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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…
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 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…
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…
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…
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…
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 the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require…
The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between…
In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the…
Large scale, streaming datasets are ubiquitous in modern machine learning. Streaming algorithms must be scalable, amenable to incremental training and robust to the presence of non-stationarity. In this work consider the problem of learning…
Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie for the selection of groups of correlated variables. To investigate on the statistical…
We present a novel approach to nonlinear constrained Tikhonov regularization from the viewpoint of optimization theory. A second-order sufficient optimality condition is suggested as a nonlinearity condition to handle the nonlinearity of…
The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data,…
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.…
Choosing the regularization parameter for inverse problems is of major importance for the performance of the regularization method. We will introduce a fast version of the Lepskij balancing principle and show that it is a valid parameter…
We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the…
This paper investigates theoretical properties and efficient numerical algorithms for the so-called elastic-net regularization originating from statistics, which enforces simultaneously l^1 and l^2 regularization. The stability of the…