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We consider choice of the regularization parameter in Tikhonov method in the case of the unknown noise level of the data. From known heuristic parameter choice rules often the best results were obtained in the quasi-optimality criterion…
We study the choice of the regularisation parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyse several…
We study the choice of the regularisation parameter for linear ill-posed problems in the presence of data noise and operator perturbations, for which a bound on the operator error is known but the data noise-level is unknown. We introduce a…
In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise…
The linear functional strategy for the regularization of inverse problems is considered. For selecting the regularization parameter therein, we propose the heuristic quasi-optimality principle and some modifications including the smoothness…
This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific…
The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always…
We investigate the convergence theory of several known as well as new heuristic parameter choice rules for convex Tikhonov regularisation. The success of such methods is dependent on whether certain restrictions on the noise are satisfied.…
In this paper we propose a quantum algorithm to determine the Tikhonov regularization parameter and solve the ill-conditioned linear equations, for example, arising from the finite element discretization of linear or nonlinear inverse…
Despite recent advances in regularisation theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov…
In this work we consider the problem of finding optimal regularization parameters for general-form Tikhonov regularization using training data. We formulate the general-form Tikhonov solution as a spectral filtered solution using the…
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…
This paper derives a new class of adaptive regularization parameter choice strategies that can be effectively and efficiently applied when regularizing large-scale linear inverse problems by combining standard Tikhonov regularization and…
In this paper, we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates. Two such rules are derived and generalize those from quadratic regularization, namely the…
A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we…
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we…
We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the…
We study multi-parameter Tikhonov regularization, i.e., with multiple penalties. Such models are useful when the sought-for solution exhibits several distinct features simultaneously. Two choice rules, i.e., discrepancy principle and…
We study the behaviour of Tikhonov regularisation on topological spaces with multiple regularisation terms. The main result of the paper shows that multi-parameter regularisation is well-posed in the sense that the results depend…
Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy…