Related papers: Convex regularization of discrete-valued inverse p…
The new version of a posteriori choice (NVAC) of the regularization parameter in the classical Tikhonov regularization method is considered. Lemmas and theorems on the error and the asymptotic convergence rate of the regularized solution…
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…
In this paper, we study an explicit Tikhonov-regularized inertial gradient algorithm for smooth convex minimization with Lipschitz continuous gradient. The method is derived via an explicit time discretization of a damped inertial system…
On normed vector spaces there is a well-known connection between the Tikhonov well-posedness of a minimisation problem and the differentiability of an associated convex conjugate function. We show how this duality naturally generalises to…
In this paper, we propose a Tikhonov-like regularization for dynamical systems associated with non-expansive operators defined in closed and convex sets of a Hilbert space. We prove the well-posedness and the strong convergence of the…
Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal…
In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the…
We study iterative regularization for linear models, when the bias is convex but not necessarily strongly convex. We characterize the stability properties of a primal-dual gradient based approach, analyzing its convergence in the presence…
This paper is concerned with a novel regularisation technique for solving linear ill-posed operator equations in Hilbert spaces from data that is corrupted by white noise. We combine convex penalty functionals with extreme-value statistics…
The analysis of Tikhonov regularization for nonlinear ill-posed equations with smoothness promoting penalties is an important topic in inverse problem theory. With focus on Hilbert scale models, the case of oversmoothing penalties, i.e.,…
An emerging new paradigm for solving inverse problems is via the use of deep learning to learn a regularizer from data. This leads to high-quality results, but often at the cost of provable guarantees. In this work, we show how…
We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence…
A version of the convexification globally convergent numerical method is constructed for a coefficient inverse problem for a wave-like partial differential equation. The presence of the Carleman Weight Function in the corresponding…
This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We focus on $B^0_{p,1}$ penalty terms which yield estimators that are sparse with respect to a wavelet…
Conditional stability estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if…
Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in $L^2$ only. For the method of transposition (sometimes called…
This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi-Tikhonov method. This new rule is essentially based on the discrepancy…
We investigate a Tikhonov regularization scheme specifically tailored for shallow neural networks within the context of solving a classic inverse problem: approximating an unknown function and its derivatives within a unit cubic domain…
We propose a novel framework for the regularised inversion of deep neural networks. The framework is based on the authors' recent work on training feed-forward neural networks without the differentiation of activation functions. The…
The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and…