On noise-induced synchronization and consensus of nonlinear network systems under input disturbances

This paper is concerned with the study of synchronization and consensus phenomena in complex networks of diffusively-coupled nodes subject to external disturbances. Specifically, we make use of stochastic Lyapunov functions to provide conditions for synchronization and consensus for networks of nonlinear, diffusively coupled nodes, where noise diffusion is not just additive but it depends on the nodes' state. The sufficient condition we provide, wich links together network topology, coupling strength and noise diffusion, offers two interesting interpretations. First, as suggested by {\em intuition}, in order for a network to achieve synchronization/consensus, its nodes need to be sufficiently well connected together. The second implication might seem, instead, counter-intuitive: if noise diffusion is {\em properly} designed, then it can drive an unsynchronized network towards synchronization/consensus. Motivated by our current research in Smart Cities and Internet of Things, we illustrate the effectiveness of our approach by showing how our results can be used to control certain collective decision processes.Comment: Preprint submitted to SIAM SICON. arXiv admin note: text overlap with arXiv:1602.0646

Nearly K\"ahler six-manifolds with two-torus symmetry

We consider nearly K\"ahler 6-manifolds with effective 2-torus symmetry. The multi-moment map for the $T^2$-action becomes an eigenfunction of the Laplace operator. At regular values, we prove the $T^2$-action is necessarily free on the level sets and determines the geometry of three-dimensional quotients. An inverse construction is given locally producing nearly K\"ahler six-manifolds from three-dimensional data. This is illustrated for structures on the Heisenberg group.Comment: 13 page

A Finite Difference Ghost-Cell Multigrid Approach for Poisson Equation with Mixed Boundary Conditions in Arbitrary Domain

In this paper we present a multigrid approach to solve the Poisson equation in arbitrary domain (identified by a level set function) and mixed boundary conditions. The discretization is based on finite difference scheme and ghost-cell method. This multigrid strategy can be applied also to more general problems where a non-eliminated boundary condition approach is used. Arbitrary domain make the definition of the restriction operator for boundary conditions hard to find. A suitable restriction operator is provided in this work, together with a proper treatment of the boundary smoothing, in order to avoid degradation of the convergence factor of the multigrid due to boundary effects. Several numerical tests confirm the good convergence property of the new method

On noise-induced synchronization and consensus

In this paper, we present new results for the synchronization and consensus of networks described by Ito stochastic differential equations. From the methodological viewpoint, our results are based on the use of stochastic Lyapunov functions. This approach allowed us to consider networks where nodes dynamics can be nonlinear and non-autonomous and where noise is not just additive but rather its diffusion can be nonlinear and depend on the network state. We first present a sufficient condition on the coupling strength and topology ensuring that a network synchronizes (fulfills consensus) despite noise. Then, we show that noise can be useful, and present a result showing how to design noise so that it induces synchronization/consensus. Motivated by our current research in Smart Cities and Internet of Things, we also illustrate the effectiveness of our approach by showing how our results can be used to analyze/control the onset of synchronization in noisy networks and to study collective decision processes.Comment: Keywords: Ito differential equations, Synchronization, Complex network

Explicit rationality of some cubic fourfolds

Recent results of Hassett, Kuznetsov and others pointed out countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor $C_{14}$ and in [arXiv:1707.00999] for the divisors $C_{26}$ and $C_{38}$. In this note we describe explicit birational maps from a general cubic fourfold in $C_{14}$, in $C_{26}$ and in $C_{38}$ to $\mathbb{P}^4$, providing concrete geometric realizations of the more abstract constructions in [arXiv:1707.00999].Comment: Shortened versio

Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$. We use congruences of 5-secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14, 26, 38$ in Hassett's notation.Comment: We added more details, improving the presentation, and modified some discursive parts. Theorem 1 has been restated in a weaker form with a hypothesis always satisfied in our application

A New Class of Conservative Large Time Step Methods for the BGK Models of the Boltzmann Equation

This work is aimed to develop a new class of methods for the BGK model of the Boltzmann equation. This technique allows to get high order of accuracy both in space and time, theoretically without CFL stability limitation. It's based on a Lagrangian formulation of the problem: information is stored on a fixed grid in space and velocity, and the equation is integrated along the characteristics. The source term is treated implicitly by using a DIRK (Diagonally Implicit Runge Kutta) scheme in order to avoid the time step restriction due to the stiff relaxation. In particular some L-stable schemes are tested by smooth and Riemann problems, both in rarefied and fully fluid regimes. Numerical results show good accuracy and efficiency of the method

Implicit-Explicit Runge-Kutta schemes for hyperbolic systems with stiff relaxation and applications

In this paper we give an overview of Implicit-Explicit Runge-Kutta schemes applied to hyperbolic systems with stiff relaxation. In particular, we focus on some recent results on the uniform accuracy for hyperbolic systems with stiff relaxation [6], and hyperbolic system with diffusive relaxation [7, 5, 4]. In the latter case, we present an original application to a model problem arising in Extended Thermodynamics.Comment: 19 page

Sensitivity and safety of fully probabilistic control

In this paper we present a sensitivity analysis for the so-called fully probabilistic control scheme. This scheme attempts to control a system modeled via a probability density function (pdf) and does so by computing a probabilistic control policy that is optimal in the Kullback-Leibler sense. Situations where a system of interest is modeled via a pdf naturally arise in the context of neural networks, reinforcement learning and data-driven iterative control. After presenting the sensitivity analysis, we focus on characterizing the convergence region of the closed loop system and introduce a safety analysis for the scheme. The results are illustrated via simulations. This is the preliminary version of the paper entitled "On robust stability of fully probabilistic control with respect to data-driven model uncertainties" that will be presented at the 2019 European Control Conference.Comment: accepted for presentation at the 2019 European Control Conference (ECC 2019

A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov-Poisson System

In this paper, we consider a finite difference grid-based semi-Lagrangian approach in solving the Vlasov-Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear advection problems, see {\em Cheng and Knorr, Journal of Computational Physics, 22(1976)}. However, such splitting is subject to the splitting error. If we consider multi-dimensional problems without splitting, difficulty arises in tracing characteristics with high order accuracy. Specifically, the evolution of characteristics is subject to the electric field which is determined globally from the distribution of particle densities via the Poisson's equation. In this paper, we propose a novel strategy of tracing characteristics high order in time via a two-stage multi-derivative prediction-correction approach and by using moment equations of the VP system. With the foot of characteristics being accurately located, we proposed to use weighted essentially non-oscillatory (WENO) interpolation to recover function values between grid points, therefore to update solutions at the next time level. The proposed algorithm does not have time step restriction as Eulerian approach and enjoys high order spatial and temporal accuracy. However, such finite difference algorithm does not enjoy mass conservation; we discuss one possible way of resolving such issue and its potential challenge in numerical stability. The performance of the proposed schemes are numerically demonstrated via classical test problems such as Landau damping and two stream instabilities