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We study filter based regularization methods for linear ill-posed problems between Hilbert spaces. We derive optimal order conditions under a-priori choice rules for the regularization parameter. Such analysis is applied to the fractional…
Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…
Tikhonov regularization is a common technique used when solving poorly behaved optimization problems. Often, and with good reason, this technique is applied by practitioners in an ad hoc fashion. In this note, we systematically illustrate…
Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix…
This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main…
This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and…
Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we…
A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized…
Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured…
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…
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…
We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable…
We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable…
We propose a comprehensive framework for solving constrained variational inequalities via various classes of evolution equations displaying multi-scale aspects. In an infinite-dimensional Hilbertian framework, the class of dynamical systems…
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…
In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
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…
In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution…
So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for…