A Boltzmann approach to mean-field sparse feedback control

sparse_boltzmannGiacomo Albi, Massimo Fornasier, and Dante Kalise

(12/11/2016 arxiv.org/abs/1611.03988)

We study the synthesis of optimal control policies for large-scale multi-agent systems. The optimal control design induces a parsimonious control intervention by means of l-1, sparsity-promoting control penalizations. We study instantaneous and infinite horizon sparse optimal feedback controllers. In order to circumvent the dimensionality issues associated to the control of large-scale agent-based models, we follow a Boltzmann approach. We generate (sub)optimal controls signals for the kinetic limit of the multi-agent dynamics, by sampling of the optimal solution of the associated two-agent dynamics. Numerical experiments assess the performance of the proposed sparse design.

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Mean field control hierarchy

imag_webGiacomo Albi, Young-Pil Choi, Massimo Fornasier, and Dante Kalise

(01/08/2016  arxiv.org/abs/1608.01728)

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrainted by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.

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Recent advances in opinion modeling: control and social influence

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.

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Discrete and Continuum Modeling of Biological Network Formation

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 formation of biological transport networks. The model describes the pressure field using a Darcy’s 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.

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Opinion dynamics over complex networks: kinetic modeling and numerical methods

site_artGiacomo Albi, Lorenzo Pareschi, Mattia Zanella

(31/03/2016 arxiv.org/abs/1604.00421)

In this paper we consider the modeling of opinion dynamics over time
dependent large scale networks. A kinetic description of the agents’
distribution over the evolving network is considered which combines an opinion
update based on binary interactions between agents with a dynamic creation and
removal process of new connections. The number of connections of each agent
influences the spreading of opinions in the network but also the way
connections are created is influenced by the agents’ opinion. The evolution of
the network of connections is studied by showing that its asymptotic behavior
is consistent both with Poisson distributions and truncated power-laws. In
order to study the large time behavior of the opinion dynamics a mean field
description is derived which allows to compute exact stationary solutions in
some simplified situations. Numerical methods which are capable to describe
correctly the large time behavior of the system are also introduced and
discussed. Finally, several numerical examples showing the influence of the
agents’ number of connections in the opinion dynamics are reported.

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Selective model-predictive control for flocking systems

All1

Giacomo Albi, Lorenzo Pareschi

(16/03/2016 arxiv.org/abs/1603.05012)

In this paper the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of the control. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we introduce a binary interaction algorithm [http://arxiv.org/abs/1203.0721] which is able to simulate efficiently the selective constrained dynamics. Consistency of the algorithm is also shown in the context of standard kinetic theory. Finally, some numerical simulations are reported to show the efficiency of the proposed techniques.

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On the optimal control of opinion dynamics on evolving networks

network2

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella

In this work we are interested in the modelling and control of opinion dynamics spreading on a time evolving network with scale-free asymptotic degree distribution. The mathematical model is formulated as a coupling of an opinion alignment system with a probabilistic description of the network. The optimal control problem aims at forcing consensus over the network, to this goal a control strategy based on the degree of connection of each agent has been designed. A numerical method based on a model predictive strategy is then developed and different numerical tests are reported. The results show that in this way it is possible to drive the overall opinion toward a desired state even if we control only a suitable fraction of the nodes.

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Biological transportation networks: modeling and simulation

menoUnorm_10n=110701
(26/08/2015, link)
We present a model for biological network formation originally introduced by Cai and Hu [Adaptation and optimization of biological transport networks, Phys. Rev. Lett. 111 (2013) 138701]. The modeling of fluid transportation (e.g., leaf venation and angiogenesis) and ion transportation networks (e.g., neural networks) is explained in detail and basic analytical features like the gradient flow structure of the fluid transportation network model and the impact of the model parameters on the geometry and topology of network formation are analyzed. We also present a numerical finite-element based discretization scheme and discuss sample cases of network formation simulations.
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Invisible control of self-organizing agents leaving unknown environments

fig_sito

Giacomo Albi, Mattia Bongini, Emiliano CristianiDante Kalise

(15/04/2015  arXiv.org/1504.04064)

In this paper we are concerned with multiscale modeling, control, and simulation of self-organizing agents leaving an unknown area under limited visibility, with special emphasis on crowds. We first introduce a new microscopic model characterized by an exploration phase and an evacuation phase. The main ingredients of the model are an alignment term, accounting for the herding effect typical of uncertain behavior, and a random walk, accounting for the need to explore the environment under limited visibility. We consider both metrical and topological interactions. Moreover, a few special agents, the leaders, not recognized as such by the crowd, are “hidden” in the crowd with a special controlled dynamics. Next, relying on a Boltzmann approach, we derive a mesoscopic model for a continuum density of followers, coupled with a microscopic description for the leaders’ dynamics. Finally, optimal control of the crowd is studied. It is assumed that leaders aim at steering the crowd towards the exits so to ease the evacuation and limit clogging effects, and locally-optimal behavior of leaders is computed. Numerical simulations show the efficiency of the control techniques in both microscopic and mesoscopic settings. We also perform a real experiment with people to study the feasibility of such a bottom-up control technique.

Page with microscopic and mesoscopic simulations.

Article on ‘WIRED.it’.

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Uncertainty Quantification in Control Problems for Flocking Models

web_imageGiacomo Albi,  Lorenzo Pareschi, Mattia Zanella.
(03/03/2015  arXiv:1503.00548)

In this paper the optimal control of flocking models with random inputs is investigated from a numerical point of view. The effect of uncertainty in the interaction parameters is studied for a Cucker-Smale type model using a generalized polynomial chaos (gPC) approach. Numerical evidence of threshold effects in the alignment dynamic due to the random parameters is given. The use of a selective model predictive control permits to steer the system towards the desired state even in unstable regimes.

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