arXiv subjects

Giovanni Russo

Publications and source records attributed to Giovanni Russo.

At least 91 records · Page 5Linked to original sources

arXiv2021

A novel spectral method for the semi-classical Schrödinger equation based on the Gaussian wave-packet transform

In this article, we develop and analyse a new spectral method to solve the semi-classical Schrödinger equation based on the Gaussian wave-packet transform (GWPT) and Hagedorn's semi-classical wave-packets (HWP). The GWPT equivalently recasts the highly oscillatory wave equation as a much less oscillatory one (the $w$ equation) coupled with a set of ordinary differential equations governing the dynamics of the so-called GWPT parameters. The Hamiltonian of the $ w $ equation consists of a quadratic part and a small non-quadratic perturbation, which is of order $ \mathcal{O}(\sqrt{\varepsilon }) $, where $ \varepsilon\ll 1 $ is the rescaled Planck's constant. By expanding the solution of the $ w $ equation as a superposition of Hagedorn's wave-packets, we construct a spectral method while the $ \mathcal{O}(\sqrt{\varepsilon}) $ perturbation part is treated by the Galerkin approximation. This numerical implementation of the GWPT avoids imposing artificial boundary conditions and facilitates rigorous numerical analysis. For arbitrary dimensional cases, we establish how the error of solving the semi-classical Schrödinger equation with the GWPT is determined by the errors of solving the $ w $ equation and the GWPT parameters. We prove that this scheme has the spectral convergence with respect to the number of Hagedorn's wave-packets in one dimension. Extensive numerical tests are provided to demonstrate the properties of the proposed method.

math.NA↗

arXiv2021

Individual- and pair-based models of epidemic spreading: master equations and analysis of their forecasting capabilities

Mathematical modeling of disease spreading plays a crucial role in understanding, controlling and preventing epidemic outbreaks. In a microscopic description of the propagation of a disease over the complex network of human contacts, the probability that an individual is in a given state (susceptible, infectious, recovered etc) depends on the state of its neighbors in the network. Thus it depends on the state of pairs of nodes, which in turns depends on triples, in a hierarchy of dynamical dependencies. In order to produce models that are at the same time reliable and manageable, one has to understand how to truncate such a hierarchy, and how the chosen order of approximation affects the ability of the model to forecast the real temporal evolution of an epidemics. In this paper we provide a systematic analysis of the reliability (under different hypotheses on the quantity and quality of available data) of the predictions obtained by truncating the hierarchy either at the level of individuals or at the level of pairs. We find that pair-based models are reliable tools both for estimating the epidemiological parameters and for forecasting the temporal evolution of the epidemics, under all conditions taken into account in our work. However, a pair-based approach provides a much better prediction of an individual-based one, only if better data, namely information on the state of node pairs, are available. Overall, our results suggest that more refined mathematical models need to be informed by improved contact tracing techniques to better support decision on policies and containment measures to adopt.

physics.soc-ph↗

arXiv2021

Intermittent non-pharmaceutical strategies to mitigate the COVID-19 epidemic in a network model of Italy via constrained optimization

This paper is concerned with the design of intermittent non-pharmaceutical strategies to mitigate the spread of the COVID-19 epidemic exploiting network epidemiological models. Specifically, by studying a variational equation for the dynamics of the infected in a network model of the epidemic spread, we derive, using contractivity arguments, a condition that can be used to guarantee that, in epidemiological terms, the effective reproduction number is less than unity. This condition has three advantages: (i) it is easily computable; (ii) it is directly related to the model parameters; (iii) it can be used to enforce a scalability condition that prohibits the amplification of disturbances within the network system. We then include satisfaction of such a condition as a constraint in a Model Predictive Control problem so as to mitigate (or suppress) the spread of the epidemic while minimizing the economic impact of the interventions. A data-driven model of Italy as a network of three macro-regions (North, Center, and South), whose parameters are identified from real data, is used to illustrate and evaluate the effectiveness of the proposed control strategy.

q-bio.PE↗

arXiv2021

Pitchfork-bifurication-based competitive and collaborative control of an E-bike system

This paper is concerned with the design of a human-in-the-loop system for deployment on a smart pedelec (e-bike). From the control-theoretic perspective, the goal is not only to use the power assistance of the e-bike to reject disturbances along the route but also to manage the possibly competitive interactions between a human and the motor intervention. Managing the competitive/cooperative nature of the interactions is crucial for applications in which we wish to control physical aspects of the cycling behavior (e.g. heart rate and breathing rate). The basis of the control is a pitchfork bifurcation system, modeling the interactions, augmented using ideas from gain-scheduling. In vivo experiments have been conducted, showing the effectiveness of the proposed control strategy.

math.OC↗

arXiv2021

A meshfree arbitrary Lagrangian-Eulerian method for the BGK model of the Boltzmann equation with moving boundaries

In this paper we present a novel technique for the simulation of moving boundaries and moving rigid bodies immersed in a rarefied gas using an Eulerian-Lagrangian formulation based on least square method. The rarefied gas is simulated by solving the Bhatnagar-Gross-Krook (BGK) model for the Boltzmann equation of rarefied gas dynamics. The BGK model is solved by an Arbitrary Lagrangian-Eulerian (ALE) method, where grid-points/particles are moved with the mean velocity of the gas. The computational domain for the rarefied gas changes with time due to the motion of the boundaries. To allow a simpler handling of the interface motion we have used a meshfree method based on a least-square approximation for the reconstruction procedures required for the scheme. We have considered a one way, as well as a two-way coupling of boundaries/rigid bodies and gas flow. The numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. Convergence studies are performed for one-dimensional and two-dimensional test-cases. Several further test problems and applications illustrate the versatility of the approach.

math.NA↗

arXiv2021

Spectral and norm estimates for matrix sequences arising from a finite difference approximation of elliptic operators

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.

math.NA↗

arXiv2021

A local velocity grid conservative semi-Lagrangian schemes for BGK model

Most numerical schemes proposed for solving BGK models for rarefied gas dynamics are based on the discrete velocity approximation. Since such approach uses fixed velocity grids, one must secure a sufficiently large domain with fine velocity grids to resolve the structure of distribution functions. When one treats high Mach number problems, the computational cost becomes prohibitively expensive. In this paper, we propose a velocity adaptation technique in the semi-Lagrangian framework for BGK model. The velocity grid will be set locally in time and space, according to mean velocity and temperature. We apply a weighted minimization approach to impose conservation. We presented several numerical tests that illustrate the effectiveness of our proposed scheme.

math.NA↗

arXiv2021

Matrix measures, stability and contraction theory for dynamical systems on time scales

This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first {formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales.} Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. {Then, we consider nonlinear time-varying dynamical systems on time scales and} establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning controllability of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.

math.DS↗

arXiv2021

On the design of autonomous agents from multiple data sources

This paper is concerned with the problem of designing agents able to dynamically select information from multiple data sources in order to tackle tasks that involve tracking a target behavior while optimizing a reward. We formulate this problem as a data-driven optimal control problem with integer decision variables and give an explicit expression for its solution. The solution determines how (and when) the data from the sources should be used by the agent. We also formalize a notion of agent's regret and, by relaxing the problem, give a regret upper bound. Simulations complement the results.

math.OC↗

arXiv2021

BGK models for inert mixtures: comparison and applications

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and numerically, for the latter using an asymptotic preserving semi-Lagrangian scheme for the BGK models. Application to the plane shock wave in a binary mixture of noble gases is also presented.

math-ph↗

arXiv2021

Lack of practical identifiability may hamper reliable predictions in COVID-19 epidemic models

Compartmental models are widely adopted to describe and predict the spreading of infectious diseases. The unknown parameters of such models need to be estimated from the data. Furthermore, when some of the model variables are not empirically accessible, as in the case of asymptomatic carriers of COVID-19, they have to be obtained as an outcome of the model. Here, we introduce a framework to quantify how the uncertainty in the data impacts the determination of the parameters and the evolution of the unmeasured variables of a given model. We illustrate how the method is able to characterize different regimes of identifiability, even in models with few compartments. Finally, we discuss how the lack of identifiability in a realistic model for COVID-19 may prevent reliable forecasting of the epidemic dynamics.

physics.soc-ph↗

arXiv2020

Tutoring Reinforcement Learning via Feedback Control

We introduce a control-tutored reinforcement learning (CTRL) algorithm. The idea is to enhance tabular learning algorithms by means of a control strategy with limited knowledge of the system model. By tutoring the learning process, the learning rate can be substantially reduced. We use the classical problem of stabilizing an inverted pendulum as a benchmark to numerically illustrate the advantages and disadvantages of the approach.

math.OC↗

arXiv2020

Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures

In this work, we propose a class of high order semi-Lagrangian scheme for a general consistent BGK model for inert gas mixtures. The proposed scheme not only fulfills indifferentiability principle, but also asymptotic preserving property, which allows us to capture the behaviors of hydrodynamic limit models. We consider two hydrodynamic closure which can be derived from the BGK model at leading order: classical Euler equations for number densities, global velocity and temperature, and a multi-velocities and temperatures Euler system. Numerical simulations are performed to demonstrate indifferentiability principle and asymptotic preserving property of the proposed conservative semi-Lagrangian scheme to the Euler limits.

math.NA↗

arXiv2020

On the crowdsourcing of behaviors for autonomous agents

This paper is concerned with the problem of designing, from data, agents that are able to craft their behavior from a number of contributors in order to fulfill some agent-specific task. This is not necessarily known to the contributors. After formalizing this crowdsourcing process as a control problem, we present a result to synthesize behaviors from the information made available by the contributors. The result is turned into an algorithmic procedure and the theoretical findings are complemented via a numerical example.

math.OC↗

arXiv2020

The Einstein condition on nearly Kähler six-manifolds

We review basic facts on the structure of nearly Kähler manifolds, focussing in particular on the six-dimensional case. A self-contained proof that nearly Kähler six-manifolds are Einstein is given by combining different known results. We finally rephrase the definition of nearly Kähler six-manifold in terms of a pair of partial differential equations.

math.DG↗

arXiv2020

Multi-moment maps on nearly Kähler six-manifolds

We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly Kähler examples.

math.DG↗

arXiv2020

Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction

In this paper, we propose and analyse a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper.

math.NA↗

arXiv2020

Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [S. Y. Cho, et al., Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, 2020]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to rigid body rotation, Vlasov Poisson system and the BGK model of rarefied gas dynamics. In the first two cases operator splitting is adopted, to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model.

math.NA↗