scholarly journals On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept

2015 ◽  
Vol 15 (3) ◽  
pp. 279-289 ◽  
Author(s):  
Jens Flemming ◽  
Bernd Hofmann ◽  
Ivan Veselić
Keyword(s):  
Inverse Problems ◽  
Variational Inequalities ◽  
Operator Equation ◽  
Convergence Rates ◽  
Cesàro Operator ◽  
Infinite Dimensional ◽  
Ill Posed ◽  
Cesaro Operator ◽  
ℓ1 Regularization

AbstractBased on the powerful tool of variational inequalities, in recent papers convergence rates results on ℓ1-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in ℓ1. In the present paper, we improve those convergence rates results and apply them to the Cesáro operator equation in ℓ2 and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals The Regularized Weak Functional Matching Pursuit for linear inverse problems

2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel
Keyword(s):  
Inverse Problems ◽  
Matching Pursuit ◽  
A Priori ◽  
Computation Time ◽  
Point Of View ◽  
Computational Point ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Frame Condition ◽  
Ill Posed

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

scholarly journals Convergence Rates for Exponentially Ill-Posed Inverse Problems with Impulsive Noise

2016 ◽  
Vol 54 (1) ◽  
pp. 341-360 ◽  
Author(s):  
Claudia König ◽  
Frank Werner ◽  
Thorsten Hohage
Keyword(s):  
Inverse Problems ◽  
Convergence Rates ◽  
Impulsive Noise ◽  
Ill Posed

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.

Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ1-regularization

2018 ◽  
Vol 26 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Jens Flemming ◽  
Daniel Gerth
Keyword(s):  
Exact Solution ◽  
Convergence Rate ◽  
Convergence Rates ◽  
Weak Continuity ◽  
Source Type ◽  
Ill Posed ◽  
Sparse Solutions ◽  
ℓ1 Regularization

AbstractWe show that the convergence rate of {\ell^{1}}-regularization for linear ill-posed equations is always {{\mathcal{O}}(\delta)} if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.

On variational regularization: Finite dimension and Hölder stability

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Gaurav Mittal ◽  
Ankik Kumar Giri
Keyword(s):  
Variational Inequalities ◽  
Convergence Rates ◽  
Finite Dimension ◽  
Noisy Data ◽  
Stability Estimates ◽  
Abstract Theory ◽  
Finite Dimensional ◽  
Ill Posed ◽  
Variational Regularization ◽  
Hölder Stability

AbstractIn this paper, we analyze the convergence rates for finite-dimensional variational regularization in Banach spaces by taking into account the noisy data and operator approximations. In particular, we determine the convergence rates by incorporating the smoothness concepts of Hölder stability estimates and the variational inequalities. Additionally, we discuss two ill-posed inverse problems to complement the abstract theory presented in our main results.

Methods of solving ill-posed inverse problems

1983 ◽  
Vol 45 (5) ◽  
pp. 1237-1245 ◽  
Author(s):  
O. M. Alifanov
Keyword(s):  
Inverse Problems ◽  
Ill Posed
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