Jerome Thai

State Estimation for Polyhedral Hybrid Systems and Applications to the Godunov Scheme for Highway Traffic Estimation

This paper investigates the problem of estimating the state of discretized hyperbolic scalar partial differential equations. It uses a Godunov scheme to discretize the so-called Lighthill–Whitham–Richards equation with a triangular flux function, and proves that the resulting nonlinear dynamical system can be decomposed in a piecewise affine manner. Using this explicit representation, the system is written as a switching dynamical system, with a state space partitioned into an exponential number of polyhedra in which one mode is active. We propose a feasible approach based on the interactive multiple model (IMM) which is a widely used algorithm for estimation of hybrid systems in the scientific community. The number of modes is reduced based on the geometric properties of the polyhedral partition. The k-means algorithm is also applied on historical data to partition modes into clusters. The performance of these algorithms are compared to the extended Kalman filter and the ensemble Kalman filter in the context of Highway Traffic State Estimation. In particular, we use sparse measurements from loop detectors along a section of the I-880 to estimate the state density for our numerical experiments.

Combined state-parameter estimation for shallow water equations

In this article, a method for assimilating data into the shallow water equations when some of the model parameters are unknown is presented. The one dimensional Saint-Venant equations are used as a model of water flow in open channels. Using these equations, a nonlinear state space model is obtained. Lagrangian measurements of the flow velocity field are used as observations or measurements. These measurements may be obtained from a group of drifters equipped with GPS receivers and communication capabilities which move with the flow and report their position at every time step. Using the derived state-space model, the extended Kalman filter is used to estimate the state and the unknown model parameters given the latest measurements. The performance of the method is evaluated using data collected from an experiment performed at the USDA-ARS Hydraulic Engineering Research Unit (HERU) in Stillwater, Oklahoma in November 2009.

State Estimation for the discretized LWR PDE using explicit polyhedral representations of the Godunov scheme

This article investigates the problem of estimating the state of discretized hyperbolic scalar partial differential equations. It uses a Godunov scheme to discretize the so-called Lighthill-Whitham-Richards equation with a triangular flux function, and proves that the resulting nonlinear dynamical system can be decomposed in a piecewise affine manner. Using this explicit representation, the system is written as a switching dynamical system (hybrid system), with an exponential number of modes. The estimation problem is posed using Kalman filtering in each of the linear mode, and the approach becomes computationally tractable by tracking the mode evolution as the estimation is performed at each time step. Numerical results are presented using the Mobile Millennium data set, and compared to results obtained using ensemble Kalman filtering, which is used for estimation in traffic monitoring.

State estimation for polyhedral hybrid systems and applications to the Godunov scheme

This paper investigates the problem of estimating the state of discretized hyperbolic scalar partial differential equations. It uses a Godunov scheme to discretize the so-called Lighthill-Whitham-Richards equation with a triangular flux function, and proves that the resulting nonlinear dynamical system can be decomposed in a piecewise affine manner. Using this explicit representation, the system is written as a switching dynamical system, with a state space partitioned into an exponential number of polyhedra in which one mode is active. We propose a feasible approach based on the interactive multiple model (IMM) which is a widely used algorithm for estimation of hybrid systems in the scientific community. The number of modes is reduced based on the geometric properties of the polyhedral partition. The k-means algorithm is also applied on historical data to partition modes into clusters. The performance of these algorithms are compared to the extended Kalman filter and the ensemble Kalman filter in the context of Highway Traffic State Estimation. In particular, we use sparse measurements from loop detectors along a section of the I-880 to estimate the state density for our numerical experiments.

Negative externalities of GPS-enabled routing applications: A game theoretical approach

This work studies the impact of the increasing penetration of routing apps on road usage. Its conclusions apply both to manned vehicles in which human drivers follow app directions, and unmanned vehicles following shortest path algorithms. To address the problem caused by the increased usage of routing apps, we model two distinct classes of users, one having limited knowledge of low-capacity road links. This approach is in sharp contrast with some previous studies assuming that each user has full knowledge of the network and optimizes his/her own travel time. We show that the increased usage of GPS routing provides a lot of benefits on the road network of Los Angeles, such as decrease in average travel times and total vehicle miles traveled. However, this global increased efficiency in urban mobility has negative impacts as well, which are not addressed by the scientific community: increase in traffic in cities bordering highway from users taking local routes to avoid congestion.

A multi-convex approach to latency inference and control in traffic equilibria from sparse data

A common behavioral assumption in the modeling of traffic networks is the user equilibrium. Since traffic volumes, resulting from the rational behavior of agents, are easily but sparsely observable, and delay functions are not directly observable, we present a mathematical program with equilibrium constraint (MPEC) framework to impute the delay functions and centrally control the system from partial observations of equilibria. We also develop a novel method for solving MPECs using multi-convex optimization. Our block descent method has an intuitive interpretation, and numerical experiments demonstrate its accuracy for structural estimation, and highlight the importance of sensor placement for toll pricing.

Inverse covariance estimation from data with missing values using the Concave-Convex Procedure

We study the problem of estimating sparse precision matrices from data with missing values. We show that the corresponding maximum likelihood problem is a Difference of Convex (DC) program by proving some new concavity results on the Schur complements. We propose a new algorithm to solve this problem based on the ConCave-Convex Procedure (CCCP), and we show that the standard EM procedure is a weaker CCCP for this problem. Numerical experiments show that our new algorithm, called m-CCCP, converges much faster than EM on both synthetic and biology datasets.

Resiliency of Mobility-as-a-Service Systems to Denial-of-Service Attacks

Mobility-as-a-Service (MaaS) systems, such as ride-sharing services, have expanded very quickly over the past years. However, the popularity of MaaS systems make them increasingly vulnerable to denial-of-service (DOS) attacks, in which attackers attempt to disrupt the system to make it unavailable to the customers. Expanding on an established queuing-theoretical model for MaaS systems, attacks are modeled as a malicious control of a fraction of vehicles in the network. We then formulate a stochastic control problem that maximizes the passenger loss in the network in steady state, and solve it as a sequence of linear and quadratic programs. Combined with a Jackson network simulation and an economic model of supply and demand for attacks, we quantify how raising the cost of attacks (via cancellation fees and higher level of security) removes economical incentives for DoS attacks. Calibrating the model on 1B taxi rides, we dynamically simulate a system under attack and estimate the passenger loss under different scenarios, such as arbitrarily depleting taxis or maximizing the passenger loss. Cost of attacks of U.S.

Approximate bilevel programming via pareto optimization for imputation and control of optimization and equilibrium models

15 protects the MaaS system against DoS attacks. The contributions are thus a theoretical framework for the analysis of the network, and practical conclusions in terms of financial countermeasures to the attacks.

Projected sub-gradient with ℓ1 or simplex constraints via isotonic regression

We consider the problem of imputing the function that describes an optimization or equilibrium process from noisy partial observations of nearly optimal (possibly non-cooperative) decisions. We generalize existing inverse optimization and variational inequality problems to construct a novel class of multi-objective optimization problems: approximate bilevel programs. In this class, the “ill” nature of the complementary condition prevalent in bilevel programming is avoided, and residual functions commonly used for the design and analysis of iterative procedures, are a powerful tool to study approximate solutions to optimization and equilibrium problems. In particular, we show that duality gaps provide stronger bounds than ℓp norms of KKT residuals. The weighted criterion method is in some sense equivalent to existing formulations in the case of full observations. Our novel approach allows to solve bilevel and inverse problems under a unifying framework, via block coordinate descent, and is demonstrated on 1) consumer utility estimation and pricing and 2) latency inference in the road network of Los Angeles.