Tigran T. Tchrakian

Real-Time Traffic Flow Forecasting Using Spectral Analysis

An algorithm for the implementation of short-term prediction of traffic with real-time updating based on spectral analysis is described. The prediction is based on the characterization of the flow based on modal functions associated with a covariance matrix constructed from historical flow data. The number of these modal functions used for prediction depends on the local traffic characteristics. Although the method works well for the examples in this paper using the lower frequency modes, it can be adapted to include modes of higher frequency, as traffic conditions dictate. This paper describes the intended online implementation of the method that predicts within-day traffic flow using a forecasting horizon of 1 h 15 min with a 15-min step. Thus, every 15 min, the traffic flow for a further 1 h 15 min is predicted. As well as forecasting to this horizon, a second algorithm incorporating a weighted averaging technique is developed, which allows the prediction of one 15-min step ahead by using current and previous predictions of traffic flows at the given time instant while placing more weight on the more recent predictions. This technique combines the features of a time-series-based prediction with spectral analysis. The development of an algorithm for the real-time implementation is described, and results are presented for a number of different schemes.

A macroscopic traffic data assimilation framework based on Fourier-Galerkin method and minimax estimation

In this paper, we propose a new framework for macroscopic traffic state estimation based on the Fourier-Galerkin projection method and minimax state estimation approach. We assign a Fourier-Galerkin reduced model to a partial differential equation describing a macroscopic model of traffic flow. Taking into account a priori estimates for the projection error, we apply the minimax method to construct the state estimate for the reduced model that gives us, in turn, the estimate of the Fourier-Galerkin coefficients associated with a solution if the original macroscopic model. We illustrate our approach with a numerical example.

Parked cars as a service delivery platform

We introduce a new view of parked cars as a massive, flexible resource that is currently wasted. Given the power supply in batteries as well as computing, communication, and sensing facilities in cars in conjunction with the precise localization they can provide, parked cars have the potential to serve as a service delivery platform with a wide range of possibilities. We describe diverse applications that can be implemented using parked cars to show the flexibility of the infrastructure. Potential user groups and service providers are discussed. As an illustrative example, a simulation study of the use case of localizing persons in need of assistance is presented. Finally, the need for new algorithms and their analysis adapted to the specifics of parked cars is also highlighted.

Parameter estimation for Euler equations with uncertain inputs

The paper presents a new state estimation algorithm for 2D incompressible Euler equations with periodic boundary conditions and uncertain but bounded inputs and initial conditions. The algorithm converges (in L2-sense) to a least squares estimator given incomplete and noisy observations. The results are illustrated by numerical examples.

A Macroscopic Traffic Data-Assimilation Framework Based on the Fourier–Galerkin Method and Minimax Estimation

In this paper, we propose a new framework for macroscopic traffic state estimation. Our approach is a robust “discretize” then “optimize” strategy, based on the Fourier–Galerkin projection method and minimax state estimation. We assign a Fourier–Galerkin reduced model to a macroscopic model of traffic flow, described by a hyperbolic partial differential equation. Taking into account a priori estimates for the projection error, we apply the minimax method to construct the state estimate for the reduced model that gives us, in turn, the estimate of the Fourier–Galerkin coefficients associated with a solution of the original macroscopic model. We illustrate our approach with a numerical example that demonstrates its shock capturing capability using only sparse measurements and under high uncertainty in initial conditions. We present implementation details for our algorithm, as well as a comparison of our method against the ensemble Kalman filter applied to a “local” discretization of the same traffic flow model.

Filter comparison for estimation on discretized PDEs modeling traffic: Ensemble Kalman Filter and minimax filter

Traffic State Estimation (TSE) refers to the estimation of the state (i.e., density, speed, or other parameters) of vehicular traffic on roads based on partial observation data (e.g., road-side detectors and on-vehicle GPS devices). It can be used as a component in applications such as traffic control systems as a means to identify and alleviate congestion. In existing studies, the Kalman Filter and its extensions, notably the Ensemble Kalman Filter (EnKF), are commonly employed for TSE. Recently, the MF has been newly adapted to this domain as a filtering algorithm for TSE. In this article, we compare the performance of the EnKF and the MF on discretized PDE models of traffic flow traffic. Specifically, for the EnKF study, the estimation is performed using stationary and mobile sensor information based on actual traffic data, by employing EnKF in conjunction with a Godunov discretization of the Lighthill-Whitham-Richards (LWR) model. For the minimax study, the discontinuous Galerkin formulation of the LWR model is used in conjunction with the implicitly-linearized MF to obtain state estimates using the same data. The advantages and disadvantages of each of the filters are empirically quantified. Insights for practical application and future research directions are presented.

Robust Traffic Density Estimation Using Discontinuous Galerkin Formulation of a Macroscopic Model

In this paper, we develop a data-assimilation algorithm for a macroscopic model of traffic flow. The algorithm is based on the Discontinuous Galerkin Method and Minimax Estimation, and is applied to a macroscopic model based on a scalar conservation law. We present numerical results which demonstrate the shock-capturing capability of the algorithm under high uncertainty in the initial traffic condition, using only sparse measurements, and under time-dependent boundary conditions. The latter makes it possible for estimation to be performed on merge/diverge sections, allowing the possibility of the deployment of the algorithm to road networks.

Exponentially convergent data assimilation algorithm for Navier-Stokes equations

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier-Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in ℝ2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data.

On Source-Term Parameter Estimation for Linear Advection-Diffusion Equations with Uncertain Coefficients

In this paper, we propose an algorithm for estimating parameters of a source term of a linear advection-diffusion equation with an uncertain advection-velocity field. First, we apply a minimax state estimation technique in order to reduce uncertainty introduced by the coefficients. Then we design a source localization algorithm which uses the state estimator as a model and estimates the parameters of the source term given incomplete and noisy data. The principal novelty of the proposed algorithm is in that it is robust with respect to uncertainty in advection coefficients. The localization algorithm is sequential; that is, it updates both the state estimate and the source estimate once a new observation arrives. To demonstrate the efficacy of the proposed algorithm, we present a numerical example of source localization in two spatial dimensions for the advection-dominated transport of a nonreactive pollutant emanating from a point source.

Source estimation for wave equations with uncertain parameters

Source estimation is a fundamental ingredient of Full Waveform Inversion (FWI). In such seismic inversion methods wavelet intensity and phase spectra are usually estimated statistically although for the FWI formulation as a nonlinear least-squares optimization problem it can naturally be incorporated to the workflow. Modern approaches for source estimation consider robust misfit functions leading to the well known robust FWI method. The present work uses synthetic data generated from a high order spectral element forward solver to produce observed data which in turn are used to estimate the intensity and the location of the point seismic source term of the original elastic wave PDE. A min-max filter approach is used to convert the original source estimation problem into a state problem conditioned to the observations and a non-standard uncertainty description. The resulting numerical scheme uses an implicit midpoint method to solve, in parallel, the chosen 2D and 3D numerical examples running on an IBM Blue Gene/Q using a grid defined by approximately sixteen thousand 5th order elements, resulting in a total of approximately 6.5 million degrees of freedom.