A reduced basis Kalman filter for parametrized partial differential equations^∗

Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
dihlmann@mathematik.uni-stuttgart.de, haasdonk}@mathematik.uni-stuttgart.de; markus.dihlmann@mathematik.uni-stuttgart.de

Received: 8 August 2014
Revised: 11 March 2015

Abstract

The Kalman filter is a widely known tool in control theory for estimating the state of a linear system disturbed by noise. However, when applying the Kalman filter on systems described by parametrerized partial differential equations (PPDEs) the calculation of state estimates can take an excessive amount of time and real-time state estimation may be infeasible. In this work we derive a low dimensional representation of a parameter dependent Kalman filter for PPDEs via the reduced basis method. Thereby rapid state estimation, and in particular the rapid estimation of a linear output of interest, will be feasible. We will also derive a posteriori error bounds for evaluating the quality of the output estimations. Furthermore we will show how to verify the stability of the filter using an observability condition. We will demonstrate the performance of the reduced order Kalman filter and the error bounds with a numerical example modeling the heat transfer in a plate.