相关论文: On the Choice of the Tikhonov Regularization Param…

A simple proof of the convergence of the variational regularization, with the regularization parameter, chosen by the discrepancy principle, is given for linear operators under suitable assumptions. It is shown that the discrepancy…

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…

In this work, we consider ill-posed inverse problems in which the forward operator is continuous and weakly closed, and the sought solution belongs to a weakly closed constraint set. We propose a regularization method based on minimizing…

Convergence rates in spectral regularization methods quantify the approximation error in inverse problems as a function of the noise level or the number of sampling points. Classical strong convergence rate results typically rely on source…

The paper addresses the problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions --…

This paper deals with an inertial proximal algorithm that contains a Tikhonov regularization term, in connection to the minimization problem of a convex lower semicontinuous function $f$. We show that for appropriate Tikhonov regularization…

The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to…

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.…

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…

We study the inverse conductivity problem with discontinuous conductivities. We consider, simultaneously, a regularisation and a discretisation for a variational approach to solve the inverse problem. We show that, under suitable choices of…

We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation…

Higher-order regularization problem formulations are popular frameworks used in machine learning, inverse problems and image/signal processing. In this paper, we consider the computational problem of finding the minimizer of the Sobolev…

We consider joint Tikhonov- and Lavrentiev-regularization of control problems with pointwise control- and state-constraints. We derive error estimates for the error which is introduced by the Tikhonov regularization. With the help of this…

Tikhonov regularization with square-norm penalty for linear forward operators has been studied extensively in the literature. However, the results on convergence theory are based on technical proofs and difficult to interpret. It is also…

Accurate determination of the regularization parameter in inverse problems still represents an analytical challenge, owing mainly to the considerable difficulty to separate the unknown noise from the signal. We present a new approach for…

Primal-dual splitting involving proximity operators in order to be able to find some approximation to the minimizer for a general form of Tikhonov type functional is in the focus of this work. This approximation is produced by a pair of…

Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here…

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…

We make some remarks on a variant of the classical Tikhonov regularization in optimal control under PDEs which allows for a certain flexibility in dealing with non-linearities and state restrictions, in the sense that differential…

In this paper we investigate the problem of identifying the source term in an elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data. A variational method of Tikhonov-type regularization with specific…