Sergiy Zhuk

DATA ASSIMILATION FOR LINEAR PARABOLIC EQUATIONS: MINIMAX PROJECTION METHOD ∗

In this paper we propose a state estimation method for linear parabolic partial differential equations (PDE) that accounts for errors in the model, truncation, and observations. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a differential-algebraic equation (DAE). The original estimation problem for the PDE is then recast as a state estimation problem for the constructed DAE using a linear continuous minimax filter. We construct a numerical time integrator that preserves the monotonic decay of a nonstationary Lyapunov function along the solution. To conclude, we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a two-dimensional fluid.

Minimax projection method for linear evolution equations

In this paper we present a minimax projection method for linear evolution equations in Hilbert space. The method extends classical Galerkin approach: it builds a differential-algebraic equation with uncertain parameters that models dynamics of exact projection coefficients representing the projection of the evolution equations solution onto a finite-dimensional subspace. The a priori ellipsoidal bounding set for uncertain parameters is also constructed. The output of the method is an ellipsoid enclosing exact projection coefficients. The ellipsoid can be constructed numerically: we illustrate this applying the method to 1D heat equation.

Minimax state estimation for linear stationary differential-algebraic equations*

Abstract This paper presents a generalization of the minimax state estimation approach for singular linear Differential-Algebraic Equations (DAE) with uncertain but bounded input and observations noise. We apply generalized Kalman Duality principle to DAE in order to represent the minimax estimate as a solution of a dual control problem for adjoint DAE. The latter is then solved converting the adjoint DAE into ODE by means of a projection algorithm. Finally, we represent the minimax estimate in the form of a linear recursive filter.

Symplectic Möbius integrators for LQ optimal control problems

The paper presents symplectic Möbius integrators for Riccati equations. All proposed methods preserve symmetry, positivity and quadratic invariants for the Riccati equations, and non-stationary Lyapunov functions. In addition, an efficient and numerically stable discretization procedure based on reinitialization for the associated linear Hamiltonian system is proposed.

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.

Divergence-free motion estimation

This paper describes an innovative approach to estimate motion from image observations of divergence-free flows. Unlike most state-of-the-art methods, which only minimize the divergence of the motion field, our approach utilizes the vorticity-velocity formalism in order to construct a motion field in the subspace of divergence free functions. A 4DVAR-like image assimilation method is used to generate an estimate of the vorticity field given image observations. Given that vorticity estimate, the motion is obtained solving the Poisson equation. Results are illustrated on synthetic image observations and compared to those obtained with state-of-the-art methods, in order to quantify the improvements brought by the presented approach. The method is then applied to ocean satellite data to demonstrate its performance on the real images.

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.

Infinite horizon control and minimax observer design for linear DAEs

In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalmans duality principle. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAE.

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.