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

A primary challenge for the reconstruction of continuous-time, continuous-amplitude distributed parameter systems is the inclusion of recent discrete-time, discrete-amplitude, 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.