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Variational Methods for the Estimation of Transport Fields with Application to the Recovery of Physics-Based Optical Flows Across Boundaries

Klinger, Matthias

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Abstract

In this thesis we develop a method for the estimation of the flow behaviour of an incom- pressible fluid based on observations of the brightness intensity of a transported visible substance which does not influence the flow. The observations are given in a subregion of the flow as a sequence of discrete images with in- and outflow across the image boundaries. The resulting mathematical problem is ill-posed and has to be regularised with information of the underlying fluid flow model. We consider a constrained optimisation problem, namely the minimisation of a tracking type data term for the brightness distribution and a regularisation term subject to a system of weakly coupled partial differential equations. The system consists of the time- dependent incompressible Navier-Stokes equations coupled by the velocity vector field to a convection-diffusion equation, which describes the transport of brightness patterns in the image sequence. Due to the flow across the boundaries of the computational domain we solve a boundary identification problem. The usage of (strong) Dirichlet boundary controls for this purpose leads to theoretical and numerical complications, so that we will instead use Robin-type controls, which allow for a more convenient theoretical and numerical framework. We will prove well-posedness and investigate the functionality of the proposed approach by means of numerical examples. Furthermore, we discuss the connection to Dirichlet-control problems, e. g. the approximation of Dirichlet-controls by the so-called penalised Neumann method, which is based on the Robin-type controls for a varying penalty parameter. We will show via numerical tests that Robin-type controls are suitable for the identifi- cation of the correct fluid flow. Moreover, the examples indicate that the underlying physical model used for the regularisation influences the flow reconstruction process. Thus appropriate knowledge of the model is essential, e. g. the viscosity parameter. For a time- independent example we will present a heuristic, which, beside the boundary identification, automatically evaluates the viscosity in case the parameter is unknown. The developed physics-based optical flow estimation approach is finally used for the data set of a prototypical application. The background of the application is the approximation of horizontal wind fields in sparsely populated areas like desert regions. A sequence of satellite images documenting the brightness intensity of an observable substance distributed by the wind (e. g. dust plumes) is thereby assumed to be the only available data. Wind field information is for example needed to simulate the distribution of other, not directly observ- able, substances in the lower atmosphere. For the prototypical example we compute a high quality reconstruction of the underlying fluid flow by a (discrete) sequence of consecutive spatially distributed brightness intensities. Thereby, we compare three different models (heat equation, Stokes system and the original fluid flow model) in the reconstruction process and show that using as much model knowledge as possible is essential for a good reconstruction result.

Item Type: Dissertation
Supervisor: Rannacher, Prof. Dr. Dr. h. c. Rolf
Date of thesis defense: 29 April 2015
Date Deposited: 08 May 2015 10:08
Date: 2015
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 500 Natural sciences and mathematics
510 Mathematics
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