arXiv subjects

Maya Briani

Publications and source records attributed to Maya Briani.

At least 19 recordsLinked to original sources

arXiv2026

Conservation Laws with Discontinuous Flux: A First Application to Traffic Flow

This work provides a traffic flow interpretation of the model recently introduced in \cite{ABS2025}. We adapt the conservation law with discontinuous gradient-dependent flux to a macroscopic traffic setting, assuming concave flux functions $f(u)$ and $g(u)$ defined on the density domain $[0,u_{\max}]$. To prevent violations of the maximal density constraint, we introduce modified Riemann Solvers tailored to this setting and generalize the existence theory for weak solutions developed in \cite{ABS2025}. Finally, we construct Riemann Solvers for a broader two-phase model incorporating a free-flow regime, in which $f(u)=g(u)$ on a subset of $[0,u_{\max}]$, and establish existence of weak solutions in this more general framework.

math.AP↗

arXiv2025

A relaxation scheme for the equations of isentropic gas dynamics on a network with jump transmission conditions

In this paper we propose a new numerical scheme of relaxation type to approximate the Euler equations of isentropic gas dynamics on the arcs of a network. At the junction mass conservation and a jump transmission condition on the density are given, and a new solver is introduced to deal with both subsonic and supersonic cases. Consistency properties of the solver are proven and numerical tests are displayed to show its good performance also with respect to other possible solvers.

math.NA↗

arXiv2024

Adapting Priority Riemann Solver for GSOM on road networks

In this paper, we present an extension of the Generic Second Order Models (GSOM) for traffic flow on road networks. We define a Riemann solver at the junction based on a priority rule and provide an iterative algorithm to construct solutions at junctions with n incoming and m outgoing roads. The logic underlying our solver is as follows: the flow is maximized while respecting the priority rule, which can be adjusted if the supply of an outgoing road exceeds the demand of a higher-priority incoming road. Approximate solutions for Cauchy problems are constructed using wave-front tracking. We establish bounds on the total variation of waves interacting with the junction and present explicit calculations for junctions with two incoming and two outgoing roads. A key novelty of this work is the detailed analysis of returning waves - waves generated at the junction that return to the junction after interacting along the roads - which, in contrast to first-order models such as LWR, can increase flux variation.

math.AP↗

arXiv2024

Numerical computation of generalized Wasserstein distances with applications to traffic model analysis

Generalized Wasserstein distances allow to quantitatively compare two continuous or atomic mass distributions with equal or different total mass. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the last years, giving some insights about their physical meaning. After that, we explore their usage in the context of the sensitivity analysis of differential models for traffic flow. The quantification of models sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

math.AP↗

arXiv2024

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↗

arXiv2023

Inverting the Fundamental Diagram and Forecasting Boundary Conditions: How Machine Learning Can Improve Macroscopic Models for Traffic Flow

In this paper, we aim at developing new methods to join machine learning techniques and macroscopic differential models for vehicular traffic estimation and forecast. It is well known that data-driven and model-driven approaches have (sometimes complementary) advantages and drawbacks. We consider here a dataset with flux and velocity data of vehicles moving on a highway, collected by fixed sensors and classified by lane and by class of vehicle. By means of a machine learning model based on an LSTM recursive neural network, we extrapolate two important pieces of information: 1) if congestion is appearing under the sensor, and 2) the total amount of vehicles which is going to pass under the sensor in the next future (30 min). These pieces of information are then used to improve the accuracy of an LWR-based first-order multi-class model describing the dynamics of traffic flow between sensors. The first piece of information is used to invert the (concave) fundamental diagram, thus recovering the density of vehicles from the flux data, and then inject directly the density datum in the model. This allows one to better approximate the dynamics between sensors, especially if an accident happens in a not monitored stretch of the road. The second piece of information is used instead as boundary conditions for the equations underlying the traffic model, to better reconstruct the total amount of vehicles on the road at any future time. Some examples motivated by real scenarios will be discussed. Real data are provided by the Italian motorway company Autovie Venete S.p.A.

cs.LG↗

arXiv2022

Estimate of traffic emissions through multiscale second order models with heterogeneous data

In this paper we propose a multiscale traffic model, based on the family of Generic Second Order Models, which integrates multiple trajectory data into the velocity function. This combination of a second order macroscopic model with microscopic information allows us to reproduce significant variations in speed and acceleration that strongly influence traffic emissions. We obtain accurate approximations even with a few trajectory data. The proposed approach is therefore a computationally efficient and highly accurate tool for calculating macroscopic traffic quantities and estimating emissions.

math.NA↗

arXiv2022

Emissions minimization on road networks via Generic Second Order Models

In this paper we consider the problem of estimating emissions due to vehicular traffic on complex networks, and minimizing their effect by regulating traffic at junctions. For the traffic evolution, we consider a Generic Second Order Model, which encompasses the majority of two-equations (i.e. second-order) models available in the literature, and extend it to road networks with merge and diverge junctions. The dynamics on the whole network is determined by selecting a solution to the Riemann Problems at junctions, i.e. the Cauchy problems with constant initial data on each incident road. The latter are solved assuming the maximization of the flow and assigning a traffic distribution coefficient for outgoing roads of diverges, and a priority rule for incoming roads of merges. A general emission model is considered and its parameters are tuned to the NOx emission rate. The minimization of emissions is then formulated in terms of the traffic distribution and priority parameters, taking into account travel times. A comparison is provided between roundabouts with optimized parameters and traffic lights, which correspond to time-varying traffic priorities. Our approach can be adapted to manage traffic in complex networks in order to reduce emissions while keeping travel time at acceptable levels.

physics.soc-ph↗

arXiv2021

Macroscopic and multi-scale models for multi-class vehicular dynamics with uneven space occupancy: a case study

In this paper we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions the creeping phenomenon can appear, i.e. one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic Follow-the-Leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena like stop & go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste-Venice and its branches, provided by Autovie Venete S.p.A.

math.DS↗

arXiv2021

Angle dependence in coupling conditions for shallow water equations at canal junctions

In this paper we propose a numerical Riemann problem solver at the junction of one dimensional shallow-water canal networks. The junction conditions take into account the angles with which the channels intersect and include the possibility of canals with different sections. The solver is illustrated with several numerical tests which underline the importance of the angle dependence to obtain reliable solutions.

math.NA↗

arXiv2020

Evaluation of $\mathrm{NO_x}$ emissions and ozone production due to vehicular traffic via second-order models

The societal impact of traffic is a long-standing and complex problem. We focus on the estimation of ozone production due to vehicular traffic. For this, we couple a system of conservation laws for vehicular traffic, an emission model, and a system of partial differential equations for the main reactions leading to ozone production and diffusion. The second-order model for traffic is obtained by choosing a special velocity function for a Collapsed Generalized Aw-Rascle-Zhang model and is tuned on NGSIM data. On the other side, the system of partial differential equations describes the main chemical reactions of $\mathrm{NO_x}$ gases with a source term provided by a general emission model applied to the output of the traffic model. We analyze the ozone impact of various traffic scenarios and describe the effect of traffic light timing. The numerical tests show the negative effect of vehicles restarts on $\mathrm{NO_x}$ emissions, suggesting to increase the length of the green phase of traffic lights to reduce them.

math.NA↗

arXiv2020

Convergence rate of Markov chains and hybrid numerical schemes to jump-diffusions with application to the Bates model

We study the rate of weak convergence of Markov chains to diffusion processes under suitable but quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the speed of convergence of this hybrid approach and we provide an example in finance, applying our results to a tree-finite difference approximation in the Heston or Bates model.

math.PR↗

arXiv2019

Numerical stability of a hybrid method for pricing options

We develop and study stability properties of a hybrid approximation of functionals of the Bates jump model with stochastic interest rate that uses a tree method in the direction of the volatility and the interest rate and a finite-difference approach in order to handle the underlying asset price process. We also propose hybrid simulations for the model, following a binomial tree in the direction of both the volatility and the interest rate, and a space-continuous approximation for the underlying asset price process coming from a Euler-Maruyama type scheme. We show that our methods allow to obtain efficient and accurate European and American option prices. Numerical experiments are provided, and show the reliability and the efficiency of the algorithms.

q-fin.CP↗

arXiv2019

Understanding Mass Transfer Directions via Data-Driven Models with Application to Mobile Phone Data

The aim of this paper is to solve an inverse problem which regards a mass moving in a bounded domain. We assume that the mass moves following an unknown velocity field and that the evolution of the mass density can be described by partial differential equations (PDEs), which is also unknown. The input data of the problems are given by some snapshots of the mass distribution at certain times, while the sought output is the velocity field that drives the mass along its displacement. To this aim, we put in place an algorithm based on the combination of two methods: first, we use the Dynamic Mode Decomposition to create a mathematical model describing the mass transfer; second, we use the notion of Wasserstein distance (also known as earth mover's distance) to reconstruct the underlying velocity field that is responsible for the displacement. Finally, we consider a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input data, density profiles (i.e. the spatial distribution) of people in given areas at different time instances. This kind of data are provided by the Italian telecommunication company TIM and are derived by mobile phone usage.

math.NA↗

arXiv2018

Open canals flow with fluvial to torrential phase transitions on networks

Network flows and specifically open canal flows can be modeled by systems of balance laws defined on topological graphs. The shallow water or Saint-Venant system of balance laws is one of the most used model and present two phases: fluvial or sub-critical and torrential or super critical. Phase transitions may occur within the same canal but transitions related to networks are less investigated. In this paper we provide a complete characterization of possible phase transitions for a simple network with two canals and one junction. Our analysis allows the study of more complicate scenarios. Moreover, we provide some numerical simulations to show the theory at work.

math.NA↗

arXiv2018

Understanding Human Mobility Flows from Aggregated Mobile Phone Data

In this paper we deal with the study of travel flows and patterns of people in large populated areas. Information about the movements of people is extracted from coarse-grained aggregated cellular network data without tracking mobile devices individually. Mobile phone data are provided by the Italian telecommunication company TIM and consist of density profiles (i.e. the spatial distribution) of people in a given area at various instants of time. By computing a suitable approximation of the Wasserstein distance between two consecutive density profiles, we are able to extract the main directions followed by people, i.e. to understand how the mass of people distribute in space and time. The main applications of the proposed technique are the monitoring of daily flows of commuters, the organization of large events, and, more in general, the traffic management and control.

physics.soc-ph↗

arXiv2017

A hybrid approach for the implementation of the Heston model

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm.

q-fin.CP↗

arXiv2016

Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance

In this paper we investigate the sensitivity of the LWR model on network to its parameters and to the network itself. The quantification of sensitivity is obtained by measuring the Wasserstein distance between two LWR solutions corresponding to different inputs. To this end, we propose a numerical method to approximate the Wasserstein distance between two density distributions defined on a network. We found a large sensitivity to the traffic distribution at junctions, the network size, and the network topology.

math.NA↗