An iteration regularization method with general convex penalty for nonlinear inverse problems in Banach spaces

2019 ◽  
Vol 361 ◽  
pp. 472-486 ◽  
Author(s):  
Jing Wang ◽  
Wei Wang ◽  
Bo Han
Keyword(s):  
Inverse Problems ◽  
Banach Spaces ◽  
Regularization Method ◽  
Nonlinear Inverse Problems ◽  
General Convex

Existence of variational source conditions for nonlinear inverse problems in Banach spaces

2018 ◽  
Vol 26 (2) ◽  
pp. 277-286 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Banach Spaces ◽  
Regularization Method ◽  
Multiple Solutions ◽  
Convergence Rates ◽  
Nonlinear Inverse Problems ◽  
Range Type ◽  
Ill Posed ◽  
Tikhonov’S Regularization

AbstractVariational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

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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