Related papers: A new interpretation of (Tikhonov) regularization
The Tikhonov regularization of linear ill-posed problems with an $\ell^1$ penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support…
We propose an efficient and flexible method for solving Abel integral equation of the first kind, frequently appearing in many fields of astrophysics, physics, chemistry, and applied sciences. This equation represents an ill-posed problem,…
In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint 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 investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the…
Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by…
Regularization plays a pivotal role in ill-posed machine learning and inverse problems. However, the fundamental comparative analysis of various regularization norms remains open. We establish a small noise analysis framework to assess the…
Measuring the error by an l^1-norm, we analyze under sparsity assumptions an l^0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed…
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…
In this paper, we study the stochastic convergence of regularized solutions for backward heat conduction problems. These problems are recognized as ill-posed due to the exponential decay of eigenvalues associated with the forward problems.…
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 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…
Regularization techniques are necessary to compute meaningful solutions to discrete ill-posed inverse problems. The well-known 2-norm Tikhonov regularization method equipped with a discretization of the gradient operator as regularization…
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…
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…
In this paper we consider convex Tikhonov regularisation for the solution of linear operator equations on Hilbert spaces. We show that standard fractional source conditions can be employed in order to derive convergence rates in terms of…
Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov…
In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on…
It is common to have to process signals or images whose values are cyclic and can be represented as points on the complex circle, like wrapped phases, angles, orientations, or color hues. We consider a Tikhonov-type regularization model to…
In this paper we provide a convergence analysis of some variational methods alternative to the classical Tikhonov regularization, namely Ivanov regularization (also called method of quasi solutions) with some versions of the discrepancy…