Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles

Burini, Diletta; Chouhad, Nadia

doi:10.1142/s0218202517400176

Cited by 47 publications

(29 citation statements)

References 36 publications

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“…This seminal approach has then been extended in several directions, with a special concern toward the derivation and qualitative analysis of different kinetic models (see [2,13,34]).…”

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confidence: 99%

Dynamics and control for multi-agent networked systems: A finite-difference approach

Biccari

Zuazua

2019

Math. Models Methods Appl. Sci.

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We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one.We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered.Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.2010 Mathematics Subject Classification. 34D20, 35B36, 65M06, 92D25, 93B05.

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“…This seminal approach has then been extended in several directions, with a special concern toward the derivation and qualitative analysis of different kinetic models (see [2,13,34]).…”

mentioning

confidence: 99%

Dynamics and control for multi-agent networked systems: A finite-difference approach

Biccari

Zuazua

2019

Math. Models Methods Appl. Sci.

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“…The first one starts with a detailed modeling of the interactions involving not only social-biological states at the microscopic scale, but also localization of the interacting active particles; applications refer to crowd dynamics [9], vehicular traffic [5,6], and multicellular systems [27]. The second approach is based on the conjecture that space dynamics is induced by a velocity jump perturbation of the spatially homogeneous activity dynamics; see for example [25,30]. The latter has been introduced by the pioneer paper [47], where the aforementioned perturbation of the transport equation is used to derive diffusion models (parabolic) and models with finite speed of propagation (hyperbolic) [48].…”

Section: Research Perspectivesmentioning

confidence: 99%

“…Some papers show how models at the macroscopic scale can be derived from the underlying description at the microscopic scale delivered by kinetic theory models [25,30]. Hence, it is quite natural to develop calculations analogous to those of the classical kinetic theory as well as to multiscale micro-macro problems.…”

Section: Introductionmentioning

confidence: 99%

On Entropy Dynamics for Active “Living” Particles

Ełaiw

Alghamdi

Bellomo

2017

Entropy

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This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living-hence, complex-systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the state of the overall system is defined by a probability distribution function over the state of the particles. The aim of this paper consists of contributing to a further development of the mathematical kinetic theory of active particles.

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“…He also calibrated the simulation model for emergency door by observing the children, adults and the elderly's walking behaviors. Recently, a mesoscopic approach based on the kinetic theory has been applied to model crowd dynamics [7][8][9]. It's proved to be able to reproduce the complex crowd behaviors such as avoiding obstacles, evacuating in a bounded domain.…”

Section: Introductionmentioning

confidence: 99%

Cluster Risk of Walking Scenarios Based on Macroscopic Flow Model and Crowding Force Analysis

Zhou

Feng

et al. 2018

Sustainability

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Abstract:In recent years, accidents always happen in confined space such as metro stations because of congestion. Various researchers investigated the patterns of dense crowd behaviors in different scenarios via simulations or experiments and proposed methods for avoiding accidents. In this study, a classic continuum macroscopic model was applied to simulate the crowded pedestrian flow in typical scenarios such as at bottlenecks or with an obstacle. The Lax-Wendroff finite difference scheme and artificial viscosity filtering method were used to discretize the model to identify high-density risk areas. Furthermore, we introduced a contact crowding force test of the interactions among pedestrians at bottlenecks. Results revealed that in the most dangerous area, the individual on the corner position bears the maximum pressure in such scenarios is 90.2 N, and there is an approximate exponential relationship between crowding force and density indicated by our data. The results and findings presented in this paper can facilitate more reasonable and precise simulation models by utilizing crowding force and crowd density and ensure the safety of pedestrians in high-density scenarios.

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Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles

Cited by 47 publications

References 36 publications

Dynamics and control for multi-agent networked systems: A finite-difference approach

Dynamics and control for multi-agent networked systems: A finite-difference approach

On Entropy Dynamics for Active “Living” Particles

Cluster Risk of Walking Scenarios Based on Macroscopic Flow Model and Crowding Force Analysis

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