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.

DOI: 10.1051/cocv/2015019

Keywords: Kalman filter, reduced order filter, partial differential equation, parameter dependent, model order reduction, error estimation, optimal filter, state estimation

^{1}; Haasdonk, Bernard

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@article{COCV_2016__22_3_625_0, author = {Dihlmann, Markus and Haasdonk, Bernard}, title = {A reduced basis {Kalman} filter for parametrized partial differential equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {625--669}, publisher = {EDP-Sciences}, volume = {22}, number = {3}, year = {2016}, doi = {10.1051/cocv/2015019}, zbl = {1346.35245}, mrnumber = {3527937}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015019/} }

TY - JOUR AU - Dihlmann, Markus AU - Haasdonk, Bernard TI - A reduced basis Kalman filter for parametrized partial differential equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 625 EP - 669 VL - 22 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015019/ DO - 10.1051/cocv/2015019 LA - en ID - COCV_2016__22_3_625_0 ER -

%0 Journal Article %A Dihlmann, Markus %A Haasdonk, Bernard %T A reduced basis Kalman filter for parametrized partial differential equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 625-669 %V 22 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015019/ %R 10.1051/cocv/2015019 %G en %F COCV_2016__22_3_625_0

Dihlmann, Markus; Haasdonk, Bernard. A reduced basis Kalman filter for parametrized partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 3, pp. 625-669. doi : 10.1051/cocv/2015019. http://archive.numdam.org/articles/10.1051/cocv/2015019/

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