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Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter.…
We modify the total-variation-regularized image segmentation model proposed by Chan, Esedoglu and Nikolova [SIAM Journal on Applied Mathematics 66, 2006] by introducing local regularization that takes into account spatial image information.…
In this paper, we consider the $\alpha\| \cdot\|_{\ell_1}-\beta\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $\alpha\geq\beta\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the…
In the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of regularizers and data-fit terms. The algorithm we propose is based on a primal-dual diagonal…
We consider linear inverse problems that are formulated in the continuous domain. The object of recovery is a function that is assumed to minimize a convex objective functional. The solutions are constrained by imposing a continuous-domain…
This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or…
We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse regularization techniques that are based on frame representations, we train an encoder-decoder network by including an $\ell^1$-penalty.…
Regularization is a popular technique in machine learning for model estimation and avoiding overfitting. Prior studies have found that modern ordered regularization can be more effective in handling highly correlated, high-dimensional data…
Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the…
The computation of sparse solutions of large-scale linear discrete ill-posed problems remains a computationally demanding task. A powerful framework in this context is the use of iteratively reweighted schemes, which are based on…
Ensemble Kalman inversion is a parallelizable methodology for solving inverse or parameter estimation problems. Although it is based on ideas from Kalman filtering, it may be viewed as a derivative-free optimization method. In its most…
In this paper we propose a new statistical stopping rule for constrained maximum likelihood iterative algorithms applied to ill-posed inverse problems. To this aim we extend the definition of Tikhonov regularization in a statistical…
When solving rank-deficient or discrete ill-posed problems by regularization methods, the choice of the regularization parameter is crucial. It is also of interest, the regularization norm used in the selection of the solution. In this…
This work deals with a regularization method enforcing solution sparsity of linear ill-posed problems by appropriate discretization in the image space. Namely, we formulate the so called least error method in an $\ell^1$ setting and perform…
The joint bidiagonalization process of a matrix pair $\{A,L\}$ can be used to develop iterative regularization algorithms for large scale ill-posed problems in general-form Tikhonov regularization…
We study the inverse problem of parameter identification in non-coercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map by using the first-order and…
Tikhonov regularization is one of the most commonly used methods of regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to…
In this paper we study general $l_p$ regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of first- and second-order stationary points, and hence also of local minimizers of the $l_p$…
Within a statistical learning setting, we propose and study an iterative regularization algorithm for least squares defined by an incremental gradient method. In particular, we show that, if all other parameters are fixed a priori, the…
All imaging modalities such as computed tomography (CT), emission tomography and magnetic resonance imaging (MRI) require a reconstruction approach to produce an image. A common image processing task for applications that utilise those…