Electronic ISBN
978-3-319-49996-3
Print ISBN
978-3-319-49994-9
DOI
https://doi.org/10.1007/978-3-319-49996-3

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