# American Institute of Mathematical Sciences

August  2010, 4(3): 463-483. doi: 10.3934/ipi.2010.4.463

## Nonstationary inversion of convection-diffusion problems - recovery from unknown nonstationary velocity fields

 1 Department of Physics and Mathematics, University of Eastern Finland, P.O.B. 1627, FI-70211 Kuopio, Finland, Finland, Finland, Finland

Received  March 2010 Revised  July 2010 Published  July 2010

In this paper, we consider the reconstruction of time-varying concentration distributions under nonstationary flow conditions. Previous studies have shown that the state estimation approach that is based on stochastic process evolution models, facilitates reconstructions of rapidly time-varying targets. However, only cases with stationary velocity fields, or cases in which the velocity field can be completely specified by a velocity profile, have been studied. While simultaneous estimation of the time-varying concentration and low-dimensional representations of the flow field itself has been shown to be possible to some extent, this would be computationally too heavy for on-line process estimation and control. On the other hand, using an incorrect flow model in the evolution model may induce intolerable estimation errors. In this paper, we consider an approach in which the state evolution model is written to correspond to a stationary flow, while the actual flow is nonstationary. The associated modelling errors are handled by constructing the state noise process to accommodate to this discrepancy. We carry out a numerical feasibility study with different Reynolds numbers and show that the approach yields significant reduction of estimation errors and simultaneously facilitates using computationally efficient reduced order models.
Citation: Antti Lipponen, Aku Seppänen, Jari Hämäläinen, Jari P. Kaipio. Nonstationary inversion of convection-diffusion problems - recovery from unknown nonstationary velocity fields. Inverse Problems & Imaging, 2010, 4 (3) : 463-483. doi: 10.3934/ipi.2010.4.463
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