arXiv subjects

Elena Resmerita

Publications and source records attributed to Elena Resmerita.

17 recordsLinked to original sources

arXiv2026

Primal-dual methods and acceleration for Morozov and equality constrained regularization

This work develops a regularization analysis of non-accelerated and accelerated primal-dual methods for solving linear inverse problems in the presence of noisy data. We investigate a Condat-Vũ algorithm and an accelerated primal-dual hybrid gradient method in Hilbert spaces, with focus on quantifying the effect of data perturbations on the reconstruction error. For the non-accelerated scheme, we derive error estimates in terms of Bregman distances, whereas for the accelerated scheme we establish error estimates in norm. The study accommodates a general class of convex data fidelities satisfying suitable perturbation conditions, which are verified explicitly for equality constrained and Morozov regularization. For the non-accelerated method, the analysis is further extended to Banach spaces, taking into account non-Euclidean geometries and including a particular non-reflexive setting tailored to nonnegative solution reconstruction. The results recover known behavior in classical settings while extending the regularization analysis to these more general frameworks. Numerical experiments with representative regularizers, including sparsity and entropic models, support the theoretical findings and illustrate practical performance under noise.

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arXiv2025

Applications of multiscale hierarchical decomposition to blind deconvolution

The blind image deconvolution is a challenging, highly ill-posed nonlinear inverse problem. We introduce a Multiscale Hierarchical Decomposition Method (MHDM) that is iteratively solving variational problems with adaptive data and regularization parameters, towards obtaining finer and finer details of the unknown kernel and image. We establish convergence of the residual in the noise-free data case, and then in the noisy data case when the algorithm is stopped early by means of a discrepancy principle. Fractional Sobolev norms are employed as regularizers for both kernel and image, with the advantage of computing the minimizers explicitly in a pointwise manner. In order to break the notorious symmetry occurring during each minimization step, we enforce a positivity constraint on the Fourier transform of the kernels. Numerical comparisons with a single-step variational method and a non-blind MHDM show that our approach produces comparable results, while less laborious parameter tuning is necessary at the price of more computations. Additionally, the scale decomposition of both reconstructed kernel and image provides a meaningful interpretation of the involved iteration steps.

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arXiv2025

Higher order error estimates for regularization of inverse problems under non-additive noise

In this work we derive higher order error estimates for inverse problems distorted by non-additive noise, in terms of Bregman distances. The results are obtained by means of a novel source condition, inspired by the dual problem. Specifically, we focus on variational regularization having the Kullback-Leibler divergence as data-fidelity, and a convex penalty term. In this framework, we provide an interpretation of the new source condition, and present error estimates also when a variational formulation of the source condition is employed. We show that this approach can be extended to variational regularization that incorporates more general convex data fidelities.

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arXiv2025

Nested Bregman Iterations for Decomposition Problems

We consider the task of image reconstruction while simultaneously decomposing the reconstructed image into components with different features. A commonly used tool for this is a variational approach with an infimal convolution of appropriate functions as a regularizer. Especially for noise corrupted observations, incorporating these functionals into the classical method of Bregman iterations provides a robust method for obtaining an overall good approximation of the true image, by stopping early the iteration according to a discrepancy principle. However, crucially, the quality of the separate components depends further on the proper choice of the regularization weights associated to the infimally convoluted functionals. Here, we propose the method of Nested Bregman iterations to improve a decomposition in a structured way. This allows to transform the task of choosing the weights into the problem of stopping the iteration according to a meaningful criterion based on normalized cross-correlation. We discuss the well-definedness and the convergence behavior of the proposed method, and illustrate its strength numerically with various image decomposition tasks employing infimal convolution functionals.

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arXiv2023

Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise

Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising.

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arXiv2023

Multiscale hierarchical decomposition methods for ill-posed problems

The Multiscale Hierarchical Decomposition Method (MHDM) was introduced as an iterative method for total variation regularization, with the aim of recovering details at various scales from images corrupted by additive or multiplicative noise. Given its success beyond image restoration, we extend the MHDM iterates in order to solve larger classes of linear ill-posed problems in Banach spaces. Thus, we define the MHDM for more general convex or even non-convex penalties, and provide convergence results for the data fidelity term. We also propose a flexible version of the method using adaptive convex functionals for regularization, and show an interesting multiscale decomposition of the data. This decomposition result is highlighted for the Bregman iteration method that can be expressed as an adaptive MHDM. Furthermore, we state necessary and sufficient conditions when the MHDM iteration agrees with the variational Tikhonov regularization, which is the case, for instance, for one-dimensional total variation denoising. Finally, we investigate several particular instances and perform numerical experiments that point out the robust behavior of the MHDM.

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arXiv2022

On Hamilton-Jacobi PDEs and image denoising models with certain non-additive noise

We consider image denoising problems formulated as variational problems. It is known that Hamilton-Jacobi PDEs govern the solution of such optimization problems when the noise model is additive. In this work, we address certain non-additive noise models and show that they are also related to Hamilton-Jacobi PDEs. These findings allow us to establish new connections between additive and non-additive noise imaging models. Specifically, we study how the solutions to these optimization problems depend on the parameters and the observed images. We show that the optimal values are ruled by some Hamilton-Jacobi PDEs, while the optimizers are characterized by the spatial gradient of the solution to the Hamilton-Jacobi PDEs. Moreover, we use these relations to investigate the asymptotic behavior of the variational model as the parameter goes to infinity, that is, when the influence of the noise vanishes. With these connections, some non-convex models for non-additive noise can be solved by applying convex optimization algorithms to the equivalent convex models for additive noise. Several numerical results are provided for denoising problems with Poisson noise or multiplicative noise.

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arXiv2021

Nonconvex flexible sparsity regularization: theory and monotone numerical schemes

Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and numerical perspectives. Namely, we show convergence of the regularization method and establish convergence properties of a couple of majorization approaches for the associated nonconvex problems. We also test a monotone algorithm for an academic example where the operator is an $M$ matrix, and on a time-dependent optimal control problem, pointing out the advantages of employing variable penalties over a fixed penalty.

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arXiv2019

An entropic Landweber method for linear ill-posed problems

The aim of this paper is to investigate the use of an entropic projection method for the iterative regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both in a case of reconstructing general nonnegative unknowns as well as for the sake of recovering probability distributions. Moreover, we discuss several variants of the algorithm and relations to other methods in the literature. The effectiveness of the approach is studied numerically in several examples.

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arXiv2018

Regularization of ill-posed problems with non-negative solutions

This survey reviews variational and iterative methods for reconstructing non-negative solutions of ill-posed problems in infinite-dimensional spaces. We focus on two classes of methods: variational methods based on entropy-minimization or constraints, and iterative methods involving projections or non-negativity-preserving multiplicative updates. We summarize known results and point out some open problems.

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arXiv2016

Regularization by Discretization in Banach Spaces

We consider ill-posed linear operator equations with operators acting between Banach spaces. For solution approximation, the methods of choice here are projection methods onto finite dimensional subspaces, thus extending existing results from Hilbert space settings. More precisely, general projection methods, the least squares method and the least error method are analyzed. In order to appropriately choose the dimension of the subspace, we consider a priori and a posteriori choices by the discrepancy principle and by the monotone error rule. Analytical considerations and numerical tests are provided for a collocation method applied to a Volterra integral equation in one space dimension.

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arXiv2016

Lavrentiev's regularization method in Hilbert spaces revisited

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.

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arXiv2016

The least error method for sparse solution reconstruction

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 convergence analysis by choosing the discretization level according to an a priori rule, as well as two a posteriori rules, via the discrepancy principle and the monotone error rule, respectively. Depending on the setting, linear or sublinear convergence rates in the $\ell^1$-norm are obtained under a source condition yielding sparsity of the solution. A part of the study is devoted to analyzing the structure of the approximate solutions and of the involved source elements.

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arXiv2016

Flexible sparse regularization

The seminal paper of Daubechies, Defrise, DeMol made clear that $\ell^p$ spaces with $p\in [1,2)$ and $p$-powers of the corresponding norms are appropriate settings for dealing with reconstruction of sparse solutions of ill-posed problems by regularization. It seems that the case $p=1$ provides the best results in most of the situations compared to the cases $p\in (1,2)$. An extensive literature gives great credit also to using $\ell^p$ spaces with $p\in (0,1)$ together with the corresponding quasinorms, although one has to tackle challenging numerical problems raised by the non-convexity of the quasi-norms. In any of these settings, either super, linear or sublinear, the question of how to choose the exponent $p$ has been not only a numerical issue, but also a philosophical one. In this work we introduce a more flexible way of sparse regularization by varying exponents. We introduce the corresponding functional analytic framework, that leaves the setting of normed spaces but works with so-called F-norms. One curious result is that there are F-norms which generate the $\ell^1$ space, but they are strictly convex, while the $\ell^1$-norm is just convex.

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arXiv2015

Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a non-convex cost function under a norm constraint, where non-convexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (non-convex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.

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arXiv2010

Morozov's principle for the augmented Lagrangian method applied to linear inverse problems

The Augmented Lagrangian Method as an approach for regularizing inverse problems received much attention recently, e.g. under the name Bregman iteration in imaging. This work shows convergence (rates) for this method when Morozov's discrepancy principle is chosen as a stopping rule. Moreover, error estimates for the involved sequence of subgradients are pointed out. The paper studies implications of these results for particular examples motivated by applications in imaging. These include the total variation regularization as well as $\ell^q$ penalties with $q\in[1,2]$. It is shown that Morozov's principle implies convergence (rates) for the iterates with respect to the metric of strict convergence and the $\ell^q$-norm, respectively.

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arXiv2008

On regularization methods of EM-Kaczmarz type

We consider regularization methods of Kaczmarz type in connection with the expectation-maximization (EM) algorithm for solving ill-posed equations. For noisy data, our methods are stabilized extensions of the well established ordered-subsets expectation-maximization iteration (OS-EM). We show monotonicity properties of the methods and present a numerical experiment which indicates that the extended OS-EM methods we propose are much faster than the standard EM algorithm.

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