arXiv subjects

Giuseppe Visconti

Publications and source records attributed to Giuseppe Visconti.

At least 19 recordsLinked to original sources

arXiv2026

Ensemble Kalman Inversion as an Inertial Interacting Particle System

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. A central limitation of its continuous-time dynamics is premature covariance collapse, which can restrict the directions accessible to the inversion and increase sensitivity to the initial ensemble. We introduce a second-order interacting particle system that combines the Kalman force with inertia, damping, attraction towards the ensemble mean, and short-range repulsion. We establish a generalized subspace property showing that nonzero initial velocities may enlarge the affine space accessible to the dynamics, while a finite-ensemble restriction remains. For linear inverse problems, we derive the mean and fluctuation dynamics, identify a regime in which fully collapsed configurations are linearly unstable, characterize asymptotic optimality on the subspace retained by the limiting covariance, and prove exponential decay for the associated frozen-covariance mean dynamics. Numerical experiments on high-dimensional linear and nonlinear inverse problems investigate subspace enlargement, discrepancy-based regularization, and ensemble collapse.

math.NA↗

arXiv2026

Micro-to-macro derivation and stability analysis of an Aw-Rascle-Zhang-type model with driver-dependent acceleration

We introduce a new traffic flow model. The main idea is to describe acceleration effects in the Aw-Rascle-Zhang model through an additional driver-dependent equation, thereby allowing individual driver characteristics to evolve dynamically rather than being prescribed solely by the traffic density. The resulting system provides a hyperbolic description of traffic dynamics and admits an explicit analysis of its characteristic structure, Riemann problem, and entropy properties. To investigate the stability of the proposed model, we perform a Chapman-Enskog expansion up to second order and derive the corresponding diffusive and higher-order corrections. This analysis reveals how the stability properties depend on the underlying velocity and pressure functions. Several examples are presented to illustrate the theoretical results.

math.NA↗

arXiv2026

Second-order multilane traffic flow models: from the microscopic to the macroscopic scale

This study addresses multilane vehicular traffic modelling, focusing on the rigorous transition from microscopic (individual vehicle-based) to macroscopic (aggregate flow-based) descriptions. While previous research on multilane traffic has largely focused on first-order models, we derive two novel multilane second-order macroscopic models by applying a microscopic-to-macroscopic limit to the multilane Bando-Follow-the-Leader model. These two models diverge fundamentally in their closure relations during the scaling limit: Model 1 directly projects the velocity dynamics, whereas Model 2 is built upon the heuristic conservation of the generalized momentum across lanes. Both models incorporate lane-changing dynamics through source terms in a hyperbolic system of balance laws, yet their structural differences lead to distinct relaxation limits, with Model 2 naturally relaxing to the first-order macroscopic multilane model recently derived by the authors, and Model 1 yielding a novel, non-standard first-order system. We propose several numerical experiments showing that the models can reproduce complex traffic phenomena, including congestion propagation, non-equilibrium effects, capacity drops, and asymmetric lane usage. Leveraging experimental datasets from real-world highways, we further construct lane-specific empirical fundamental diagrams and compare them with their simulated counterparts. The results demonstrate that the structural assumptions of Model 2 provide a superior quantitative fit with empirical data, faithfully capturing critical density values, traffic scattering, and characteristic lane-dependent patterns, thus offering a robust and generalizable tool for realistic traffic flow analysis.

physics.soc-ph↗

arXiv2026

Agent-Based Optimal Control for Image Processing

We investigate the use of multi-agent systems to solve classical image processing tasks, such as colour quantization and segmentation. We frame the task as an optimal control problem, where the objective is to steer the multi-agent dynamics to obtain colour clusters that segment the image. To do so, we balance the total variation of the colour field and fidelity to the original image. The solution is obtained resorting to primal-dual splitting and the method of multipliers. Numerical experiments, implemented in parallel with CUDA, demonstrate the efficacy of the approach and its potential for high-dimensional data.

math.OC↗

arXiv2025

Relaxation and stability analysis of a third-order multiclass traffic flow model

Traffic flow modeling spans a wide range of mathematical approaches, from microscopic descriptions of individual vehicle dynamics to macroscopic models based on aggregate quantities. A fundamental challenge in macroscopic modeling lies in the closure relations, particularly in the specification of a traffic hesitation function in second-order models like Aw-Rascle-Zhang. In this work, we propose a third-order hyperbolic traffic model in which the hesitation evolves as a driver-dependent dynamic quantity. Starting from a microscopic formulation, we relax the standard assumption by introducing an evolution law for the hesitation. This extension allows to incorporate hysteresis effects, modeling the fact that drivers respond differently when accelerating or decelerating, even under identical local traffic conditions. Furthermore, various relaxation terms are introduced. These allow us to establish relations to the Aw-Rascle-Zhang model and other traffic flow models.

math.OC↗

arXiv2025

Dissipation-dispersion analysis of fully-discrete implicit discontinuous Galerkin methods and application to stiff hyperbolic problems

The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical dissipation-dispersion. As far as we know, no analysis of these artifacts has been considered for implicit integration of DG methods. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta impacts deeply on the quality of the solution. We analyze one-dimensional dissipation-dispersion to select the best combination of the space-time discretization for high Courant numbers. Then, we apply our findings to the integration of one-dimensional stiff hyperbolic systems. Implicit schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. Numerical experiments explore the performance of this technique on scalar equations and systems.

math.NA↗

arXiv2025

Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function

This work is devoted to the Galerkin projection of highly nonlinear random quantities. The dependency on a random input is described by Haar-type wavelet systems. The classical Haar sequence has been used by Pettersson, Iaccarino, Nordstroem (2014) for a hyperbolic stochastic Galerkin formulation of the one-dimensional Euler equations. This work generalizes their approach to several multi-dimensional systems with Lipschitz continuous and non-polynomial flux functions. Theoretical results are illustrated numerically by a genuinely multidimensional CWENO reconstruction.

math.NA↗

arXiv2024

Quinpi: Integrating stiff hyperbolic systems with implicit high order finite volume schemes

Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g.~because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm.~Appl.~Math.~\& Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler's system of gas-dynamics.

math.NA↗

arXiv2024

Controllability of continuous networks and a kernel-based learning approximation

Residual deep neural networks are formulated as interacting particle systems leading to a description through neural differential equations, and, in the case of large input data, through mean-field neural networks. The mean-field description allows also the recast of the training processes as a controllability problem for the solution to the mean-field dynamics. We show theoretical results on the controllability of the linear microscopic and mean-field dynamics through the Hilbert Uniqueness Method and propose a computational approach based on kernel learning methods to solve numerically, and efficiently, the training problem. Further aspects of the structural properties of the mean-field equation will be reviewed.

math.OC↗

arXiv2022

Numerical boundary control for semilinear hyperbolic systems

This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.

math.OC↗

arXiv2022

Recent Trends on Nonlinear Filtering for Inverse Problems

Among the class of nonlinear particle filtering methods, the Ensemble Kalman Filter (EnKF) has gained recent attention for its use in solving inverse problems. We review the original method and discuss recent developments in particular in view of the limit for infinitely particles and extensions towards stability analysis and multi--objective optimization. We illustrate the performance of the method by using test inverse problems from the literature.

math.NA↗

arXiv2022

A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion

The Ensemble Kalman Filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In this context, the EnKF is known as Ensemble Kalman Inversion (EKI). In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stability analysis reveals possible drawbacks in the phase space of moments provided by the continuous limits of the EKI, but observed also in the multi--dimensional setting. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version by using test inverse problems from the literature and comparing it with the classical continuous limit formulation of the method.

math.NA↗

arXiv2021

Mean-field limit of a hybrid system for multi-lane multi-class traffic

This article aims to study coupled mean-field equation and ODEs with discrete events motivated by vehicular traffic flow. Precisely, multi-lane traffic flow in presence of human-driven and autonomous vehicles is considered, with the autonomous vehicles possibly influenced by external policy makers. First a finite-dimensional hybrid system is developed based on the continuous Bando-Follow-the-Leader dynamics coupled with discrete events due to lane-change maneuvers. Then the mean-field limit of the finite-dimensional hybrid system is rigorously derived for the dynamics of the human-driven vehicles. The microscopic lane-change maneuvers of the human-driven vehicles generates a source term to the mean-field PDE. This leads to an infinite-dimensional hybrid system, which is described by coupled Vlasov-type PDE, ODEs and discrete events.

math.AP↗

arXiv2020

Continuous Limits for Constrained Ensemble Kalman Filter

The Ensemble Kalman Filter method can be used as an iterative particle numerical scheme for state dynamics estimation and control--to--observable identification problems. In applications it may be required to enforce the solution to satisfy equality constraints on the control space. In this work we deal with this problem from a constrained optimization point of view, deriving corresponding optimality conditions. Continuous limits, in time and in the number of particles, allows us to study properties of the method. We illustrate the performance of the method by using test inverse problems from the literature.

math.NA↗

arXiv2020

Mean field models for large data-clustering problems

We consider mean-field models for data--clustering problems starting from a generalization of the bounded confidence model for opinion dynamics. The microscopic model includes information on the position as well as on additional features of the particles in order to develop specific clustering effects. The corresponding mean--field limit is derived and properties of the model are investigated analytically. In particular, the mean--field formulation allows the use of a random subsets algorithm for efficient computations of the clusters. Applications to shape detection and image segmentation on standard test images are presented and discussed.

math.NA↗

arXiv2020

Efficient implementation of adaptive order reconstructions

Including polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. In this paper we address this issue with a new Central WENOZ (CWENOZ) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the CWENOZ parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order WENO reconstructions [Balsara, Garain and Shu, J.Comput.Phys., 2016], [Arbogast, Huang and Zhao, SIAM J.Numer.Anal., 2018]. The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell.

math.NA↗

arXiv2019

The BGK approximation of kinetic models for traffic

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.

physics.soc-ph↗

arXiv2019

Macroscopic modeling of multi-lane motorways using a two-dimensional second-order model of traffic flow

Lane changing is one of the most common maneuvers on motorways. Although, macroscopic traffic models are well known for their suitability to describe fast moving crowded traffic, most of these models are generally developed in one dimensional framework, henceforth lane changing behavior is somehow neglected. In this paper, we propose a macroscopic model, which accounts for lane-changing behavior on motorway, based on a two-dimensional extension of the Aw and Rascle [Aw and Rascle, SIAM J.Appl.Math., 2000] and Zhang [Zhang, Transport.Res.B-Meth., 2002] macroscopic model for traffic flow. Under conditions, when lane changing maneuvers are no longer possible, the model "relaxes" to the one-dimensional Aw-Rascle-Zhang model. Following the same approach as in [Aw, Klar, Materne and Rascle, SIAM J.Appl.Math., 2002], we derive the two-dimensional macroscopic model through scaling of time discretization of a microscopic follow-the-leader model with driving direction. We provide a detailed analysis of the space-time discretization of the proposed macroscopic as well as an approximation of the solution to the associated Riemann problem. Furthermore, we illustrate some features of the proposed model through some numerical experiments.

math.NA↗