scholarly journals Influence of dimension on the convergence of level-sets in total variation regularization

2020 ◽  
Vol 26 ◽  
pp. 52 ◽  
Author(s):  
José A. Iglesias ◽  
Gwenael Mercier
Keyword(s):  
Inverse Problems ◽  
Banach Spaces ◽  
Total Variation ◽  
Level Sets ◽  
Total Variation Regularization ◽  
Practical Application ◽  
Linear Inverse Problems ◽  
Hausdorff Convergence ◽  
Natural Choice ◽  
Fidelity Term

We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation between the dimension and the assumed integrability of the solution that makes such an extension possible. We also give some counterexamples of practical application scenarios where the natural choice of fidelity term makes such a convergence fail.

scholarly journals A note on convergence of solutions of total variation regularized linear inverse problems

2018 ◽  
Vol 34 (5) ◽  
pp. 055011 ◽  
Author(s):  
José A Iglesias ◽  
Gwenael Mercier ◽  
Otmar Scherzer
Keyword(s):  
Inverse Problems ◽  
Total Variation ◽  
Linear Inverse Problems ◽  
Convergence Of Solutions

scholarly journals Sequential subspace optimization for nonlinear inverse problems

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Anne Wald ◽  
Thomas Schuster
Keyword(s):  
Inverse Problems ◽  
Banach Spaces ◽  
Noise Level ◽  
Initial Value ◽  
Linear Inverse Problems ◽  
Nonlinear Inverse Problems ◽  
Subspace Optimization ◽  
Linear Problems ◽  
Forward Operator ◽  
Search Directions

AbstractIn this work we discuss a method to adapt sequential subspace optimization (SESOP), which has so far been developed for linear inverse problems in Hilbert and Banach spaces, to the case of nonlinear inverse problems. We start by revising the technique for linear problems. In a next step, we introduce a method using multiple search directions that are especially designed to fit the nonlinearity of the forward operator. To this end, we iteratively project the initial value onto stripes whose width is determined by the search direction, the nonlinearity of the operator and the noise level. We additionally propose a fast algorithm that uses two search directions. Finally, we will show convergence and regularization properties for the presented method.

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