arXiv subjects

Benedetto Piccoli

Publications and source records attributed to Benedetto Piccoli.

85 records · Page 5Linked to original sources

arXiv2012

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 $L^1$ spaces are not natural for such equations, since we lose uniqueness of the solution.

math.AP↗

arXiv2012

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.

math-ph↗

arXiv2011

Left invertibility of discrete-time output-quantized systems: the linear case with finite inputs

This paper studies left invertibility of discrete-time linear output-quantized systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility, i.e. injectivity of I/O map, is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is much easier to detect. The condition under which left invertibility and left D-invertibility are equivalent is that the elements of the dynamic matrix of the system form an algebraically independent set. Our main result is a method to compute left D-invertibility for all linear systems with no eigenvalue of modulus one. Therefore we are able to check left invertibility of output-quantized linear systems for a full measure set of matrices. Some examples are presented to show the application of the proposed method.

math.OC↗

arXiv2010

Sensor Deployment for Network-like Environments

This paper considers the problem of optimally deploying omnidirectional sensors, with potentially limited sensing radius, in a network-like environment. This model provides a compact and effective description of complex environments as well as a proper representation of road or river networks. We present a two-step procedure based on a discrete-time gradient ascent algorithm to find a local optimum for this problem. The first step performs a coarse optimization where sensors are allowed to move in the plane, to vary their sensing radius and to make use of a reduced model of the environment called collapsed network. It is made up of a finite discrete set of points, barycenters, produced by collapsing network edges. Sensors can be also clustered to reduce the complexity of this phase. The sensors' positions found in the first step are then projected on the network and used in the second finer optimization, where sensors are constrained to move only on the network. The second step can be performed on-line, in a distributed fashion, by sensors moving in the real environment, and can make use of the full network as well as of the collapsed one. The adoption of a less constrained initial optimization has the merit of reducing the negative impact of the presence of a large number of local optima. The effectiveness of the presented procedure is illustrated by a simulated deployment problem in an airport environment.

math.OC↗

arXiv2010

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.

math-ph↗

arXiv2010

Time-evolving measures and macroscopic modeling of pedestrian flow

This paper deals with the early results of a new model of pedestrian flow, conceived within a measure-theoretical framework. The modeling approach consists in a discrete-time Eulerian macroscopic representation of the system via a family of measures which, pushed forward by some motion mappings, provide an estimate of the space occupancy by pedestrians at successive time steps. From the modeling point of view, this setting is particularly suitable to treat nonlocal interactions among pedestrians, obstacles, and wall boundary conditions. In addition, analysis and numerical approximation of the resulting mathematical structures, which is the main target of this work, follow more easily and straightforwardly than in case of standard hyperbolic conservation laws, also used in the specialized literature by some Authors to address analogous problems.

math-ph↗

arXiv2010

Effects of anisotropic interactions on the structure of animal groups

This paper proposes an agent-based model which reproduces different structures of animal groups. The shape and structure of the group is the effect of simple interaction rules among individuals: each animal deploys itself depending on the position of a limited number of close group mates. The proposed model is shown to produce clustered formations, as well as lines and V-like formations. The key factors which trigger the onset of different patterns are argued to be the relative strength of attraction and repulsion forces and, most important, the anisotropy in their application.

math.DS↗

arXiv2009

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.

math-ph↗

arXiv2009

Detection of Gaussian signals via hexagonal sensor networks

This paper considers a special case of the problem of identifying a static scalar signal, depending on the location, using a planar network of sensors in a distributed fashion. Motivated by the application to monitoring wild-fires spreading and pollutants dispersion, we assume the signal to be Gaussian in space. Using a network of sensors positioned to form a regular hexagonal tessellation, we prove that each node can estimate the parameters of the Gaussian from local measurements. Moreover, we study the sensitivity of these estimates to additive errors affecting the measurements. Finally, we show how a consensus algorithm can be designed to fuse the local estimates into a shared global estimate, effectively compensating the measurement errors.

math.OC↗

arXiv2009

Entropy type conditions for Riemann solvers at nodes

This paper deals with conservation laws on networks, represented by graphs. Entropy-type conditions are considered to determine dynamics at nodes. Since entropy dispersion is a local concept, we consider a network composed by a single node $J$ with $n$ incoming and $m$ outgoing arcs. We extend at $J$ the classical Kružkov entropy obtaining two conditions, denoted by (E1) and (E2): the first requiring entropy condition for all Kružkov entropies, the second only for the value corresponding to sonic point. First we show that in case $n \ne m$, no Riemann solver can satisfy the strongest condition. Then we characterize all the Riemann solvers at $J$ satisfying the strongest condition (E1), in the case of nodes with at most two incoming and two outgoing arcs. Finally we focus three different Riemann solvers, introduced in previous papers. In particular, we show that the Riemann solver introduced for data networks is the only one always satisfying (E2).

math.AP↗

arXiv2008

Pedestrian flows in bounded domains with obstacles

In this paper we systematically apply the mathematical structures by time-evolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discrete-time Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Radon positive measures generated by a push-forward recursive relation. We assume that two fundamental aspects of pedestrian behavior rule the dynamics of the system: On the one hand, the will to reach specific targets, which determines the main direction of motion of the walkers; on the other hand, the tendency to avoid crowding, which introduces interactions among the individuals. The resulting model is able to reproduce several experimental evidences of pedestrian flows pointed out in the specialized literature, being at the same time much easier to handle, from both the analytical and the numerical point of view, than other models relying on nonlinear hyperbolic conservation laws. This makes it suitable to address two-dimensional applications of practical interest, chiefly the motion of pedestrians in complex domains scattered with obstacles.

math-ph↗

arXiv1994

Time-Optimal Control Problem for the Swing and the Ski

This paper is concerned with a class of time-optimal control problems for the swing and the ski. We first consider the motion of a man standing on a swing. For simplicity, we neglect friction and air resistance and assume that the mass of the swinger is concentrated in his baricenter $B$. For the ski, we use the model considered by Aldo Bressan (1991), with a special approximation of the skier. Our main concern is the existence and the structure of time-optimal controls. As first step, we introduce an auxiliary control system with control entering linearly and show the relationships between this and the original system. We then perform a detailed study of the auxiliary system, using geometric techniques. In turn this provides accurate information on the time-optimal controls for the original system.

funct-an↗

arXiv1992

A Baire Category Approach to the Bang-Bang Property

Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bang-bang property is proved for a new class of ``concave" multifunctions, characterized by the existence of suitable linear selections. The proofs rely on Lyapunov's theorem in connection with a Baire category argument.

funct-an↗