On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept
-
Jens Flemming
,
Bernd Hofmann
Abstract
Based 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.
Funding source: DFG
Award Identifier / Grant number: FL 832/1-1, HO 1454/8-2
Funding source: DAAD
Award Identifier / Grant number: PPP-grant 56266051
Funding source: Ministry of Science of the Republic of Croatia
We appreciate the fruitful discussion with Radu I. Boţ (University of Vienna) and are particularly grateful that he brought the paper [Proc. Amer. Math. Soc. 14 (1963), 334–336] to our attention. We also express our thanks to Peter Stollmann and Thomas Kalmes (TU Chemnitz) for valuable hints.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Numerical Simulation of Surface Acoustic Wave Actuated Separation of Rigid Enantiomers by the Fictitious Domain Lagrange Multiplier Method
- An Optimal Adaptive Finite Element Method for an Obstacle Problem
- On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept
- Shape Optimization by Pursuing Diffeomorphisms
- Analysis of a Stabilized CNLF Method with Fast Slow Wave Splittings for Flow Problems
- Some Recent Progress on Numerical Methods for Controlled Regime-Switching Models with Applications to Insurance and Risk Management
- Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems
- M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator
- Dual Weighted Residual Error Control for Frictional Contact Problems
Articles in the same Issue
- Frontmatter
- Numerical Simulation of Surface Acoustic Wave Actuated Separation of Rigid Enantiomers by the Fictitious Domain Lagrange Multiplier Method
- An Optimal Adaptive Finite Element Method for an Obstacle Problem
- On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept
- Shape Optimization by Pursuing Diffeomorphisms
- Analysis of a Stabilized CNLF Method with Fast Slow Wave Splittings for Flow Problems
- Some Recent Progress on Numerical Methods for Controlled Regime-Switching Models with Applications to Insurance and Risk Management
- Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems
- M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator
- Dual Weighted Residual Error Control for Frictional Contact Problems
