Intersection modeling using a convergent scheme based on Hamilton-Jacobi equa...Guillaume Costeseque
The document presents an intersection modeling approach using a Hamilton-Jacobi equation. It proposes modeling traffic flow at intersections as the maximization of total flow, without internal states. A numerical scheme is presented to solve the HJ equation based on conservation of vehicles and FIFO principles. Numerical simulations are shown for a diverging intersection with one incoming and two outgoing roads, demonstrating the propagation of density waves over time.
Road junction modeling using a scheme based on Hamilton-Jacobi equationsGuillaume Costeseque
The document summarizes a numerical scheme for modeling traffic flow at road junctions using Hamilton-Jacobi equations. The scheme models traffic on each branch of the junction as well as at the junction point where branches meet. It introduces Hamiltonians representing traffic flow and establishes gradient estimates and existence/uniqueness results for the numerical solution. The scheme is shown to converge to the unique viscosity solution of the underlying partial differential equations as the grid is refined.
This document discusses numerical methods for variational principles in traffic modeling. It begins with an introduction to variational principles in physical systems and their application to traffic modeling. It then provides an overview of macroscopic traffic flow models, including first-order Lighthill-Whitham-Richards models and higher-order Generic Second Order Models. The document explains that traffic models can be formulated as variational problems and solved using principles like minimum action, Hamilton-Jacobi equations, and dynamic programming. Numerical methods are needed to solve the resulting variational problems in modeling real-world traffic flows.
Mixed Integer Linear Programming Formulation for the Taxi Sharing Problemjfrchicanog
The document presents a mixed integer linear programming (MILP) formulation for solving the taxi sharing problem. The taxi sharing problem aims to optimize taxi routes by allowing passengers with similar pick-up and drop-off locations to share taxis. The formulation models the problem as sequences of passenger locations that represent taxi rides. Experiments on real-world taxi trip data show the MILP formulation finds lower cost solutions than a parallel evolutionary algorithm, especially on medium and large problem instances, demonstrating the benefits of the exact MILP approach.
The document reviews optimal speed traffic flow models. It discusses macroscopic and microscopic models, with a focus on car-following models and the optimal velocity model (OVM). The OVM describes how each vehicle tries to travel at an optimal speed based on the distance to the preceding vehicle. Several improved models are presented, including the comprehensive optimal velocity model which considers both distance and speed difference, and the optimal velocity forecast model which incorporates anticipated speed changes. While the models reviewed consider single-lane traffic, the conclusion recommends including non-car vehicles and driver behaviors more common to Nigeria for greater applicability.
Invited presentation at SIAM PP 18: Communication Hiding Through Pipelining i...wvanroos
The performance of Krylov method suffers in the strong scaling limit due to the synchronization latencies mainly associated with the dot-products. We review the development of pipelined Krylov methods that exploit asynchronous communication to overlap communication and computation. For example, by reordering the operations in the conjugate gradients it is possible to execute the dot-product simultaneously with the sparse matrix vector products. This gives the dot-product more time to complete while doing useful work.
We also introduce deep pipelines, where the dot-products are overlapping with multiple sparse matrix-vector products. This gives the dot-product even more time to complete and these deep pipelines solve most of the scaling problems. However, due to this reorganization of the algorithm the propagation of rounding errors is altered resulting in different final accuracy, requiring a detailed numerical analysis.
We conclude with experiments that illustrate the improved scalability.
This document discusses the transportation problem and methods for solving it. It begins with an example of distributing goods from 3 canneries to 4 warehouses. It then defines the transportation problem as distributing goods from sources to destinations at minimum cost. It provides the formulation for the transportation problem as a linear program. It discusses properties of transportation problems and methods for constructing an initial basic feasible solution, including the Northwest Corner Rule and Vogel's Approximation Method. It also discusses using the transportation simplex method to solve transportation problems.
Intersection modeling using a convergent scheme based on Hamilton-Jacobi equa...Guillaume Costeseque
The document presents an intersection modeling approach using a Hamilton-Jacobi equation. It proposes modeling traffic flow at intersections as the maximization of total flow, without internal states. A numerical scheme is presented to solve the HJ equation based on conservation of vehicles and FIFO principles. Numerical simulations are shown for a diverging intersection with one incoming and two outgoing roads, demonstrating the propagation of density waves over time.
Road junction modeling using a scheme based on Hamilton-Jacobi equationsGuillaume Costeseque
The document summarizes a numerical scheme for modeling traffic flow at road junctions using Hamilton-Jacobi equations. The scheme models traffic on each branch of the junction as well as at the junction point where branches meet. It introduces Hamiltonians representing traffic flow and establishes gradient estimates and existence/uniqueness results for the numerical solution. The scheme is shown to converge to the unique viscosity solution of the underlying partial differential equations as the grid is refined.
This document discusses numerical methods for variational principles in traffic modeling. It begins with an introduction to variational principles in physical systems and their application to traffic modeling. It then provides an overview of macroscopic traffic flow models, including first-order Lighthill-Whitham-Richards models and higher-order Generic Second Order Models. The document explains that traffic models can be formulated as variational problems and solved using principles like minimum action, Hamilton-Jacobi equations, and dynamic programming. Numerical methods are needed to solve the resulting variational problems in modeling real-world traffic flows.
Mixed Integer Linear Programming Formulation for the Taxi Sharing Problemjfrchicanog
The document presents a mixed integer linear programming (MILP) formulation for solving the taxi sharing problem. The taxi sharing problem aims to optimize taxi routes by allowing passengers with similar pick-up and drop-off locations to share taxis. The formulation models the problem as sequences of passenger locations that represent taxi rides. Experiments on real-world taxi trip data show the MILP formulation finds lower cost solutions than a parallel evolutionary algorithm, especially on medium and large problem instances, demonstrating the benefits of the exact MILP approach.
The document reviews optimal speed traffic flow models. It discusses macroscopic and microscopic models, with a focus on car-following models and the optimal velocity model (OVM). The OVM describes how each vehicle tries to travel at an optimal speed based on the distance to the preceding vehicle. Several improved models are presented, including the comprehensive optimal velocity model which considers both distance and speed difference, and the optimal velocity forecast model which incorporates anticipated speed changes. While the models reviewed consider single-lane traffic, the conclusion recommends including non-car vehicles and driver behaviors more common to Nigeria for greater applicability.
Invited presentation at SIAM PP 18: Communication Hiding Through Pipelining i...wvanroos
The performance of Krylov method suffers in the strong scaling limit due to the synchronization latencies mainly associated with the dot-products. We review the development of pipelined Krylov methods that exploit asynchronous communication to overlap communication and computation. For example, by reordering the operations in the conjugate gradients it is possible to execute the dot-product simultaneously with the sparse matrix vector products. This gives the dot-product more time to complete while doing useful work.
We also introduce deep pipelines, where the dot-products are overlapping with multiple sparse matrix-vector products. This gives the dot-product even more time to complete and these deep pipelines solve most of the scaling problems. However, due to this reorganization of the algorithm the propagation of rounding errors is altered resulting in different final accuracy, requiring a detailed numerical analysis.
We conclude with experiments that illustrate the improved scalability.
This document discusses the transportation problem and methods for solving it. It begins with an example of distributing goods from 3 canneries to 4 warehouses. It then defines the transportation problem as distributing goods from sources to destinations at minimum cost. It provides the formulation for the transportation problem as a linear program. It discusses properties of transportation problems and methods for constructing an initial basic feasible solution, including the Northwest Corner Rule and Vogel's Approximation Method. It also discusses using the transportation simplex method to solve transportation problems.
This document discusses multi-modal journey planning and describes a proposed solution approach. It summarizes the multi-modal journey planning problem, characteristics, previous work, and proposes a hybrid approach using a mathematical programming model combined with heuristic methods like Dijkstra's algorithm. The approach involves using the programming model to solve the multi-modal journey planning problem after applying Dijkstra's algorithm and graph techniques to pre-process the data.
Numerical approach for Hamilton-Jacobi equations on a network: application to...Guillaume Costeseque
The document describes a numerical approach for solving Hamilton-Jacobi equations on networks and its application to modeling traffic flow. It presents a Hamilton-Jacobi model for traffic flow on a network that views the network as a graph with edges and vertices. A numerical scheme is developed that discretizes the Hamilton-Jacobi equations in space and time and couples them at junction points using a maximum principle. Theoretical results proving gradient bounds, existence and uniqueness of solutions, and convergence of the numerical solution are also presented.
Likelihood-free methods provide techniques for approximating Bayesian computations when the likelihood function is unavailable or computationally intractable. Monte Carlo methods like importance sampling and iterated importance sampling generate samples from an approximating distribution to estimate integrals. Population Monte Carlo is an iterative Monte Carlo algorithm that propagates a population of particles over time to explore the target distribution. Approximate Bayesian computation uses simulation-based methods to approximate posterior distributions when the likelihood is unavailable.
We all make choices between alternatives every day in many contexts - not just transport. There is theory to help planners forecast those decisions, but it is generally poorly understood. The aim of this presentation is to be of particular relevance to all PhD students and early career researchers - who should know something about DCM even if not planning to work in that area. No prior knowledge necessary.
Tony Fowkes first joined ITS in September 1976, coming from the University's School of Economic Studies, where he had been lecturing. Initially he worked on Car Ownership Forecasting, before working in a wide variety of areas of Transport Planning. In 1982 he joined the first UK Value of Time study, as well as a parallel project on Business Travel which led to pioneering work on Business Value of Time. On both those projects he helped to develop the new technique of Stated Preference estimation. In 1984 he began 4 years here as British Railways Senior Rail Research Fellow. He then moved to a mix of teaching and research, jointly with LUBS. He has published widely and contributed to many influential reports for government bodies. He retired in October 2016 as Reader in Transport Econometrics, and is now a Visiting Reader at ITS.
This document discusses the Hofstadter butterfly model for the honeycomb lattice structure of graphene. It shows that the Hall conductivity σH in an energy gap must satisfy a Diophantine equation relating σH, the magnetic flux per unit cell p/q, and an integer s. For the honeycomb lattice, the conjecture is that σH lies in the window (-q,q) rather than the typical (-q/2,q/2). The bulk-edge correspondence relates σH to the number of edge state crossings in the Brillouin zone. Numerical results for σH calculated from the edge state spectrum agree with the Diophantine equation in 99.8% of cases.
Contribution à l'étude du trafic routier sur réseaux à l'aide des équations d...Guillaume Costeseque
The document discusses traffic flow modeling on road networks. It begins by motivating the use of Hamilton-Jacobi equations to model traffic at a macroscopic scale on networks. It then provides an introduction to traffic modeling, including microscopic and macroscopic models. It focuses on the Lighthill-Whitham-Richards model and discusses higher-order models. It also discusses how microscopic models can be homogenized to derive macroscopic models using Hamilton-Jacobi equations. Finally, it discusses multi-anticipative traffic models and numerical schemes for solving the equations.
This document provides an overview of discrete choice analysis and nested logit models. It begins with a review of binary and multinomial logit models and the independence from irrelevant alternatives property. It then introduces nested logit models as a way to address IIA violations when alternatives are correlated or choices are multidimensional. The document provides an example of a nested logit specification and calculation of choice probabilities. It concludes with extensions like mixed logit models and an appendix on additional model specifications.
The document summarizes a presentation on using Hamilton-Jacobi equations to model traffic flow, specifically the moving bottleneck problem. It introduces traffic flow models including Lighthill-Whitham-Richards and extensions to mesoscopic and multiclass multilane models. It describes the moving bottleneck theory for modeling a slower vehicle generating queues. The talk outlines formulating the problem using partial differential equations coupling traffic flow with the moving bottleneck trajectory and discusses numerical methods for solving the equations.
This document discusses vehicle routing and scheduling models and algorithms. It introduces basic models like the Traveling Salesman Problem (TSP), Vehicle Routing Problem (VRP), and Pickup and Delivery Problem with Time Windows (PDPTW). Construction heuristics like savings, insertion, and set covering algorithms are presented to find initial feasible solutions that can then be improved using local search methods. The document outlines practical considerations and recent variants like dynamic and stochastic routing problems.
The document discusses using an inoculation approach to solve the problem of rescheduling railway trains following a small perturbation like a delay. The inoculation approach involves first running the algorithm with an "empty" incident to solve initial constraints and find a reasonable starting permutation, then using that solution to initialize the population for solving the actual problem. Experimental results show the inoculation approach finds good solutions faster than alternatives for both easy and hard instances of the scheduling problem.
Hamilton-Jacobi approach for second order traffic flow modelsGuillaume Costeseque
This document summarizes a presentation on using a Hamilton-Jacobi approach for second order traffic flow models. It begins with an introduction to traffic modeling, discussing both Eulerian and Lagrangian representations of traffic. It then discusses using a variational principle to apply to generic second order traffic flow models (GSOM), which account for additional driver attributes beyond just density. Specifically, it discusses formulating GSOM models in Lagrangian coordinates using a Hamilton-Jacobi framework. The document outlines solving the HJ PDE using characteristics, and decomposing problems into elementary blocks defined by piecewise affine initial, upstream and internal boundary conditions.
APassengerKnockOnDelayModelForTimetableOptimisation_beamerPeter Sels
The document presents a model for optimizing passenger travel time in train timetabling by accounting for knock-on delays. It develops a stochastic goal function to minimize expected passenger transfer time considering primary delays and knock-on effects. Graph-based approaches are used to derive knock-on time and linearize it for optimization. Results show the optimized schedule reduces expected passenger time by 2.44% compared to the original planned schedule.
Traffic flow modeling on road networks using Hamilton-Jacobi equationsGuillaume Costeseque
This document discusses traffic flow modeling using Hamilton-Jacobi equations on road networks. It motivates the use of macroscopic traffic models based on conservation laws and Hamilton-Jacobi equations to describe traffic flow. These models can capture traffic behavior at a aggregate level based on density, flow and speed. The document outlines different orders of macroscopic traffic models, from first order Lighthill-Whitham-Richards models to higher order models that account for additional traffic attributes. It also discusses the relationship between microscopic car-following models and the emergence of macroscopic behavior through homogenization.
Hamilton-Jacobi equation on networks: generalized Lax-Hopf formulaGuillaume Costeseque
The document discusses Hamilton-Jacobi equations on networks to model flows such as traffic. It presents an initial framework developed by Imbert-Monneau from 2010-2014 that models flows on a network as a single Hamilton-Jacobi equation with implicit junction conditions. This approach has good mathematical properties but lacks an explicit solution and does not model traffic flows satisfactorily. The document then outlines improvements to the approach including introducing time and space dependencies to better model traffic flows.
Differential game theory for Traffic Flow ModellingSerge Hoogendoorn
Lecture given at the INdAM symposium in Rome, 2017. The lecture shows how you can use differential games to model traffic flows, focussing on pedestrian simulation.
My first presentation as a PhD student in which I outline the background to my research project. This presentation was given as part of the University of Southampton Transportation Research Group seminar programme.
This document provides an overview of basic theory and formulae for small hydro projects. It reviews mathematical fundamentals like area, volume, trigonometry, and algebra. It then covers commonly applied formulae for discharge equations, deflection calculations, and physics of compressed air. The document concludes with the process for sizing a small hydro site, including estimating the flow duration curve and picking the appropriate turbine based on integrating power potential.
This document proposes a method for continuous time series alignment in human action recognition. It defines continuous versions of time series, warping paths, and the dynamic time warping (DTW) distance. The method finds the optimal continuous warping path by approximating solutions to a cost minimization problem. An experiment applies the continuous DTW to classify human activities from accelerometer data, achieving classification accuracy close to the discrete DTW method. The continuous approach solves issues with resampling data and has potential for improved approximations and optimization methods.
This document discusses predicting uncertainty in traffic forecasts. It describes different types of uncertainty that can affect forecasts, including input uncertainty, model uncertainty, and unexpected events. The main method discussed for quantifying uncertainty is Monte Carlo simulation, where inputs and model parameters are varied in repeated model runs to produce a range of forecast outcomes. Examples are provided of applying this method to models in the Netherlands, Paris, and for the Fréjus tunnel, showing how uncertainty increases over time and is affected by factors like major infrastructure projects.
Representation formula for traffic flow estimation on a networkGuillaume Costeseque
This document discusses representation formulas for traffic flow estimation on networks using Hamilton-Jacobi equations. It begins by motivating the use of HJ equations, noting advantages like smooth solutions and physically meaningful quantities. It then presents the basic ideas of Lax-Hopf formulas for solving HJ equations on networks, including a simple case study of a junction. The document outlines its topics which include notations from traffic flow modeling, basic recalls on Lax-Hopf formulas, HJ equations on networks, and a new approach.
Sampling-Based Planning Algorithms for Multi-Objective MissionsMd Mahbubur Rahman
multiobjective path planning has Increasing demand in military missions, rescue operations, construction job-sites.
There is Lack of robotic path planning algorithm that compromises multiple
objectives. Commonly no solution that optimizes all the objective functions. Here we modify RRT, RRT* sampling based algorithm.
This document discusses queue length estimation on urban corridors using an optimization-based framework that accounts for bounded vehicle acceleration. The approach formulates the problem as a mixed integer linear program (MILP) that finds traffic conditions satisfying macroscopic flow models and sparse GPS data constraints. Optimal solutions are used to compute traffic states and deduce queue lengths over time by identifying regions where density is near the jam density. Model constraints enforce compatibility between initial, boundary, and internal traffic conditions.
This document discusses multi-modal journey planning and describes a proposed solution approach. It summarizes the multi-modal journey planning problem, characteristics, previous work, and proposes a hybrid approach using a mathematical programming model combined with heuristic methods like Dijkstra's algorithm. The approach involves using the programming model to solve the multi-modal journey planning problem after applying Dijkstra's algorithm and graph techniques to pre-process the data.
Numerical approach for Hamilton-Jacobi equations on a network: application to...Guillaume Costeseque
The document describes a numerical approach for solving Hamilton-Jacobi equations on networks and its application to modeling traffic flow. It presents a Hamilton-Jacobi model for traffic flow on a network that views the network as a graph with edges and vertices. A numerical scheme is developed that discretizes the Hamilton-Jacobi equations in space and time and couples them at junction points using a maximum principle. Theoretical results proving gradient bounds, existence and uniqueness of solutions, and convergence of the numerical solution are also presented.
Likelihood-free methods provide techniques for approximating Bayesian computations when the likelihood function is unavailable or computationally intractable. Monte Carlo methods like importance sampling and iterated importance sampling generate samples from an approximating distribution to estimate integrals. Population Monte Carlo is an iterative Monte Carlo algorithm that propagates a population of particles over time to explore the target distribution. Approximate Bayesian computation uses simulation-based methods to approximate posterior distributions when the likelihood is unavailable.
We all make choices between alternatives every day in many contexts - not just transport. There is theory to help planners forecast those decisions, but it is generally poorly understood. The aim of this presentation is to be of particular relevance to all PhD students and early career researchers - who should know something about DCM even if not planning to work in that area. No prior knowledge necessary.
Tony Fowkes first joined ITS in September 1976, coming from the University's School of Economic Studies, where he had been lecturing. Initially he worked on Car Ownership Forecasting, before working in a wide variety of areas of Transport Planning. In 1982 he joined the first UK Value of Time study, as well as a parallel project on Business Travel which led to pioneering work on Business Value of Time. On both those projects he helped to develop the new technique of Stated Preference estimation. In 1984 he began 4 years here as British Railways Senior Rail Research Fellow. He then moved to a mix of teaching and research, jointly with LUBS. He has published widely and contributed to many influential reports for government bodies. He retired in October 2016 as Reader in Transport Econometrics, and is now a Visiting Reader at ITS.
This document discusses the Hofstadter butterfly model for the honeycomb lattice structure of graphene. It shows that the Hall conductivity σH in an energy gap must satisfy a Diophantine equation relating σH, the magnetic flux per unit cell p/q, and an integer s. For the honeycomb lattice, the conjecture is that σH lies in the window (-q,q) rather than the typical (-q/2,q/2). The bulk-edge correspondence relates σH to the number of edge state crossings in the Brillouin zone. Numerical results for σH calculated from the edge state spectrum agree with the Diophantine equation in 99.8% of cases.
Contribution à l'étude du trafic routier sur réseaux à l'aide des équations d...Guillaume Costeseque
The document discusses traffic flow modeling on road networks. It begins by motivating the use of Hamilton-Jacobi equations to model traffic at a macroscopic scale on networks. It then provides an introduction to traffic modeling, including microscopic and macroscopic models. It focuses on the Lighthill-Whitham-Richards model and discusses higher-order models. It also discusses how microscopic models can be homogenized to derive macroscopic models using Hamilton-Jacobi equations. Finally, it discusses multi-anticipative traffic models and numerical schemes for solving the equations.
This document provides an overview of discrete choice analysis and nested logit models. It begins with a review of binary and multinomial logit models and the independence from irrelevant alternatives property. It then introduces nested logit models as a way to address IIA violations when alternatives are correlated or choices are multidimensional. The document provides an example of a nested logit specification and calculation of choice probabilities. It concludes with extensions like mixed logit models and an appendix on additional model specifications.
The document summarizes a presentation on using Hamilton-Jacobi equations to model traffic flow, specifically the moving bottleneck problem. It introduces traffic flow models including Lighthill-Whitham-Richards and extensions to mesoscopic and multiclass multilane models. It describes the moving bottleneck theory for modeling a slower vehicle generating queues. The talk outlines formulating the problem using partial differential equations coupling traffic flow with the moving bottleneck trajectory and discusses numerical methods for solving the equations.
This document discusses vehicle routing and scheduling models and algorithms. It introduces basic models like the Traveling Salesman Problem (TSP), Vehicle Routing Problem (VRP), and Pickup and Delivery Problem with Time Windows (PDPTW). Construction heuristics like savings, insertion, and set covering algorithms are presented to find initial feasible solutions that can then be improved using local search methods. The document outlines practical considerations and recent variants like dynamic and stochastic routing problems.
The document discusses using an inoculation approach to solve the problem of rescheduling railway trains following a small perturbation like a delay. The inoculation approach involves first running the algorithm with an "empty" incident to solve initial constraints and find a reasonable starting permutation, then using that solution to initialize the population for solving the actual problem. Experimental results show the inoculation approach finds good solutions faster than alternatives for both easy and hard instances of the scheduling problem.
Hamilton-Jacobi approach for second order traffic flow modelsGuillaume Costeseque
This document summarizes a presentation on using a Hamilton-Jacobi approach for second order traffic flow models. It begins with an introduction to traffic modeling, discussing both Eulerian and Lagrangian representations of traffic. It then discusses using a variational principle to apply to generic second order traffic flow models (GSOM), which account for additional driver attributes beyond just density. Specifically, it discusses formulating GSOM models in Lagrangian coordinates using a Hamilton-Jacobi framework. The document outlines solving the HJ PDE using characteristics, and decomposing problems into elementary blocks defined by piecewise affine initial, upstream and internal boundary conditions.
APassengerKnockOnDelayModelForTimetableOptimisation_beamerPeter Sels
The document presents a model for optimizing passenger travel time in train timetabling by accounting for knock-on delays. It develops a stochastic goal function to minimize expected passenger transfer time considering primary delays and knock-on effects. Graph-based approaches are used to derive knock-on time and linearize it for optimization. Results show the optimized schedule reduces expected passenger time by 2.44% compared to the original planned schedule.
Traffic flow modeling on road networks using Hamilton-Jacobi equationsGuillaume Costeseque
This document discusses traffic flow modeling using Hamilton-Jacobi equations on road networks. It motivates the use of macroscopic traffic models based on conservation laws and Hamilton-Jacobi equations to describe traffic flow. These models can capture traffic behavior at a aggregate level based on density, flow and speed. The document outlines different orders of macroscopic traffic models, from first order Lighthill-Whitham-Richards models to higher order models that account for additional traffic attributes. It also discusses the relationship between microscopic car-following models and the emergence of macroscopic behavior through homogenization.
Hamilton-Jacobi equation on networks: generalized Lax-Hopf formulaGuillaume Costeseque
The document discusses Hamilton-Jacobi equations on networks to model flows such as traffic. It presents an initial framework developed by Imbert-Monneau from 2010-2014 that models flows on a network as a single Hamilton-Jacobi equation with implicit junction conditions. This approach has good mathematical properties but lacks an explicit solution and does not model traffic flows satisfactorily. The document then outlines improvements to the approach including introducing time and space dependencies to better model traffic flows.
Differential game theory for Traffic Flow ModellingSerge Hoogendoorn
Lecture given at the INdAM symposium in Rome, 2017. The lecture shows how you can use differential games to model traffic flows, focussing on pedestrian simulation.
My first presentation as a PhD student in which I outline the background to my research project. This presentation was given as part of the University of Southampton Transportation Research Group seminar programme.
This document provides an overview of basic theory and formulae for small hydro projects. It reviews mathematical fundamentals like area, volume, trigonometry, and algebra. It then covers commonly applied formulae for discharge equations, deflection calculations, and physics of compressed air. The document concludes with the process for sizing a small hydro site, including estimating the flow duration curve and picking the appropriate turbine based on integrating power potential.
This document proposes a method for continuous time series alignment in human action recognition. It defines continuous versions of time series, warping paths, and the dynamic time warping (DTW) distance. The method finds the optimal continuous warping path by approximating solutions to a cost minimization problem. An experiment applies the continuous DTW to classify human activities from accelerometer data, achieving classification accuracy close to the discrete DTW method. The continuous approach solves issues with resampling data and has potential for improved approximations and optimization methods.
This document discusses predicting uncertainty in traffic forecasts. It describes different types of uncertainty that can affect forecasts, including input uncertainty, model uncertainty, and unexpected events. The main method discussed for quantifying uncertainty is Monte Carlo simulation, where inputs and model parameters are varied in repeated model runs to produce a range of forecast outcomes. Examples are provided of applying this method to models in the Netherlands, Paris, and for the Fréjus tunnel, showing how uncertainty increases over time and is affected by factors like major infrastructure projects.
Representation formula for traffic flow estimation on a networkGuillaume Costeseque
This document discusses representation formulas for traffic flow estimation on networks using Hamilton-Jacobi equations. It begins by motivating the use of HJ equations, noting advantages like smooth solutions and physically meaningful quantities. It then presents the basic ideas of Lax-Hopf formulas for solving HJ equations on networks, including a simple case study of a junction. The document outlines its topics which include notations from traffic flow modeling, basic recalls on Lax-Hopf formulas, HJ equations on networks, and a new approach.
Sampling-Based Planning Algorithms for Multi-Objective MissionsMd Mahbubur Rahman
multiobjective path planning has Increasing demand in military missions, rescue operations, construction job-sites.
There is Lack of robotic path planning algorithm that compromises multiple
objectives. Commonly no solution that optimizes all the objective functions. Here we modify RRT, RRT* sampling based algorithm.
This document discusses queue length estimation on urban corridors using an optimization-based framework that accounts for bounded vehicle acceleration. The approach formulates the problem as a mixed integer linear program (MILP) that finds traffic conditions satisfying macroscopic flow models and sparse GPS data constraints. Optimal solutions are used to compute traffic states and deduce queue lengths over time by identifying regions where density is near the jam density. Model constraints enforce compatibility between initial, boundary, and internal traffic conditions.
MODIFIED VOGEL APPROXIMATION METHOD FOR BALANCED TRANSPORTATION MODELS TOWARD...IAEME Publication
This paper is built on a study in relation to transportation problem as it affects most organisational decision in a decomposed setting. The case study used in this work is Dangote cement factory (in Ibese, Nigeria) with three sources and four destinationscentres. The factory is supported by increasing number of cement delivery trucks. Some models for solving balanced transportation problems (TPs) are considered in order to determine the optimal and initial basic feasible solutions (IBFS). From the analysis, it is observed that Modified Vogel Approximation Method (MVAM) is a better method. This is partly because MVAM considers each unit cost in its solution algorithm and minimises total cost comparatively with Vogel Approximation Method (VAM). The results arefurther justified and validated using windows version 2.00 Tora package.
Poster for Bayesian Statistics in the Big Data Era conferenceChristian Robert
The document proposes a new version of Hamiltonian Monte Carlo (HMC) sampling that is essentially calibration-free. It achieves this by learning the optimal leapfrog scale from the distribution of integration times using the No-U-Turn Sampler algorithm. Compared to the original NUTS algorithm on benchmark models, this new enhanced HMC (eHMC) exhibits significantly improved efficiency with no hand-tuning of parameters required. The document tests eHMC on a Susceptible-Infected-Recovered model of disease transmission.
Eco-friendly Reduction of Travel Times in European Smart Cities (GECCO'14)Daniel H. Stolfi
This article proposes an innovative solution for reducing polluting gas emissions from road traffic in modern cities. It is based on our new Red Swarm architecture which is composed of a series of intelligent spots with WiFi connections that can suggest a customized route to drivers. We have tested our proposal in four different case studies corresponding to actual European smart cities. To this end, we first import the city information from OpenStreetMap into the SUMO road traffic micro-simulator, propose a Red Swarm architecture based on intelligent spots located at traffic lights, and then optimize the resulting system in terms of travel times and gas emissions by using an evolutionary algorithm. Our results show that an important quantitative reduction in gas emissions as well as in travel times can be achieved when vehicles are rerouted according to our Red Swarm indications. This represents a promising result for the low cost implementation of an idea that could engage the interest of both citizens and municipal authorities.
http://dx.doi.org/10.1145/2576768.2598317
The lecture outline discusses network models and network flow problems. It introduces key concepts like the maximum flow problem and minimum cost flow problem. It provides examples of solving the maximum flow problem using the Ford-Fulkerson method and concepts like residual networks and augmenting paths. The document also provides a sample problem solving the maximum flow problem on a network transporting water.
Similar to Schéma numérique basé sur une équation d'Hamilton-Jacobi : modélisation des intersections (20)
Analyse des données du Registre de preuve de covoiturage à l'échelle régional...Guillaume Costeseque
Nous présentons des analyses spatio-temporelles ainsi qu'une modélisation du volume de déplacements en covoiturage à l'échelle régionale à partir des données issues du Registre de preuve de covoiturage (RPC) en France.
Support de cours "Transports intelligents" donné à l’École Centrale de Nantes le 12 mars 20201, auprès d'étudiants en 2eme année filière ingénierie urbaine, option Aménagement et Transport. Présentation des enjeux, des acteurs, du contexte et de quelques tendances des systèmes de transport intelligents avec quelques exemples d'application à Nantes
Support de cours "Transports intelligents" donné à l’École Centrale de Nantes le 9 mars 2020, auprès d'étudiants en 2eme année filière ingénierie urbaine, option Aménagement et Transport. Présentation des enjeux, des acteurs, du contexte et de quelques tendances des systèmes de transport intelligents avec quelques exemples d'application à Nantes
A multi-objective optimization framework for a second order traffic flow mode...Guillaume Costeseque
This slides deal with a common work with Paola Goatin (Inria Sophia-Antipolis), Simone Göttlich and Oliver Kolb (Universität Mannheim) about Riemann solvers for the ARZ model on a junction
Cette présentation a été réalisée lors du congrès ATEC ITS de janvier 2020. Elle aborde l'évaluation réalisée par le Cerema concernant l'expérimentation d'une navette Navya "Autonom Shuttle" en route ouverte sur le territoire de Nantes Métropole, entre mars et mai 2019.
Some recent developments in the traffic flow variational formulationGuillaume Costeseque
This document summarizes recent developments in modeling traffic flow using Hamilton-Jacobi equations. It discusses using Hamilton-Jacobi equations to model cumulative vehicle counts on highways with entrance and exit ramps. Source terms are added to the Hamilton-Jacobi equations to account for the effects of exogenous lateral inflows and outflows of vehicles onto the highway. Analytical solutions are presented for cases with constant inflow rates, and for an extended Riemann problem with piecewise constant boundary and inflow conditions.
Hamilton-Jacobi equations and Lax-Hopf formulae for traffic flow modelingGuillaume Costeseque
The document discusses using Hamilton-Jacobi equations and Lax-Hopf formulas to model traffic flow. It introduces the Lighthill-Whitham-Richards traffic model in both Eulerian and Lagrangian coordinates. In the Eulerian framework, the cumulative vehicle count satisfies a Hamilton-Jacobi equation, and Lax-Hopf formulas provide representations involving minimizing cost along trajectories. Similarly in the Lagrangian framework, vehicle position satisfies a Hamilton-Jacobi equation, and Lax-Hopf formulas involve minimizing cost along characteristic curves. The document outlines applying variational principles and optimal control interpretations to these traffic models.
Mesoscopic multiclass traffic flow modeling on multi-lane sectionsGuillaume Costeseque
This document presents a mesoscopic multiclass traffic flow model for multi-lane highway sections. It formulates the model using a system of coupled Hamilton-Jacobi partial differential equations to represent different vehicle classes. The model accounts for non-FIFO behavior and capacity drop at lane changes using a parameter. It is solved numerically using a Lax-Hopf scheme, with coupling conditions to represent interactions between classes.
The impact of source terms in the variational representation of traffic flowGuillaume Costeseque
This document examines the impact of source terms in the variational representation of traffic flow. It finds that variational theory solutions are valid in Eulerian coordinates for exogenous source terms but not for endogenous source terms that depend on traffic conditions. However, in discrete time steps, endogenous source terms from the previous step become exogenous for the current step, allowing improved numerical solution methods. Specifically, it presents Godunov's method and variational networks methods for solving the kinematic wave and Hamilton-Jacobi models of traffic flow in discrete time steps.
The document discusses second-order traffic flow models on networks. It provides motivation for higher-order models by noting limitations of first-order models like LWR in capturing phenomena observed in real traffic flows. It introduces the Generic Second-Order Model (GSOM) family, which incorporates an additional driver attribute variable beyond density to obtain more accurate descriptions of traffic. GSOM reduces to LWR when this additional variable is ignored. Several examples of GSOM models are also presented.
The document discusses second order traffic flow models on networks. It begins with an introduction to traffic modeling, including macroscopic representations of traffic flow using density, flow, and speed. First order models like Lighthill-Whitham-Richards (LWR) are introduced, as well as higher order Generic Second Order Models (GSOM) that can account for driver behavior and vehicle interactions. The document then discusses applying a variational principle and Hamilton-Jacobi formulation to both LWR and GSOM models, allowing them to be analyzed using tools from optimal control and viability theory.
Numerical approach for Hamilton-Jacobi equations on a network: application to...Guillaume Costeseque
This document presents a numerical scheme for solving Hamilton-Jacobi equations on networks to model traffic flow. It describes applying a Godunov-type scheme using finite differences on networks consisting of branches connected at junctions. The scheme computes numerical solutions of the Hamilton-Jacobi equations on each branch and couples them at junctions using maximum operations. Gradient estimates, existence and uniqueness, and convergence properties of the numerical solutions are proven. The document also interprets the numerical solutions in terms of discrete car densities on the branches and shows the scheme is consistent with classical macroscopic traffic models.
This document discusses second order traffic flow models on networks. It provides motivation for higher order models by noting limitations of first order models like the Lighthill-Whitham-Richards model. It introduces the Generic Second Order Model (GSOM) family which incorporates additional driver attributes beyond just density. GSOM models are presented in both Eulerian and Lagrangian coordinates. The variational principle and Hamilton-Jacobi formulation are discussed for applying dynamic programming and minimum principles to these models.
This document discusses dynamic road traffic modeling. It explains that dynamic models are important because they consider time as a factor. Dynamic traffic models can be used for both long-term infrastructure planning and short-term traffic management. They aim to predict departure times, route choices, and how traffic loads networks over time based on traffic flow theory. Dynamic models can take a microscopic approach tracking individual vehicles or a macroscopic approach using averaged quantities. The document also outlines some common uses of dynamic macroscopic models, including estimating the current traffic state, forecasting future states, and optimizing control measures to achieve objectives.
The ORESTE project aims to optimize traffic flow in corridors using ramp metering and optimal rerouting strategies. It is a collaboration between Inria and UC Berkeley. The team has developed macroscopic traffic flow models and an adjoint method to compute gradients for optimal control problems related to ramp metering and rerouting compliant users. Numerical results on test cases show that ramp metering and rerouting can significantly reduce congestion. The team has published several journal and conference papers on their work and is pursuing additional research on junction models and controls based on autonomous vehicles.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...
Schéma numérique basé sur une équation d'Hamilton-Jacobi : modélisation des intersections
1. Sch´ema num´erique bas´e sur une ´equation
d’Hamilton-Jacobi: mod´elisation des intersections
Guillaume Costeseque
Joint work with R. Monneau & JP. Lebacque
Ecole des Ponts ParisTech, CERMICS
& IFSTTAR, GRETTIA
GdT ”Math´ematiques pour la mod´elisation des Transports”
29 Novembre 2012 - Marne-la-Vall´ee
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 1 / 40
3. Introduction
Outline
1 Introduction
Traffic flow modelling
Link models
Overview of intersection modelling
2 Proposed intersection model
3 Numerical simulations
4 Concluding remarks
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 3 / 40
4. Introduction Traffic flow modelling
Objectives
From Papageorgiou and Gentile keynote speeches,
Dynamic Traffic Assignment (DTA) models useful for:
Transport planning (off-line)
Traffic monitoring and control (real-time)
Dynamic Loading Network (DLN) model =
Link model (propagate flows on arcs)
+ Node model (manage flows at intersection)
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 4 / 40
5. Introduction Link models
LWR model
Introduced by Lighhill, Whitham (1955) and Richards (1956):
Flow dynamics analogy: Kinematic Wave (KW) theory
Conservation law for the vehicles densities:
ρt(x, t) + (Q(x, t))x = 0
Q(x, t) = ρ(x, t)V (x, t)
(1.1)
Assumption of equilibrium & Fundamental Diagram (FD) :
V (x, t) = Ve (ρ(x, t)) (1.2)
x x + ∆x
ρ(x, t)∆x
Q(x, t)∆t Q(x + ∆x, t)∆t
Speed V
ρcrit ρmax
Density ρ
Vmax
Vcrit
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 5 / 40
6. Introduction Overview of intersection modelling
Intersection modelling
Main congestion points on networks
Still an interesting research point
Jn+l
Jn+2
Jj
Jn+m
J1
Ji
Jn
Many approaches: point-wise VS spatial extended models, equilibrium
VS optimization models, with or without internal states etc.
[Lebacque and Khoshyaran, (2002, 2005, 2009)]
State-of-the-art review:
See [Tamp`ere, Corthout, Catterysse, Immers (2011)]
and [Fl¨otter¨od and Rohde (2011)]
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 6 / 40
7. Proposed intersection model
Outline
1 Introduction
2 Proposed intersection model
Hamilton-Jacobi equation
Numerical Scheme
Some mathematical results
3 Numerical simulations
4 Concluding remarks
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 7 / 40
8. Proposed intersection model Hamilton-Jacobi equation
HJ equation: settings
Point-wise intersection without internal state
No distinction on incoming or outgoing roads
No consideration on multiclasses or multilanes
J3
J1
J2
e1
e2
eN
JN
e3
uα := primitive of ρα solution of LWR equation on branch α...
...modulo γα the proportion of flow sent or received by branch α
uα solution of:
uα
t + Hα(uα
x ) = 0 outside the node,
ut + max
α=1,...,N
H−
α (uα
x ) = 0 at the node.
(2.3)
Flow maximization at the node
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 8 / 40
9. Proposed intersection model Hamilton-Jacobi equation
HJ versus conservation laws
Conservation laws Hamilton-Jacobi
Variable ρα(x, t) uα(x, t)
Concentration in (x, t) Vehicle index in (x, t)
Equation ρα
t + (Qα(ρα))x = 0 uα
t + Hα (uα
x ) = 0
FD / Hamilt. Demand (∆) Decreasing part (H−
α )
Supply (Σ) Increasing part (H+
α )
Qmax
Density ρ
ρcrit ρmax
Flow Q
Qmax
Supply Σ
Density ρ
Demand ∆
Density ρ
ρcrit
ρcrit
ρmax
Qmax
H+
α (p)
pα
0 p
H−
α (p)
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 9 / 40
10. Proposed intersection model Numerical Scheme
Introduction to the scheme
Assumptions:
Conservation of vehicles
FIFO (First-In-First-Out) sequel
Maximization on the total though-flow
Coefficients γα fixed once for all
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 10 / 40
11. Proposed intersection model Numerical Scheme
Presentation of the NS
∆x and ∆t = space and time steps satisfying CFL condition
pα,n
i,± (resp. Wα,n
i ) discrete space (resp. time) derivative
Proposition (Numerical Scheme)
Wα,n
i = max (−H+
α (pα,n
i,− ), −H−
α (pα,n
i,+ )), for i ≥ 1, for all α
Un
0 := Uα,n
0 , for all α
Wα,n
0 = max
α=1,...,N
(−H−
α (pα,n
i,+ )),
for i = 0
(2.4)
Passing flow = Minimum between Upstream Demand and
Downstream Supply [Lebacque (1996)]
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 11 / 40
12. Proposed intersection model Some mathematical results
Some results
From [Costeseque, Monneau, Lebacque (in progress)]:
First main result is Time and Space Gradient estimates
Second main result is the convergence of the numerical solution to
the HJ viscosity solution when (∆t, ∆x) → 0 (under suitable
assumptions)
From [Imbert, Monneau, Zidani (2011)]:
Existence and uniqueness of the viscosity solution for the HJ equation
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 12 / 40
13. Numerical simulations
Outline
1 Introduction
2 Proposed intersection model
3 Numerical simulations
Example 1: Diverge
Example 2: Merge
4 Concluding remarks
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 13 / 40
14. Numerical simulations
Initial statements
We assume that the FD is bi-parabolic and we take:
Link flow capacity = free flow speed × critical density
Jam critical density = 20 veh/km/lane
Maximal density = 160 veh/km/lane
Focus on highway on or off-ramps modelling
Coefficients γα for incoming roads are capacity-proportional: see
[Cassidy and Ahn, (2005)], [Bar-Gera and Ahn, (2010)] and [Ni and
Leonard, (2005)]
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 14 / 40
15. Numerical simulations Example 1: Diverge
Diverge: Simulations
Representation of an off-ramp:
x > 0x < 0
x = 0
ρ1
ρ2
ρ3
Simplest case: one incoming road and two outgoing roads;
Single condition: u1(0, t) = u2(0, t) = u3(0, t) (continuity of the
index at the junction point)
Configuration:
Branch Number of lanes Maximal speed γα
1 2 90 km/h 1
2 2 90 km/h 0.75
3 1 50 km/h 0.25
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 15 / 40
16. Numerical simulations Example 1: Diverge
Diverge: Fundamental Diagrams
0 50 100 150 200 250 300 350
0
500
1000
1500
2000
2500
3000
3500
4000
Density (veh/km)
Flow(veh/h)
Fundamental diagrams per branch
1
2
3
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 16 / 40
22. Numerical simulations Example 1: Diverge
Diverge: Cumulative Vehicles Count
0 5 10 15 20 25
0
5
10
15
20
25
30
35
Time (s)
CumulativeNumberofVehicles
CVC on road n°1
Downstream station
Upstream station
−10 0 10 20 30
0
5
10
15
20
25
30
35
Time (s)
CumulativeNumberofVehicles
CVC on road n°2
Downstream station
Upstream station
−10 0 10 20 30
0
5
10
15
20
25
30
35
Time (s)
CumulativeNumberofVehicles
CVC on road n°3
Downstream station
Upstream station
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 22 / 40
23. Numerical simulations Example 1: Diverge
Diverge: Trajectories
5.42868
6.514427.60015
8.68589
9.77163
10.8574
11.9431
13.0288
14.1146
15.2003
16.286
17.3718
18.4575
19.5433
20.629
21.7147
Trajectories on road n° 1
Time (s)
Position(m)
0 10 20 30
−200
−150
−100
−50
0
−1.47006−0.2934180.8832282.059873.236524.413165.589816.766467.94319.11975
10.2964
11.473
12.6497
13.8263
15.003
16.1796
17.356318.5329
Trajectories on road n° 2
Time (s)
Position(m)
0 10 20 30
0
50
100
150
200
−1.47006−0.2934180.8832282.059873.236524.413165.589816.766467.94319.1197510.2964
11.473
12.6497
13.8263
15.00316.1796
17.3563
18.5329
Trajectories on road n° 3
Time (s)
Position(m)
0 10 20 30
0
50
100
150
200
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 23 / 40
24. Numerical simulations Example 2: Merge
Merge: Simulations
Representation of an on-ramp:
x > 0x < 0
x = 0
ρ3
ρ2
ρ1
Simplest case: two incoming roads and one outgoing road
Configuration:
Branch Number of lanes Maximal speed γα
1 3 90 km/h 0.8
2 1 70 km/h 0.2
3 3 90 km/h 1
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 24 / 40
25. Numerical simulations Example 2: Merge
Merge: Fundamental Diagrams
Branch Initial state Final state
1 50 veh/km 4900 veh/h 180 veh/km 4300 veh/h
2 20 veh/km 1400 veh/h 70 veh/km 1100 veh/h
3 30 veh/km 3300 veh/h 605 veh/km 5400 veh/h
0 50 100 150 200 250 300 350 400 450 500
0
1000
2000
3000
4000
5000
6000
Density (veh/km)
Flow(veh/h)
Fundamental diagrams per branch
1
2
3
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 25 / 40
34. Concluding remarks
Concluding remarks
Balance
Convergent numerical scheme
Special configurations (diverge / merge)
Use of the Cumulative Vehicles Count
Perspectives:
High order models (GSOM family)
Integrate internal node supplies (urban intersection)
Micro-Macro passage (conflicts modelling)
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 34 / 40
35. Concluding remarks
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 35 / 40
36. Complements
Some references
G. Fl¨otter¨od and J. Rohde, Operational macroscopic modeling of
complex urban intersections, Transportation Research Part B:
Methodological 45(6), (2011), pp. 903-922.
C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach
to junction problems and application to traffic flows, Working paper,
Universit´e Paris-Dauphine, Paris, (2011), 38 pages.
M.M. Khoshyaran and J.P. Lebacque, Internal state models for
intersections in macroscopic traffic flow models, TGF09, Shanghai
2009.
J.P. Lebacque and M.M. Koshyaran, First-order macroscopic traffic
flow models: intersection modeling, network modeling, 2005, pp.
365-386.
C. Tamp`ere, R. Corthout, D. Cattrysse and L. Immers, A generic class
of first order node models for dynamic macroscopic simulations of
traffic flows, Transportation Research Part B, 45 (1) (2011), pp.
289-309.
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 36 / 40
37. Complements
References for conservation laws
Hyperbolic equation of conservation laws with discontinuous flux:
ρt + (Q(x, ρ))x = 0 with Q(x, p) = 1{x<0}Q1(p) + 1{x≥0}Q2(p)
Uniqueness result for two branches:
See [Garavello, Natalini, Piccoli, Terracina (2007)]
and [Andreianov, Karlsen, Risebro (2010)]
Book of [Garavello, Piccoli (2006)] for conservation laws on networks:
Construction of a solution using the ”wave front tracking method”
No proof of the uniqueness of the solution on a general networks
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 37 / 40
38. Complements
Literature review
State-of-the-art review from [Tamp`ere, Corthout, Catterysse, Immers
(2011)] and [Fl¨otter¨od and Rohde (2011)]: list of seven requirements:
General applicability for any n, m ≥ 1
Non-negativity of flows
Demand and Supply constraints
Vehicles conservation through the junction
Conservation of Turning Fractions (FIFO)
Maximization of Flows from an User perspective
Compliance to the invariance principle [Lebacque, Khoshyaran (2005)]
(Opt.) Internal node supplies: non-uniqueness! [Corthout et al., 2012]
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 38 / 40
39. Complements
HJ equation: settings
Point-wise intersection without internal state
No distinction on incoming or outgoing roads
No consideration on multiclasses or multilanes
J3
J1
J2
e1
e2
eN
JN
e3
Primitive of ρα solution of classical LWR equation on branch α:
uα(x, t) = uα(0, t) −
1
γα
x
0
ρα
(y, t)dy, for α ∈ {outgoing roads}
uα(x, t) = uα(0, t) +
1
γα
x
0
ρα
(y, t)dy, for α ∈ {incoming roads}
with γα the proportion of flow sent or received by branch α
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 39 / 40
40. Complements
HJ formulation
Search of suitable continuous functions u, viscosity solutions of:
uα
t + Hα(uα
x ) = 0 on (0, T) × J∗
α,
ut + max
α=1,...,N
H−
α (uα
x ) = 0 on (0, T) × {0}.
(5.5)
submitted to an initial condition (globally Lipschitz continuous)
u(0, x) = u0(x), with x ∈ J. (5.6)
The Hamiltonians Hα are convex functions such that:
The function Hα : R → R is continuous and lim
|p|→+∞
Hα(p) = +∞;
There exists pα
0 ∈ R such that Hα is non-increasing on (−∞, pα
0 ] and
non-decreasing on [pα
0 , +∞).
We denote by H−
α and H+
α the corresponding functions.
G. Costeseque (Universit´e ParisEst) D5 Traffic flow models Marne-la-Vall´ee, Nov. 2012 40 / 40