This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data

Traffic state estimation is a challenging problem for the transportation community due to the limited deployment of sensing infrastructure. However, recent trends in the mobile phone industry suggest that GPS equipped devices will become standard in the next few years. Leveraging these GPS equipped devices as traffic sensors will fundamentally change the type and the quality of traffic data collected on large scales in the near future. New traffic models and data assimilation algorithms must be developed to efficiently transform this data into usable traffic information.In this work, we introduce a new partial differential equation (PDE) based on the Lighthill-Whitham-Richards PDE, which serves as a flow model for velocity. We formulate a Godunov discretization scheme to cast the PDE into a Velocity Cell Transmission Model (CTM-v), which is a nonlinear dynamical system with a time varying observation matrix. The Ensemble Kalman Filtering (EnKF) technique is applied to the CTMv to estimate the velocity field on the highway using data obtained from GPS devices, and the method is illustrated in microsimulation on a fully calibrated model of I880 in California. Experimental validation is performed through the unprecedented 100-vehicle Mobile Century experiment, which used a novel privacy-preserving traffic monitoring system to collect GPS cell phone data specifically for this research.

Electrical impedance tomography (EIT) is a diffuse imaging modality in which the resistivity distribution inside the object is estimated based on electrical measurements made on the boundary. Several applications can be found in geophysics, medicine and industry. Image reconstruction is an iterative procedure in which the norm between the computed and measured voltages is minimized. Also, an additional regularization term is included in the minimized functional due to the ill-posedness of the problem. In the reconstruction process, the geometry of the object is assumed to be known. Geometry is known in many cases but there are various situations in which the shape of the domain is unknown. For example, in medical applications the shape of the domain, a part of the human body on which the measurement electrodes are attached, is unknown unless some other imaging modality is used for receiving the shape. In industrial applications, such as in the imaging of a stirrer vessel for detecting air distribution or detecting large air bubbles in pipelines, the free surface between the liquid and air is unknown and should be estimated. Within the domain we may also have ‘voids’ having zero conductivity and we might be interested in detecting the shapes and locations of the voids. An example could be the detection of corrosion faults in metallic plates. In this paper, two approaches for shape and free surface estimation are proposed. The approaches taken here are based on the idea used in shape optimization problems. In the first approach, the unknown shape is parametrized using the mesh nodes as parameters. In the second approach, we define new ‘design variables’ which are used as parameters. These design variables are the coefficients of a Bézier curve that defines the shape of the surface. In this paper, we show results of the free surface estimation from both computer simulations and tank measurements. Also, results of simultaneous reconstruction of the resistivity distribution and the free surface are shown. Comparison of the results between these two approaches will be given. The comparison shows better performance in the Bézier curve approach.