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.
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.
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.
A 3hrs intro lecture to Approximate Bayesian Computation (ABC), given as part of a PhD course at Lund University, February 2016. For sample codes see http://www.maths.lu.se/kurshemsida/phd-course-fms020f-nams002-statistical-inference-for-partially-observed-stochastic-processes/
We provide a comprehensive convergence analysis of the asymptotic preserving implicit-explicit particle-in-cell (IMEX-PIC) methods for the Vlasov–Poisson system with a strong magnetic field. This study is of utmost importance for understanding the behavior of plasmas in magnetic fusion devices such as tokamaks, where such a large magnetic field needs to be applied in order to keep the plasma particles on desired tracks.
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.
The document summarizes algorithms for solving min-cost linear problems, including min-cost flow problems and min-cost potential problems. It describes how these problems can be formulated and solved using descent methods, where the search direction is chosen as a negative cycle or cut with minimum cost. Iteratively, an optimal solution is found by moving in the direction of negative cycles or cuts and updating residual graphs and data structures. Duality between the min-cost flow and potential problems is also discussed.
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.
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.
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.
A 3hrs intro lecture to Approximate Bayesian Computation (ABC), given as part of a PhD course at Lund University, February 2016. For sample codes see http://www.maths.lu.se/kurshemsida/phd-course-fms020f-nams002-statistical-inference-for-partially-observed-stochastic-processes/
We provide a comprehensive convergence analysis of the asymptotic preserving implicit-explicit particle-in-cell (IMEX-PIC) methods for the Vlasov–Poisson system with a strong magnetic field. This study is of utmost importance for understanding the behavior of plasmas in magnetic fusion devices such as tokamaks, where such a large magnetic field needs to be applied in order to keep the plasma particles on desired tracks.
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.
The document summarizes algorithms for solving min-cost linear problems, including min-cost flow problems and min-cost potential problems. It describes how these problems can be formulated and solved using descent methods, where the search direction is chosen as a negative cycle or cut with minimum cost. Iteratively, an optimal solution is found by moving in the direction of negative cycles or cuts and updating residual graphs and data structures. Duality between the min-cost flow and potential problems is also discussed.
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.
My data are incomplete and noisy: Information-reduction statistical methods f...Umberto Picchini
We review parameter inference for stochastic modelling in complex scenario, such as bad parameters initialization and near-chaotic dynamics. We show how state-of-art methods for state-space models can fail while, in some situations, reducing data to summary statistics (information reduction) enables robust estimation. Wood's synthetic likelihoods method is reviewed and the lecture closes with an example of approximate Bayesian computation methodology.
Accompanying code is available at https://github.com/umbertopicchini/pomp-ricker and https://github.com/umbertopicchini/abc_g-and-k
Readership lecture given at Lund University on 7 June 2016. The lecture is of popular science nature hence mathematical detail is kept to a minimum. However numerous links and references are offered for further reading.
Inference for stochastic differential equations via approximate Bayesian comp...Umberto Picchini
Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/
A likelihood-free version of the stochastic approximation EM algorithm (SAEM)...Umberto Picchini
I show how to obtain approximate maximum likelihood inference for "complex" models having some latent (unobservable) component. With "complex" I mean models having a so-called intractable likelihood, where the latter is unavailable in closed for or is too difficult to approximate. I construct a version of SAEM (and EM-type algorithm) that makes it possible to conduct inference for complex models. Traditionally SAEM is implementable only for models that are fairly tractable analytically. By introducing the concept of synthetic likelihood, where information is captured by a series of user-defined summary statistics (as in approximate Bayesian computation), it is possible to automatize SAEM to run on any model having some latent-component.
This document discusses density operators and their use in quantum information and computing. It begins by introducing density operators and how they can be used to describe quantum systems whose state is not precisely known or composite systems. The key properties of density operators are that they must have a trace of 1 and be positive operators. The document then covers reduced density operators which describe subsystems by taking the partial trace. Finally, it discusses how the reduced density operator gives the correct measurement statistics for observations on a subsystem.
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.
This document discusses compactness estimates for nonlinear partial differential equations (PDEs), specifically Hamilton-Jacobi equations. It provides background on Kolmogorov entropy measures of compactness and covers recent results estimating the Kolmogorov entropy of solutions to scalar conservation laws and Hamilton-Jacobi equations, showing it is on the order of 1/ε. The document outlines applications of these estimates and open questions regarding extending the estimates to non-convex fluxes and non-uniformly convex Hamiltonians.
The document provides an overview of the EM algorithm and its application to outlier detection. It begins with introducing the EM algorithm and explaining its iterative process of estimating parameters via E-step and M-step. It then proves properties of the EM algorithm such as non-decreasing log-likelihood and convergence. An example of using EM for Gaussian mixture modeling is provided. Finally, the document discusses directly and indirectly applying EM to outlier detection.
Optimal order a posteriori error bounds in L∞(L2) norm are derived for semidiscrete semilinear parabolic problems. Standard continuous Galerkin (conforming) finite element method is employed. Our main tools in deriving these error estimates are the elliptic reconstruction technique which is first introduced by Makridakis and Nochetto [5], with the aid of Gronwall’s lemma and continuation argument.
This document discusses macrocanonical models for texture synthesis. It begins by introducing the goal of texture synthesis and providing a brief history. It then describes the parametric question of combining randomness and structure in images. Specifically, it discusses maximizing entropy under geometric constraints. The document goes on to discuss links to statistical physics, defining microcanonical and macrocanonical models. It focuses on studying the macrocanonical model, describing how to find optimal parameters through gradient descent and how to sample from the model using Langevin dynamics. The document provides examples of texture synthesis and compares results to other methods.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
Approximate Bayesian computation for the Ising/Potts modelMatt Moores
This document provides an introduction to Approximate Bayesian Computation (ABC). ABC is a likelihood-free method for approximating posterior distributions when the likelihood function is intractable or expensive to evaluate. The document outlines the basic ABC rejection sampling algorithm and discusses extensions like using summary statistics, ABC-MCMC, and ABC sequential Monte Carlo. It also applies ABC to parameter inference for a hidden Potts model used in Bayesian image segmentation.
EM algorithm and its application in probabilistic latent semantic analysiszukun
The document discusses the EM algorithm and its application in Probabilistic Latent Semantic Analysis (pLSA). It begins by introducing the parameter estimation problem and comparing frequentist and Bayesian approaches. It then describes the EM algorithm, which iteratively computes lower bounds to the log-likelihood function. Finally, it applies the EM algorithm to pLSA by modeling documents and words as arising from a mixture of latent topics.
The document discusses two main topics:
1) The electrostatic interpretation of zeros of orthogonal polynomials, which describes how the zeros represent equilibrium positions of charged particles. It examines how canonical perturbations like Christoffel, Uvarov, and Geronimus transformations affect the zeros.
2) The asymptotic behavior of ratios of Laguerre orthogonal polynomials. It provides strong asymptotic expansions that are valid in different regions of the complex plane. It also examines the asymptotics of ratios of Laguerre polynomials.
short course at CIRM, Bayesian Masterclass, October 2018Christian Robert
Markov Chain Monte Carlo (MCMC) methods generate dependent samples from a target distribution using a Markov chain. The Metropolis-Hastings algorithm constructs a Markov chain with a desired stationary distribution by proposing moves to new states and accepting or rejecting them probabilistically. The algorithm is used to approximate integrals that are difficult to compute directly. It has been shown to converge to the target distribution as the number of iterations increases.
We are interested in finding a permutation of the entries of a given square matrix so that the maximum number of its nonzero entries are moved to one of the corners in a L-shaped fashion.
If we interpret the nonzero entries of the matrix as the edges of a graph, this problem boils down to the so-called core–periphery structure, consisting of two sets: the core, a set of nodes that is highly connected across the whole graph, and the periphery, a set of nodes that is well connected only to the nodes that are in the core.
Matrix reordering problems have applications in sparse factorizations and preconditioning, while revealing core–periphery structures in networks has applications in economic, social and communication networks.
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 discusses using graphics processing units (GPUs) to perform approximate Bayesian computation (ABC) for parameter estimation of complex models. It describes how GPUs are well-suited for ABC due to their ability to perform linear computations on many threads in parallel. The document provides examples of applying ABC to GPUs for problems involving dynamical systems, network evolution models, and parameter estimation for protein interaction networks.
This document summarizes results on analyzing stochastic gradient descent (SGD) algorithms for minimizing convex functions. It shows that a continuous-time version of SGD (SGD-c) can strongly approximate the discrete-time version (SGD-d) under certain conditions. It also establishes that SGD achieves the minimax optimal convergence rate of O(t^-1/2) for α=1/2 by using an "averaging from the past" procedure, closing the gap between previous lower and upper bound results.
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 provides an overview of advanced econometrics techniques including simulations, bootstrap methods, and penalization. It discusses how computers allow for numerical standard errors and testing procedures through simulations and resampling rather than relying on asymptotic formulas. Specific techniques covered include the linear regression model, nonlinear transformations, asymptotics versus finite samples using bootstrap, and moving from least squares to other regressions like quantile regression. Historical references for techniques like permutation methods, the jackknife, and bootstrapping are also provided.
My data are incomplete and noisy: Information-reduction statistical methods f...Umberto Picchini
We review parameter inference for stochastic modelling in complex scenario, such as bad parameters initialization and near-chaotic dynamics. We show how state-of-art methods for state-space models can fail while, in some situations, reducing data to summary statistics (information reduction) enables robust estimation. Wood's synthetic likelihoods method is reviewed and the lecture closes with an example of approximate Bayesian computation methodology.
Accompanying code is available at https://github.com/umbertopicchini/pomp-ricker and https://github.com/umbertopicchini/abc_g-and-k
Readership lecture given at Lund University on 7 June 2016. The lecture is of popular science nature hence mathematical detail is kept to a minimum. However numerous links and references are offered for further reading.
Inference for stochastic differential equations via approximate Bayesian comp...Umberto Picchini
Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/
A likelihood-free version of the stochastic approximation EM algorithm (SAEM)...Umberto Picchini
I show how to obtain approximate maximum likelihood inference for "complex" models having some latent (unobservable) component. With "complex" I mean models having a so-called intractable likelihood, where the latter is unavailable in closed for or is too difficult to approximate. I construct a version of SAEM (and EM-type algorithm) that makes it possible to conduct inference for complex models. Traditionally SAEM is implementable only for models that are fairly tractable analytically. By introducing the concept of synthetic likelihood, where information is captured by a series of user-defined summary statistics (as in approximate Bayesian computation), it is possible to automatize SAEM to run on any model having some latent-component.
This document discusses density operators and their use in quantum information and computing. It begins by introducing density operators and how they can be used to describe quantum systems whose state is not precisely known or composite systems. The key properties of density operators are that they must have a trace of 1 and be positive operators. The document then covers reduced density operators which describe subsystems by taking the partial trace. Finally, it discusses how the reduced density operator gives the correct measurement statistics for observations on a subsystem.
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.
This document discusses compactness estimates for nonlinear partial differential equations (PDEs), specifically Hamilton-Jacobi equations. It provides background on Kolmogorov entropy measures of compactness and covers recent results estimating the Kolmogorov entropy of solutions to scalar conservation laws and Hamilton-Jacobi equations, showing it is on the order of 1/ε. The document outlines applications of these estimates and open questions regarding extending the estimates to non-convex fluxes and non-uniformly convex Hamiltonians.
The document provides an overview of the EM algorithm and its application to outlier detection. It begins with introducing the EM algorithm and explaining its iterative process of estimating parameters via E-step and M-step. It then proves properties of the EM algorithm such as non-decreasing log-likelihood and convergence. An example of using EM for Gaussian mixture modeling is provided. Finally, the document discusses directly and indirectly applying EM to outlier detection.
Optimal order a posteriori error bounds in L∞(L2) norm are derived for semidiscrete semilinear parabolic problems. Standard continuous Galerkin (conforming) finite element method is employed. Our main tools in deriving these error estimates are the elliptic reconstruction technique which is first introduced by Makridakis and Nochetto [5], with the aid of Gronwall’s lemma and continuation argument.
This document discusses macrocanonical models for texture synthesis. It begins by introducing the goal of texture synthesis and providing a brief history. It then describes the parametric question of combining randomness and structure in images. Specifically, it discusses maximizing entropy under geometric constraints. The document goes on to discuss links to statistical physics, defining microcanonical and macrocanonical models. It focuses on studying the macrocanonical model, describing how to find optimal parameters through gradient descent and how to sample from the model using Langevin dynamics. The document provides examples of texture synthesis and compares results to other methods.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
Approximate Bayesian computation for the Ising/Potts modelMatt Moores
This document provides an introduction to Approximate Bayesian Computation (ABC). ABC is a likelihood-free method for approximating posterior distributions when the likelihood function is intractable or expensive to evaluate. The document outlines the basic ABC rejection sampling algorithm and discusses extensions like using summary statistics, ABC-MCMC, and ABC sequential Monte Carlo. It also applies ABC to parameter inference for a hidden Potts model used in Bayesian image segmentation.
EM algorithm and its application in probabilistic latent semantic analysiszukun
The document discusses the EM algorithm and its application in Probabilistic Latent Semantic Analysis (pLSA). It begins by introducing the parameter estimation problem and comparing frequentist and Bayesian approaches. It then describes the EM algorithm, which iteratively computes lower bounds to the log-likelihood function. Finally, it applies the EM algorithm to pLSA by modeling documents and words as arising from a mixture of latent topics.
The document discusses two main topics:
1) The electrostatic interpretation of zeros of orthogonal polynomials, which describes how the zeros represent equilibrium positions of charged particles. It examines how canonical perturbations like Christoffel, Uvarov, and Geronimus transformations affect the zeros.
2) The asymptotic behavior of ratios of Laguerre orthogonal polynomials. It provides strong asymptotic expansions that are valid in different regions of the complex plane. It also examines the asymptotics of ratios of Laguerre polynomials.
short course at CIRM, Bayesian Masterclass, October 2018Christian Robert
Markov Chain Monte Carlo (MCMC) methods generate dependent samples from a target distribution using a Markov chain. The Metropolis-Hastings algorithm constructs a Markov chain with a desired stationary distribution by proposing moves to new states and accepting or rejecting them probabilistically. The algorithm is used to approximate integrals that are difficult to compute directly. It has been shown to converge to the target distribution as the number of iterations increases.
We are interested in finding a permutation of the entries of a given square matrix so that the maximum number of its nonzero entries are moved to one of the corners in a L-shaped fashion.
If we interpret the nonzero entries of the matrix as the edges of a graph, this problem boils down to the so-called core–periphery structure, consisting of two sets: the core, a set of nodes that is highly connected across the whole graph, and the periphery, a set of nodes that is well connected only to the nodes that are in the core.
Matrix reordering problems have applications in sparse factorizations and preconditioning, while revealing core–periphery structures in networks has applications in economic, social and communication networks.
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 discusses using graphics processing units (GPUs) to perform approximate Bayesian computation (ABC) for parameter estimation of complex models. It describes how GPUs are well-suited for ABC due to their ability to perform linear computations on many threads in parallel. The document provides examples of applying ABC to GPUs for problems involving dynamical systems, network evolution models, and parameter estimation for protein interaction networks.
This document summarizes results on analyzing stochastic gradient descent (SGD) algorithms for minimizing convex functions. It shows that a continuous-time version of SGD (SGD-c) can strongly approximate the discrete-time version (SGD-d) under certain conditions. It also establishes that SGD achieves the minimax optimal convergence rate of O(t^-1/2) for α=1/2 by using an "averaging from the past" procedure, closing the gap between previous lower and upper bound results.
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 provides an overview of advanced econometrics techniques including simulations, bootstrap methods, and penalization. It discusses how computers allow for numerical standard errors and testing procedures through simulations and resampling rather than relying on asymptotic formulas. Specific techniques covered include the linear regression model, nonlinear transformations, asymptotics versus finite samples using bootstrap, and moving from least squares to other regressions like quantile regression. Historical references for techniques like permutation methods, the jackknife, and bootstrapping are also provided.
1) The document discusses simulation-based techniques and the bootstrap method for economics.
2) It provides historical references for permutation methods dating back to Fisher (1935) and jackknife and bootstrapping which started with Monte Carlo algorithms in the 1940s.
3) Bootstrapping is introduced as an asymptotic refinement based on computer simulations that generates additional samples from the original data to reduce uncertainty, rather than collecting more observations.
Wavelet-based Reflection Symmetry Detection via Textural and Color HistogramsMohamed Elawady
This document presents a methodology for wavelet-based reflection symmetry detection using textural and color histograms. It extracts multiscale edge segments using Log-Gabor filters and measures symmetry based on edge orientations, local texture histograms, and color histograms. Evaluation on public datasets shows it outperforms previous methods in detecting single and multiple symmetries, with quantitative and qualitative results presented. Future work could improve the detection using continuous maximal-seeking.
1) Mathematicians use statistical models to predict future trends and values based on historical data. Both continuous and discrete univariate and multivariate models are explored.
2) Specific models examined include the Ornstein-Uhlenbeck process, Euler-Maruyama and Milstein schemes for numerical approximations of continuous processes, and autoregressive AR(p) models for discrete processes.
3) The models are fitted to inflation rate data to predict future inflation values based on parameter estimation techniques like maximum likelihood estimation. Model outputs like predicted values and distributions are examined.
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.
Large strain solid dynamics in OpenFOAMJibran Haider
The document describes a numerical methodology for simulating large strain solid dynamics using OpenFOAM. It proposes using a total Lagrangian formulation and first-order conservation laws similar to computational fluid dynamics to model solid mechanics problems involving large deformations. A cell-centered finite volume method is used for spatial discretization along with Riemann solvers and linear reconstruction to capture fluxes. A two-stage Runge-Kutta scheme is employed for time integration. Results are presented demonstrating the method's ability to handle problems involving mesh convergence, enhanced reconstruction, highly nonlinear behavior, plasticity, contact, unstructured meshes, and complex geometries.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
Large strain computational solid dynamics: An upwind cell centred Finite Volu...Jibran Haider
Presented our research at the 12th World Congress on Computational Mechanics (WCCM) and 6th Asia Pacific Congress on Computational Mechanics (APCOM) at the COEX Convention Center in Seoul, Korea.
1) Machine learning draws on areas of mathematics including probability, statistical inference, linear algebra, and optimization theory.
2) While there are easy-to-use machine learning packages, understanding the underlying mathematics is important for choosing the right algorithms, making good parameter and validation choices, and interpreting results.
3) Key concepts in probability and statistics that are important for machine learning include random variables, probability distributions, expected value, variance, covariance, and conditional probability. These concepts allow quantification of relationships and uncertainties in data.
A copula model to analyze minimum admission scores Mariela Fernández
This document discusses using a copula model to analyze minimum admission scores. It introduces the motivation to set minimum scores efficiently using E[Language|Mathematics ≥ m0] and E[Mathematics|Language ≥ l0]. It then provides an overview of copula theory and defines the asymmetric cubic section copula used in the model. The document applies the copula model to admission data from 2010-2011 and examines the results, including final remarks on future work.
A Numerical Method For Solving The Problem U T - Delta F (U) 0Kim Daniels
This document presents a numerical method for solving nonlinear evolution equations of the form ut - Δf(u) = 0. The method involves solving a corresponding linear problem and performing simple algebraic operations at each time step. It is proven that the method is stable and convergent. Numerical experiments demonstrating the method are also presented. The method generalizes an existing technique for solving the Stefan problem and is related to another existing method. The document provides details on implementing the method to solve a specific nonlinear evolution equation with Dirichlet boundary conditions.
Sequential and parallel algorithm to find maximum flow on extended mixed netw...csandit
The problem of finding maximum flow in network
graph is extremely interesting and
practically applicable in many fields in our daily
life, especially in transportation. Therefore, a
lot of researchers have been studying this problem
in various methods. Especially in 2013, we
has developed a new algorithm namely, postflow-pull
algorithm to find the maximum flow on
traditional networks. In this paper, we revi
sed postflow-push methods to solve this
problem of finding maximum flow on extended mixed
network. In addition, to take more
advantage of multi-core architecture of t
he parallel computing system, we build this
parallel algorithm. This is a completely new method
not being announced in the world. The
results of this paper are basically systematized an
d proven. The idea of this algorithm is using
multi processors to work in parallel by postflow_p
ush algorithm. Among these processors,
there is one main processor managing data, sending
data to the sub processors, receiving data
from the sub-processors. The sub-processors simulta
neously execute their work and send their
data to the main processor until the job is finishe
d, the main processor will show the results of
the problem.
SEQUENTIAL AND PARALLEL ALGORITHM TO FIND MAXIMUM FLOW ON EXTENDED MIXED NETW...cscpconf
This document summarizes an algorithm for finding maximum flow on extended mixed networks using revised postflow-push methods and parallel processing. The key points are:
1. The authors revise postflow-push methods to solve the maximum flow problem on extended mixed networks, which model problems more accurately than traditional networks.
2. To take advantage of parallel computing, the authors develop a parallel algorithm where multiple processors work simultaneously using postflow-push until the problem is solved.
3. The algorithm balances inflow and outflow at vertices by pushing flow along outgoing edges and against incoming edges in the residual network until no unbalanced vertices remain.
Markov chain Monte Carlo methods and some attempts at parallelizing themPierre Jacob
Markov chain Monte Carlo (MCMC) methods are commonly used to approximate properties of target probability distributions. However, MCMC estimators are generally biased for any fixed number of samples. The document discusses various techniques for constructing unbiased estimators from MCMC output, including regeneration, sequential Monte Carlo samplers, and coupled Markov chains. Specifically, running two Markov chains in parallel and taking the difference in their values at meeting times can yield an unbiased estimator, though certain conditions must hold.
An Importance Sampling Approach to Integrate Expert Knowledge When Learning B...NTNU
The introduction of expert knowledge when learning Bayesian Networks from data is known to be an excellent approach to boost the performance of automatic learning methods, specially when the data is scarce. Previous approaches for this problem based on Bayesian statistics introduce the expert knowledge modifying the prior probability distributions. In this study, we propose a new methodology based on Monte Carlo simulation which starts with non-informative priors and requires knowledge from the expert a posteriori, when the simulation ends. We also explore a new Importance Sampling method for Monte Carlo simulation and the definition of new non-informative priors for the structure of the network. All these approaches are experimentally validated with five standard Bayesian networks.
Read more:
http://link.springer.com/chapter/10.1007%2F978-3-642-14049-5_70
An Exact Exponential Branch-And-Merge Algorithm For The Single Machine Total ...Joe Andelija
This document summarizes a research paper that proposes a new exact exponential branch-and-merge algorithm for solving the single machine total tardiness scheduling problem. The algorithm improves upon the best known dynamic programming approach that has a time complexity of O*(2n) by avoiding redundant computations through a node merging operation. The branch-and-merge technique decomposes the problem when assigning the longest job to different positions, generating two subproblems to schedule jobs before and after that job. It achieves a time complexity that converges to O*(2n) while keeping polynomial space complexity, improving the state-of-the-art for this NP-hard problem.
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.
This document presents a novel algebraic method called multidimensional rank reduction estimator (MD RARE) for simultaneously estimating parameters of a parametric MIMO channel model from channel sounder measurements. The MD RARE algorithm estimates propagation delay, direction of arrival, and other parameters sequentially by exploiting the Vandermonde structure of the data model. This reduces dimensionality and complexity compared to existing methods. The performance of MD RARE is illustrated using simulated and measured data from a vector channel sounder.
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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
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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.
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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 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.
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Hamilton-Jacobi equation on networks: generalized Lax-Hopf formula
1. Hamilton-Jacobi equations on networks:
Generalized Lax-Hopf formula
Guillaume Costeseque
(PhD with supervisors R. Monneau & J-P. Lebacque)
Universit´e Paris Est, Ecole des Ponts ParisTech & IFSTTAR
Pr´esentation S´eminaire “R´eseaux” - Institut Henri Poincar´e,
June 03, 2014
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 1 / 62
2. Flows on a network
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 2 / 62
3. Flows on a network
A network is like a (oriented) graph
made of edges and vertices
Examples: traffic flow, gas pipelines,
blood vessels, shallow water, internet
communications...
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 3 / 62
4. Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traffic
6 Concluding remarks
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 4 / 62
5. Introduction
Motivation
Classical approaches:
Macroscopic modeling on (homogeneous) sections
Coupling conditions at (pointwise) junction
For instance, consider
ρt + (Q(ρ))x = 0, scalar conservation law,
ρ(., t = 0) = ρ0(.), initial conditions,
ψ(ρ(x = 0−, t), ρ(x = 0+, t)) = 0, coupling condition.
(1)
See Garavello, Piccoli [5], Lebacque, Khoshyaran [9] and Bressan et al. [2]
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 5 / 62
6. Introduction
Motivation
Our aim
Macroscopic modeling in a single framework
Implicit condition at junctions
Analytical solution & tractable numerical methods
Our tools: Hamilton-Jacobi equations (HJ)
ut + H(x, t, Du) = 0
Class of nonlinear first-order Partial Differential Equations (PDEs) that
naturally arise from mechanics (see Arnold’s book)
Arnol’d, Vladimir Igorevich, Mathematical methods of classical mechanics,
Vol. 60. Springer, 1989.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 6 / 62
7. “First” attempt: Imbert-Monneau’s framework (2010-2014)
Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traffic
6 Concluding remarks
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 7 / 62
8. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Motivation: the simple divergent road
x > 0
x > 0γl
γrx < 0
Il
Ir
γe
Ie
γe = 1,
0 ≤ γl , γr ≤ 1,
γl + γr = 1
LWR model [Lighthill, Whitham ’55; Richards ’56]:
ρt + (Q(ρ))x = 0
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 8 / 62
9. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Qmax
ρcrit ρmax
Density ρ
Flow Q(ρ)
Q(ρ) = ρV (ρ) with V (ρ) = speed of the equilibrium flow
DFs
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 9 / 62
10. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Getting the Hamilton-Jacobi equation
LWR model on each branch (outside the junction point)
ρα
t + (Qα
(ρα
))x = 0 on branch α
Primitive:
Uα(x, t) = Uα(0, t) +
1
γα
x
0
ρα
(y, t)dy,
Uα(0, t) = g(t) = index of the single car at the junction point
x > 0
x > 09
11
8
10
12
6420 1 3 5
7
−1
x < 0
Uα
t +
1
γα
Qα
(γα
Uα
x ) = g′
(t) +
1
γα
Qα
(ρα
(0, t))
= 0 for a good choice of g
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 10 / 62
11. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Settings
JN
J1
J2
branch Jα
x
x
0
x
x
New functions uα:
uα(x, t) = −Uα(x, t), x > 0, for outgoing roads
uα(x, t) = −Uα(−x, t), x > 0
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 11 / 62
12. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Junction model
Proposition (Junction model)
That leads to the following junction model (see [8, 7])
uα
t + Hα(uα
x ) = 0, x > 0, α = 1, . . . , N
uα = uβ =: u, x = 0,
ut + H(u1
x , . . . , uN
x ) = 0, x = 0
(2)
with initial condition uα(0, x) = uα
0 (x) and
H(u1
x , . . . , uN
x ) = max L
Flux limiter
, max
α=1,...,N
H−
α (uα
x )
Minimum between
demand and supply
Imbert, C., and R. Monneau, Level-set convex Hamilton-Jacobi equations on
networks, preprint (2014).
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 12 / 62
13. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Pros & Cons
Good mathematical properties (existence and uniqueness of the
viscosity solution)...
...under weak assumptions on the Hamiltonian (“bi-monotone”
instead of strictly convex)
Possibility to compute the solution thanks to convergent finite
difference scheme [4]
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 13 / 62
14. “First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Pros & Cons
Good mathematical properties (existence and uniqueness of the
viscosity solution)...
...under weak assumptions on the Hamiltonian (“bi-monotone”
instead of strictly convex)
Possibility to compute the solution thanks to convergent finite
difference scheme [4]
No explicit / analytical solution (Lax-Hopf formula)
Not very satisfactory in traffic: (γα)α fixed!
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 13 / 62
15. “First” attempt: Imbert-Monneau’s framework (2010-2014) Numerical scheme
Presentation of the scheme
Proposition (Numerical Scheme)
Let us consider the discrete space and time derivatives:
pα,n
i :=
Uα,n
i+1 − Uα,n
i
∆x
and (DtU)α,n
i :=
Uα,n+1
i − Uα,n
i
∆t
Then we have the following numerical scheme:
(DtU)α,n
i + max{H+
α (pα,n
i−1), H−
α (pα,n
i )} = 0, i ≥ 1
Un
0 := Uα,n
0 , i = 0, α = 1, ..., N
(DtU)n
0 + max
α=1,...,N
H−
α (pα,n
0 ) = 0, i = 0
(3)
With the initial condition Uα,0
i := uα
0 (i∆x).
∆x and ∆t = space and time steps satisfying a CFL condition
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 14 / 62
16. “First” attempt: Imbert-Monneau’s framework (2010-2014) Numerical scheme
CFL condition
The natural CFL condition is given by:
∆x
∆t
≥ sup
α=1,...,N
i≥0, 0≤n≤nT
|H′
α(pα,n
i )| (4)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 15 / 62
17. “First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Basic assumptions
For all α = 1, . . . , N,
(A0) The initial condition uα
0 is Lipschitz continuous.
(A1) The Hamiltonians Hα are C1(R) and convex such that:
p
H−
α (p) H+
α (p)
pα
0
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 16 / 62
18. “First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Gradient estimates
Theorem (Time and Space Gradient estimates)
Assume (A0)-(A1). If the CFL condition (4) is satisfied, then we have
that:
(i) Considering Mn = sup
α,i
(DtU)α,n
i and mn = inf
α,i
(DtU)α,n
i , we have the
following time derivative estimate:
m0
≤ mn
≤ mn+1
≤ Mn+1
≤ Mn
≤ M0
(ii) Considering pα
= (H−
α )−1(−m0) and pα = (H+
α )−1(−m0), we have
the following gradient estimate:
pα
≤ pα,n
i ≤ pα, for all i ≥ 0, n ≥ 0 and α = 1, ..., N
Proof
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 17 / 62
19. “First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Stronger CFL condition
−m0
pα
p
Hα(p)
pα
As for any α = 1, . . . , N, we have that:
pα
≤ pα,n
i ≤ pα for all i, n ≥ 0
Then the CFL condition becomes:
∆x
∆t
≥ sup
α=1,...,N
pα∈[pα
,pα]
|H′
α(pα)| (5)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 18 / 62
20. “First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 19 / 62
21. “First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α
Theorem (Existence and uniqueness [IMZ, ’11])
Under (A0)-(A1)-(A2), there exists a unique viscosity solution u of (2) on
the junction, satisfying for some constant CT > 0
|u(t, y) − u0(y)| ≤ CT for all (t, y) ∈ JT .
Moreover the function u is Lipschitz continuous with respect to (t, y).
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 19 / 62
22. “First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Convergence
Theorem (Convergence from discrete to continuous [CML, ’13])
Assume that (A0)-(A1)-(A2) and the CFL condition (5) are satisfied.
Then the numerical solution converges uniformly to u the unique viscosity
solution of (2) when ε → 0, locally uniformly on any compact set K:
lim sup
ε→0
sup
(n∆t,i∆x)∈K
|uα
(n∆t, i∆x) − Uα,n
i | = 0
Proof
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 20 / 62
23. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Setting
J1
JNI
JNI +1
JNI +NO
x < 0 x = 0 x > 0
Jβ
γβ Jλ
γλ
NI incoming and NO outgoing roads
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 21 / 62
24. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Car densities
The car density ρα solves the LWR equation on branch α:
ρα
t + (Qα
(ρα
))x = 0
By definition
ρα
= γα
∂x Uα
on branch α
And
uα(x, t) = −Uα(−x, t), x > 0, for incoming roads
uα(x, t) = −Uα(x, t), x > 0, for outgoing roads
where the car index uα solves the HJ equation on branch α:
uα
t + Hα
(uα
x ) = 0, for x > 0
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 22 / 62
25. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Flow
Hα(p) :=
−
1
γα
Qα(γαp) for α = 1, ..., NI
−
1
γα
Qα(−γαp) for α = NI + 1, ..., NI + NO
Incoming roads Outgoing roads
ρcrit
γα
ρmax
γα
p
−
Qmax
γα
p
−
Qmax
γα
HαHα
H−
α H−
α H+
αH+
α
−
ρmax
γα
−
ρcrit
γα
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 23 / 62
26. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Links with “classical” approach
Definition (Discrete car density)
The discrete car density ρα,n
i with n ≥ 0 and i ∈ Z is given by:
ρα,n
i :=
γαpα,n
|i|−1 for α = 1, ..., NI , i ≤ −1
−γαpα,n
i for α = NI + 1, ..., NI + NO, i ≥ 0
(6)
J1
JNI
JNI +1
JNI +NO
x < 0 x > 0
−2
−1
2
1
0
−2
−2
−1
−1
1
1
2
2
Jβ
Jλ
ρλ,n
1
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 24 / 62
27. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Traffic interpretation
Proposition (Scheme for vehicles densities)
The scheme deduced from (3) for the discrete densities is given by:
∆x
∆t
{ρα,n+1
i − ρα,n
i } =
Fα(ρα,n
i−1, ρα,n
i ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0, −1
Fα
0 (ρ·,n
0 ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0
Fα(ρα,n
i−1, ρα,n
i ) − Fα
0 (ρ·,n
0 ) for i = −1
With
Fα(ρα,n
i−1, ρα,n
i ) := min Qα
D(ρα,n
i−1), Qα
S (ρα,n
i )
Fα
0 (ρ·,n
0 ) := γα min min
β≤NI
1
γβ
Qβ
D(ρβ,n
0 ), min
λ>NI
1
γλ
Qλ
S (ρλ,n
0 )
incoming outgoing
ρλ,n
0ρβ,n
−1ρβ,n
−2 ρλ,n
1
x
x = 0
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 25 / 62
28. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Supply and demand functions
Remark
It recovers the seminal Godunov scheme with passing flow = minimum
between upstream demand QD and downstream supply QS.
Density ρ
ρcrit ρmax
Supply QS
Qmax
Density ρ
ρcrit ρmax
Flow Q
Qmax
Density ρ
ρcrit
Demand QD
Qmax
From [Lebacque ’93, ’96]
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 26 / 62
29. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Supply and demand VS Hamiltonian
H−
α (p) =
−
1
γα
Qα
D(γαp) for α = 1, ..., NI
−
1
γα
Qα
S (−γαp) for α = NI + 1, ..., NI + NO
And
H+
α (p) =
−
1
γα
Qα
S (γαp) for α = 1, ..., NI
−
1
γα
Qα
D(−γαp) for α = NI + 1, ..., NI + NO
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 27 / 62
30. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Some references for conservation laws
ρt + (Q(x, ρ))x = 0 with Q(x, p) = 1{x<0}Qin
(p) + 1{x≥0}Qout
(p)
Uniqueness results only for restricted configurations:
See [Garavello, Natalini, Piccoli, Terracina ’07]
and [Andreianov, Karlsen, Risebro ’11]
Book of [Garavello, Piccoli ’06] 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 network
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 28 / 62
31. “First” attempt: Imbert-Monneau’s framework (2010-2014) Traffic interpretation
Numerics on networks
Godunov scheme mainly used for conservation laws:
[Bretti, Natalini, Piccoli ’06, ’07]: Godunov scheme compared to
kinetic schemes / fast algorithms
[Blandin, Bretti, Cutolo, Piccoli ’09]: Godunov scheme adapted for
Colombo model (only tested for 1 × 1 junctions)
For Hamilton-Jacobi equations on networks:
[G¨ottlich, Ziegler, Herty ’13]: Lax-Freidrichs scheme outside the
junction + coupling conditions (density) at the junction
[Han, Piccoli, Friesz, Yao ’12]: Lax-Hopf formula for HJ equation
coupled with a Riemann solver at junction
[Camilli, Festa, Schieborn ’13]: semi-Lagrangian scheme only
designed for Eikonal equations
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 29 / 62
32. Lax-Hopf: classical approach
Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traffic
6 Concluding remarks
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 30 / 62
33. Lax-Hopf: classical approach
Basic idea
Consider the simplest first order Hamilton-Jacobi equation
ut + H(Du) = 0 (7)
with u : Rn × (0, +∞) → R and H : Rn → R.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 31 / 62
34. Lax-Hopf: classical approach
Basic idea
Family of simple linear functions
ˆu(x, t) = α.x − tH(α) + β, for any α ∈ Rn
, β ∈ R
which are solutions of (7).
Idea: build more general solutions,
taking an envelope of elementary solutions (E. Hopf 1965 [6])
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 32 / 62
35. Lax-Hopf: classical approach
Lax-Hopf formulæ
Consider the Cauchy problem
ut + H(Du) = 0, in Rn × (0, +∞),
u(., 0) = u0(.), on Rn.
(8)
Two cases according to the assumptions on the smoothness of
the Hamiltonian H and
the initial data u0
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 33 / 62
36. Lax-Hopf: classical approach
Lax-Hopf formulæ
Assumptions: case 1
(A1) H : Rn → R is convex
(A2) u0 : Rn → R is uniformly Lipschitz
Theorem (First Lax-Hopf formula)
If (A1)-(A2) hold true, then
u(x, t) := inf
z∈Rn
sup
y∈Rn
[u0(z) + y.(x − z) − tH(y)] (9)
is the unique uniformly continuous viscosity solution of (8).
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 34 / 62
37. Lax-Hopf: classical approach
Notice that analogue formula of (9) has been used by
Hopf in 1950 for a specific hyperbolic equation
ut + uux = µuxx
Lax in 1957 for hyperbolic systems
and Oleinik in 1963
From where the name “Lax-Hopf-Oleinik” formula
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 35 / 62
38. Lax-Hopf: classical approach
Lax-Hopf formulæ
(Continued)
Assumptions: case 2
(A3) H : Rn → R is continuous
(A4) u0 : Rn → R is uniformly Lipschitz and convex
Theorem (Second Lax-Hopf formula)
If (A3)-(A4) hold true, then
u(x, t) := sup
y∈Rn
inf
z∈Rn
[u0(z) + y.(x − z) − tH(y)] (10)
is the unique uniformly continuous viscosity solution of (8).
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 36 / 62
39. Lax-Hopf: classical approach
Proofs of Lax-Hopf formulæ are viscosity solutions
Classical results: existence and uniqueness of viscosity solution of (8)
Two ways of proving that (9) and (10) are viscosity solutions of (8)
PDE approach (sub- and supersolution inequalities)
Optimal control approach (dynamic programming equation)
Bardi, M., and L. C. Evans, On Hopf’s formulas for solutions of
Hamilton-Jacobi equations, Nonlinear Analysis: Theory, Methods &
Applications 8.11 (1984): 1373-1381.
Evans, L. C., Partial differential equations, Graduate studies in
mathematics, vol. 19, American Mathematics Society, 2009
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 37 / 62
40. Lax-Hopf: classical approach
Some naive remarks
First Lax-Hopf formula (9) can be recast as
u(x, t) := inf
z∈Rn
u0(z) − tH∗ x − z
t
thanks to Legendre-Fenchel transform
L(z) = H∗
(z) := sup
y∈Rn
(y.z − H(y)) .
If H is strictly convex, 1-coercive i.e. lim
|p|→∞
H(p)
|p|
= +∞,
then H∗ is also convex and
(H∗
)∗
= H.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 38 / 62
41. Lax-Hopf: classical approach
Legendre-Fenchel transform
Illustration for the density-flow Fundamental Diagram
F(p) = −H(−p), for any p ∈ R,
and
M(q) := sup
ρ
[F(ρ) − ρq] , for any q ∈ R
M(q)
u
w
Density ρ
q
q
Flow F
w u
q
Transform M
−wρmax
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 39 / 62
42. Lax-Hopf: classical approach
Interpretation of Lax-Hopf formula
Consider the cost function c that embeds
initial conditions (at time t = 0),
boundary conditions (at locations xmin, xmax),
internal conditions (along trajectories x∗(t) e.g.)
Theorem
If (A1), (A4) hold true
Then Lax-Hopf formulæ (9) and (10) are equivalent
and we have
u(xT , T) := inf
(x0,t0)∈Rn×(0,T)
c(x0, t0) − (T − t0)H∗ xT − x0
T − t0
. (11)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 40 / 62
43. Lax-Hopf: classical approach
Interpretation of Lax-Hopf formula
(Continued)
Lax-Hopf formula (11) reads as an optimal control problem
Time
Space
J
(T, xT )˙X(τ)
(t0, x0)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 41 / 62
44. Lax-Hopf: classical approach
Interpretation of Lax-Hopf formula
(Continued)
Time
Space
J
(T, xT )˙X(τ)
(t0, x0)
Lax-Hopf formula (11) reads as an optimal
control problem
u(xT , T) := inf
(x0,t0)∈Rn×(0,T)
infimum over all
the feasible trajectories
starting from (x0,t0)
initial cost
c(x0, t0) −(T − t0)
actualized cost
H∗ xT − x0
T − t0
mean speed of
the trajectory
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 42 / 62
45. Introducing time and space into Hamiltonian
Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traffic
6 Concluding remarks
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 43 / 62
46. Introducing time and space into Hamiltonian
Time-space dependent Hamiltonian
Simplicity: x ∈ R (i.e. n = 1)
but...
ut + H(x, t, ux ) = 0, on R × (0, +∞). (12)
Extension of the Lax-Hopf formula?
Classical formulæ (9) and (10) do not work
No simple linear solutions for (12)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 44 / 62
47. Introducing time and space into Hamiltonian
Time-space dependent Hamiltonian
Simplicity: x ∈ R (i.e. n = 1)
but...
ut + H(x, t, ux ) = 0, on R × (0, +∞). (12)
Extension of the Lax-Hopf formula?
Classical formulæ (9) and (10) do not work
No simple linear solutions for (12)
Very recent generalization: Jean-Pierre Aubin & Luxi Chen [1, 3]
through viability tools
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 44 / 62
48. Introducing time and space into Hamiltonian
Survival kit in viability theory
Dynamical systems controlled by differential inclusions
Viability kernel Capture basin
Aubin, J.-P., and A. M. Bayen, and P. Saint-Pierre, Viability theory, new
directions, Springer, 2011.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 45 / 62
49. Introducing time and space into Hamiltonian
Settings
Let (xT , T) the target point.
Denote Ω the aperture time between initial and final points
Ω := T − t0 with t0 ∈ (0, T).
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 46 / 62
50. Introducing time and space into Hamiltonian
“Coarse” Lax-Hopf formula
We still can write
u(xT , T) := inf
Ω≥0
inf
{x(.)∈Al(xT ,T)}
c(x(T − Ω), T − Ω) +
T
T−Ω
l(τ, x(τ), x′
(τ))
which is an optimal control problem very hard to solve,
optimal trajectories (characteristics) are not straight lines
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 47 / 62
51. Introducing time and space into Hamiltonian
Settings
(Continued)
We set µ the average speed on the trajectory from initial to final points
µ :=
1
Ω
T
T−Ω
x′
(τ)dτ,
and the average cost on the optimal trajectory x⋆
Γ(xT , T, Ω, µ) :=
1
Ω
T
T−Ω
l(τ, x⋆(τ), x′
⋆(τ))dτ.
(Recall that l = H∗)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 48 / 62
52. Introducing time and space into Hamiltonian
Generalized Lax-Hopf formula
Theorem (Generalized Lax-Hopf formula)
If H : (x, t, p) → H(x, t, p) is convex and lower semi-continuous w.r.t. p,
Then
u(xT , T) := inf
Ω≥0, µ∈X
[c(x(T − Ω), T − Ω) + ΩΓ(xT , T, Ω, µ)]
is the (unique) lower semi-continuous solution of
ut + H(t, x, ux ) = 0, on R × (0, +∞),
u(t, x) ≤ c(t, x), for any (t, x)
(13)
(in Barron-Jensen / Frankowska sense).
Notice that we have defined X the set of speeds such that l(µ) < ∞.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 49 / 62
53. Introducing time and space into Hamiltonian
Drawback
Actually,
Γ(xT , T, Ω, µ) := inf
x(.)∈Al(xT ,T) T
T−Ω x′(τ)dτ=Ωµ
1
Ω
T
T−Ω
l(τ, x(τ), x′
(τ))dτ
is still a optimal control problem to solve...
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 50 / 62
54. Introducing time and space into Hamiltonian
Drawback
Actually,
Γ(xT , T, Ω, µ) := inf
x(.)∈Al(xT ,T) T
T−Ω x′(τ)dτ=Ωµ
1
Ω
T
T−Ω
l(τ, x(τ), x′
(τ))dτ
is still a optimal control problem to solve...
Alternatives?
Viability kernel algorithms (see Patrick Saint-Pierre works) more
useful for simple conditions c (e.g. only Cauchy problems)
Compute some characteristics equations (system of coupled ODEs)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 50 / 62
55. Discussions: back to traffic
Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traffic
6 Concluding remarks
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 51 / 62
56. Discussions: back to traffic
Network Model
A simple case study: generalization of the three-detector problem
Shock information
u(t, x)
Kinematic information
Kinematic information = upstream demand advected by kinematic
waves moving forward
Shock information = downstream supply affected by shock waves
moving backward
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 52 / 62
57. Discussions: back to traffic
How to...
Shock information
u(t, x)
Kinematic information
Open question: How to compute the
Cumulative Flow defined as
u(t, x) := −
x
x0
ρ(t, y)dy
at point x, and time t > 0, knowing the
traffic states at different points of this
simple network?
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 53 / 62
58. Concluding remarks
Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traffic
6 Concluding remarks
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 54 / 62
59. Concluding remarks
In brief
Heightened interest in traffic modeling for Lax-Hopf formula
Potentialities of the generalized Lax-Hopf formula
Deducing an implicit traffic flow model at junction is not trivial
A lot remains to do but the impact must be important...
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 55 / 62
60. Concluding remarks
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 56 / 62
61. Complements References
Some references I
J.-P. Aubin and L. Chen, Generalized lax-hopf formulas for cournot maps and
hamilton-jacobi-mckendrik equations, (2014).
A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, Flows
on networks: recent results and perspectives, EMS Surveys in Mathematical
Sciences, (2014).
L. Chen, Computation of the” enrichment” of a value function of an optimization
problem on cumulated transaction-costs through a generalized lax-hopf formula,
arXiv preprint arXiv:1401.1610, (2014).
G. Costeseque, J.-P. Lebacque, and R. Monneau, A convergent scheme for
hamilton-jacobi equations on a junction: application to traffic, arXiv preprint
arXiv:1306.0329, (2013).
M. Garavello and B. Piccoli, Traffic flow on networks, American institute of
mathematical sciences Springfield, MO, USA, 2006.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 57 / 62
62. Complements References
Some references II
E. Hopf, Generalized solutions of non-linear equations of first order, Journal of
Mathematics and Mechanics, 14 (1965), pp. 951–973.
C. Imbert and R. Monneau, Level-set convex hamilton-jacobi equations on
networks, (2014).
C. Imbert, R. Monneau, and H. Zidani, A hamilton-jacobi approach to
junction problems and application to traffic flows, ESAIM: Control, Optimisation
and Calculus of Variations, 19 (2013), pp. 129–166.
J.-P. Lebacque and M. M. Khoshyaran, First-order macroscopic traffic flow
models: Intersection modeling, network modeling, in Transportation and Traffic
Theory. Flow, Dynamics and Human Interaction. 16th International Symposium on
Transportation and Traffic Theory, 2005.
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 58 / 62
63. Complements References
Fundamental diagram
Fundamental diagram: multi-valued in congested case
[S. Fan, M. Herty, B. Seibold, 2013], NGSIM dataset
Back
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 59 / 62
64. Complements Proofs of the main results
Sketch of the proof (gradient estimates):
Time derivative estimate:
1. Estimate on mα,n = inf
i
(DtU)α,n
i and partial result for mn = inf
α
mα,n
2. Similar estimate for Mn
3. Conclusion
Space derivative estimate:
1. New bounded Hamiltonian ˜Hα(p) for p ≤ pα
and p ≥ pα
2. Time derivative estimate from above
3. Lemma: if for any (i, n, α), (DtU)α,n
i ≥ m0 then
pα
≤ pα,n
i ≤ pα
4. Conclusion as ˜Hα = Hα on [pα
, pα]
Back
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 60 / 62
65. Complements Proofs of the main results
Convergence with uniqueness assumption
Sketch of the proof: (Comparison principle very helpful)
1. uα(t, x) := lim sup
ε
Uα,n
i is a subsolution of (2) (contradiction on
Definition inequality with a test function ϕ)
2. Similarly, uα is a supersolution of (2)
3. Conclusion: uα = uα viscosity solution of (2)
Back
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 61 / 62
66. Complements Proofs of the main results
Convergence without uniqueness assumption
Sketch of the proof: (No comparison principle)
1. Discrete Lipschitz bounds on uα
ε (n∆t, i∆x) := Uα,n
i
2. Extension by continuity of uα
ε
3. Ascoli theorem (convergent subsequence on every compact set)
4. The limit of one convergent subsequence (uα
ε )ε is super and
sub-solution of (2)
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 62 / 62