In this paper, we consider the efficient numerical minimization of Tikhonov functionals resulting from total-variation (TV) regularization of linear inverse problems. Since the TV penalty is non-smooth, this is typically done either via smooth approximations, which are inexact, or using non-smooth optimization techniques, which can often be numerically expensive, in particular for large-scale problems. Here, we present a numerically efficient minimization approach based on the recently proposed semismooth* Newton method, which employs a novel concept of graphical derivatives and exhibits locally superlinear convergence. The proposed approach is specifically tailored to TV regularization, suitable for large-scale inverse problems, and supported by strong mathematical convergence guarantees. Furthermore, we demonstrate its performance on two (large-scale) tomographic imaging problems and compare our results to those obtained via other state-of-the-art TV regularization approaches.

We consider the identification of a scalar coefficient in a PDE-based parameter estimation problem with contact constraints. The considered problem can be used as an idealized model of a membrane under forces, constrained by a barrier or indenter. More generally, it serves as a benchmark for the analysis of more complex contact problems and the development of corresponding reconstruction algorithms. In this paper, we discuss both the forward and inverse parameter estimation problems, as well as uniqueness and non-uniqueness issues caused by the contact constraints. Furthermore, we consider the design and implementation of reconstruction approaches which we test on numerical examples, illustrating both uniqueness and non-uniqueness as well as parameter identifyability.

For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large.

For operators representing ill-posed problems, an ordering by ill-posedness is proposed, where one operator is considered more ill-posed than another one if the former can be expressed as a concatenation of bounded operators involving the latter. This definition is motivated by a recent one introduced by Math & eacute; and Hofmann [Adv. Oper. Theory, 10 (2025), Paper No. 36] that utilizes bounded and orthogonal operators, and we show the equivalence of our new definition with this one for the case of compact and non-compact linear operators in Hilbert spaces. We compare our ordering with other measures of ill-posedness such as the decay of the singular values, norm estimates, and range inclusions. Furthermore, as the new definition does not depend on the notion of orthogonal operators, it can be extended to the case of linear operators in Banach spaces, and it also provides ideas for applications to nonlinear problems in Hilbert spaces. In the latter context, certain nonlinearity conditions can be interpreted as ordering relations between a nonlinear operator and its linearization.

We consider the classic Fourier phase retrieval problem of reconstructing a function H from intensity measurements |ℱ^-1 H|^2 in a finite-dimensional setting. In particular, we consider the case of differential measurements and investigate in which form classic results for the continuous case, such as the transport of intensity equation (TIE), transfer to the practically relevant finite-dimensional case. Finally, we discuss the implications of our results on numerical reconstruction methods.

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.

In this article, concepts of well- and ill-posedness for linear operators in Hilbert and Banach spaces are discussed. While these concepts are well understood in Hilbert spaces, this is not the case in Banach spaces, as there are several competing definitions, related to the occurrence of uncomplemented subspaces. We provide an overview of the various definitions and, based on this, discuss the classification of type I and type II ill-posedness in Banach spaces. Furthermore, a discussion of borderline (hybrid) cases in this classification is given together with several example instances of operators.

We consider different norms for the Radon transform Rf of a function f and investigate under which conditions they can be estimated from above or below by some standard norms for f. We define Fourier-based norms for Rf which can be related to Bessel-potential space norms for f. Furthermore, we define a variant of a total-variation norm for Rf and provide conditions under which it is equivalent to the total-variation norm of f. To illustrate potential applications of these results, we propose a novel nonlinear backprojection method for inverting the Radon transform and present numerical results on simulated and experimental data.

Estimating the ratio of two probability densities from finitely many observations of the densities is a central problem in machine learning and statistics with applications in two-sample testing, divergence estimation, generative modeling, covariate shift adaptation, conditional density estimation, and novelty detection. In this work, we analyze a large class of density ratio estimation methods that minimize a regularized Bregman divergence between the true density ratio and a model in a reproducing kernel Hilbert space (RKHS). We derive new finite-sample error bounds, and we propose a Lepskii type parameter choice principle that minimizes the bounds without knowledge of the regularity of the density ratio. In the special case of quadratic loss, our method adaptively achieves a minimax optimal error rate. A numerical illustration is provided.

We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Ces & agrave;ro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Ces & agrave;ro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one.

Motivated by the aggregation method, we present an iterative method for finding approximate solutions of least-squares problems for linear ill-posed problems over (mixed) rational Krylov spaces.The mixed rational Krylov spaces where the solution is sought consist of Tikhonov-regularized solutions mixed with usual Krylov space elements from the normal equations.We present an algorithm based on the Arnoldi-Lanczos iteration, and, as main result, derive the rational CG method, a short-term iteration that, similar as the usual conjugate gradient method, does not requires orthogonalization or saving of the Krylov basis vectors.Some numerical experiments illustrate the performance of the method.

We consider infinite-dimensional generalized Hilbert matrices of the form Hi,j=didjxi+xj, where di are nonnegative weights and xi are pairwise distinct positive numbers. We state sufficient and, for monotonically rearrangeable xi, also necessary conditions for di, xi such that the induced operator from l2 -> l2 and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.

We consider linear ill-conditioned operator equations in a Hilbert space setting. Motivated by the aggregation method, we consider approximate solutions constructed from linear combinations of Tikhonov regularization, which amounts to finding solutions in a rational Krylov space. By mixing these with usual Krylov spaces, we consider least-squares problem in these mixed rational spaces. Applying the Arnoldi method leads to a sparse, pentadiagonal representation of the forward operator, and we introduce the Lanczos method for solving the least-squares problem by factorizing this matrix. Finally, we present an equivalent conjugate-gradient-type method that does not rely on explicit orthogonalization but uses short-term recursions and Tikhonov regularization in each second step. We illustrate the convergence and regularization properties by some numerical examples.

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.

The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well-understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landweber iteration for various nonlinear inverse problems. These are chosen from different practically relevant fields such as integral equations, parameter estimation, and tomography.

We state some sufficient criteria for the tangential cone conditions to hold for the electrical impedance tomography problem. The results are based on an estimate for the first-order Taylor residual for the forward operator and some convexity results in the Löwner order. As a consequence, for conductivities satisfying certain monotonicity properties, the tangential cone condition is verified.

The Hausdorf moment problem (HMP) over the unit interval in an L 2 -setting is a classical example of an ill-posed inverse problem. Since various applications can be rewritten in terms of the HMP, it has gathered significant attention in the literature. From the point of view of regularization it is of special interest because of the occurrence of a non-compact forward operator with non-closed range. Consequently, HMP constitutes one of few examples of a linear ill-posed problem of type I in the sense of Nashed. In this paper we highlight this property and its consequences, for example, the existence of an infinite-dimensional subspace of stability. On the other hand, we show conditional stability estimates for the HMP in Sobolev spaces that indicate severe ill-posedness for the full recovery of a function from its moments, because H ̈older-type stability can be excluded. However, the associated recovery of the function value at the rightmost point of the unit interval is stable of H ̈older-type in an H1 -setting. We moreover discuss stability estimates for the truncated HMP, where the forward operator becomes compact. Some numerical case studies illustrate the theoretical results and complete the paper.

We show that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems provided that a parameter is chosen accordingly to the smoothness of the solution. This result is proven both for an a priori stopping rule and for the discrepancy principle. The essential tool to obtain this result is a representation of the residual polynomials via Gegenbauer polynomials.

We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization properties and also obtain rates of convergence for our methods. A numerical example concerning a dynamical electrical impedance tomography (EIT) problem is used to illustrate the theoretical results.

We consider denoising and deblurring problems for tensors. While images can be discretized as matrices, the analogous procedure for color images or videos leads to a tensor formulation. We extend the classical ROF functional for variational denoising and deblurring to the tensor case by employing multi-dimensional total variation regularization. Furthermore, the resulting minimization problem is calculated by the FISTA method generalized to the tensor case. We provide some numerical experiments by applying the scheme to the denoising, the deblurring, and the recoloring of color images as well as to the deblurring of videos.