Robust regression and outlier detection
Robust regression and outlier detection
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Convex analysis and variational problems
Convex analysis and variational problems
An Analysis of the Zero-Crossing Method for Choosing Regularization Parameters
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Semismooth Newton Methods for Operator Equations in Function Spaces
SIAM Journal on Optimization
The Primal-Dual Active Set Strategy as a Semismooth Newton Method
SIAM Journal on Optimization
A Variational Approach to Remove Outliers and Impulse Noise
Journal of Mathematical Imaging and Vision
Efficient Minimization Methods of Mixed l2-l1 and l1-l1 Norms for Image Restoration
SIAM Journal on Scientific Computing
Handbook of Image and Video Processing (Communications, Networking and Multimedia)
Handbook of Image and Video Processing (Communications, Networking and Multimedia)
Lagrange Multiplier Approach to Variational Problems and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise
SIAM Journal on Scientific Computing
Efficient minimization method for a generalized total variation functional
IEEE Transactions on Image Processing
An Efficient Primal-Dual Method for $L^1$TV Image Restoration
SIAM Journal on Imaging Sciences
An efficient two-phase L1-TV method for restoring blurred images with impulse noise
IEEE Transactions on Image Processing
A property of the minimum vectors of a regularizing functionaldefined by means of the absolute norm
IEEE Transactions on Signal Processing
Heuristic Parameter-Choice Rules for Convex Variational Regularization Based on Error Estimates
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
A variational Bayesian method to inverse problems with impulsive noise
Journal of Computational Physics
A Regularization Parameter for Nonsmooth Tikhonov Regularization
SIAM Journal on Scientific Computing
Karush-Kuhn-Tucker Conditions for Nonsmooth Mathematical Programming Problems in Function Spaces
SIAM Journal on Control and Optimization
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This paper considers the numerical solution of inverse problems with an $\mathrm{L}^1$ data fitting term, which is challenging due to the lack of differentiability of the objective functional. Utilizing convex duality, the problem is reformulated as minimizing a smooth functional with pointwise constraints, which can be efficiently solved using a semismooth Newton method. In order to achieve superlinear convergence, the dual problem requires additional regularization. For both the primal and the dual problems, the choice of the regularization parameters is crucial. We propose adaptive strategies for choosing these parameters. The regularization parameter in the primal formulation is chosen according to a balancing principle derived from the model function approach, whereas the one in the dual formulation is determined by a path-following strategy based on the structure of the optimality conditions. Several numerical experiments confirm the efficiency and robustness of the proposed method and adaptive strategy.