Variational Regularization Strategy for Atmospheric Tomography

by Erdem Altuntac
Doctoral thesis
Date of Examination:2016-04-04
Date of issue:2016-07-22
Advisor:Prof. Dr. Russell Luke
Referee:Prof. Dr. Russell Luke
Referee:Prof. Dr. Thorsten Hohage
Persistent Address: http://dx.doi.org/10.53846/goediss-5760

 

 

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Abstract

English

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.
Keywords: atmospheric tomography, variational regularization, total variation, loss of contrast
 
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