dc.contributor.advisor | Luke, Russell Prof. Dr. | |
dc.contributor.author | Altuntac, Erdem | |
dc.date.accessioned | 2016-07-22T09:57:41Z | |
dc.date.available | 2016-07-22T09:57:41Z | |
dc.date.issued | 2016-07-22 | |
dc.identifier.uri | http://hdl.handle.net/11858/00-1735-0000-0028-87D4-7 | |
dc.identifier.uri | http://dx.doi.org/10.53846/goediss-5760 | |
dc.language.iso | eng | de |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | |
dc.subject.ddc | 510 | de |
dc.title | Variational Regularization Strategy for Atmospheric Tomography | de |
dc.type | doctoralThesis | de |
dc.contributor.referee | Luke, Russell Prof. Dr. | |
dc.date.examination | 2016-04-04 | |
dc.description.abstracteng | The main focus of this dissertation is to establish
the necessary theory with numerical illustrations
for solving an atmospheric tomography problem.
The inverse problem is the reconstruction of some volume data from
sparsely and non-uniformly distributed tomographic measurement.
For a given linear, compact and injective forward operator $\mathcal{T}$
defined between some reflexive Banach space $\mathcal{V}$ and Hilbert space
$\mathcal{H},$ $\mathcal{T} : \mathcal{V} \rightarrow \mathcal{H},$ we seek the regularized minimizer for the general Tikhonov type objective functional \begin{eqnarray} F_{\alpha}:&\mathcal{V} \times \mathcal{H}& \longmapsto \mathbb{R}_{+} \nonumber\\ &(\varphi , f^{\delta})& \longmapsto
F_{\alpha}(\varphi , f^{\delta})
:= \frac{1}{2} \Vert{\mathcal{T}\varphi - f^{\delta}}\Vert_{\mathcal{H}}^2 + \alpha J(\varphi) ,
\nonumber
\end{eqnarray}
where the smooth and convex penalizer is
$J : \mathcal{V} \rightarrow \mathbb{R}_{+}.$ We analyse the stable convergence of the expected
regularized solution to the true solution by a posteriori strategy for the
choice of the regularization parameter, $\alpha = \alpha(\delta , f^{\delta}).$
Of particular interest in terms of
the penalty term $J,$ we choose a smoothed total variation functional
\begin{eqnarray}
J_{\beta}^{TV}(\varphi) := \int_{\Omega} \sqrt{\vert\nabla\varphi(x)\vert_2^2 + \beta} d x ,
\nonumber
\end{eqnarray}
where $0 < \beta < 1$ and $\vert \cdot \vert_2$ is the usual Euclidean norm. A new lower bound for the
Bregman distance particularly associated with
the penalty term $J_{\beta}^{TV}$ is estimated. We further investigate
the impact of TV regularization on inverse ill-posed
problems and convey the phenomenon of loss of contrast in TV regularization.
We demonstrate our regularization on simulated data, employing a novel
reverse-communication large-scale nonlinear optimization software and
also compare the result against traditional algorithms. | de |
dc.contributor.coReferee | Hohage, Thorsten Prof. Dr. | |
dc.subject.eng | atmospheric tomography, variational regularization, total variation, loss of contrast | de |
dc.identifier.urn | urn:nbn:de:gbv:7-11858/00-1735-0000-0028-87D4-7-7 | |
dc.affiliation.institute | Fakultät für Mathematik und Informatik | de |
dc.subject.gokfull | Mathematics (PPN61756535X) | de |
dc.identifier.ppn | 863710115 | |