Prediction and reconstruction of distributed dynamic phenomena characterized by linear partial differential equations

Abstract

A primary challenge for the reconstruction of continuous-time, continuous-amplitude distributed parameter systems is the inclusion of recent discrete-time, discreteamplitude, spatially discrete measurements. Hence, a systematic method for data processing is required that also handles incomplete and noisy data, e.g. data from a sensor network. This article presents two approaches to the reconstruction of distributed parameter systems that can be described by linear partial differential equations (PDEs) and involve one or several discrete measurement points. In both approaches, the linear PDE is first converted into a bank of linear lumped systems by means of modal analysis. In addition, a measurement equation relating state and (sensor) data is derived. In the second step, a Kalman filter (KF) is used to dynamically estimate the state of the lumped systems, which provides an approximation of the solution of the underlying PDE. The first approach uses Fourier Analysis. The second approach uses Fourier Analysis and the collocation method. The approaches are both demonstrated for a simple linear inhomogeneous PDE, the one-dimensional heat equation.

Parameter identification and reconstruction for distributed phenomena based on hybrid density filter

“…That means, the physical model characterizing the distributed phenomena has to be converted from a distributed-parameter form to a lumped-parameter form. This conversion can be achieved by methods for solving partial differential equations, such as finite-difference method, finite-element method [7], modal analysis [8] and finite-spectral method [4], [9].…”

Passive localization based on local observations in wireless sensor network using finite element method and Kalman filter

“…Keeping in the view of error minimization, this optimal estimator find the best estimate from noisy data amounts to filtering out the noise [3,11].…”

Bayesian Estimation of Distributed Phenomena Using Discretized Representations of Partial Differential Equations

“…Here, we extend and generalize our previous research work (Roberts and Hanebeck, 2005) in such a way that both the system model and the measurement model are derived by the finite-spectral method. Using this method, it turns out that nonlinear phenomena with complex boundary conditions can be reconstructed and predicted in a systematic manner.…”

“…A large number of distributed phenomena, such as irrotational fluid flow, heat conduction, and wave propagation (Roberts and Hanebeck, 2005), can be described by means of a set of linear partial differential equations.…”

“…These methods just need a few parameters for characterizing a smooth solution of the partial differential equation (Roberts and Hanebeck, 2005). The drawback, however, is that the modal analysis works only for linear partial differential equations and global expansion functions can be found only for problems with simple boundary conditions.…”