It is a common belief that Tikhonov scheme with ‖ · ‖L2 -penalty fails to reconstruct a sparse structure with respect to a given system {φi}. However, in this paper we present a procedure for sparsity reconstruction, which is totally based on the standard Tikhonov method. This procedure consists of two steps. At first Tikhonov scheme is used as a sieve to find the coefficients near φi, which ar...

We develop a hybrid iterative approach for computing solutions to large-scale illposed inverse problems via Tikhonov regularization. We consider a hybrid LSMR algorithm, where Tikhonov regularization is applied to the LSMR subproblem rather than the original problem. We show that, contrary to standard hybrid methods, hybrid LSMR iterates are not equivalent to LSMR iterates on the directly regul...

For a parameter-dependent system of ordinary differential equations we present a systematic approach to the determination of parameter values near which singular perturbation scenarios (in the sense of Tikhonov and Fenichel) arise. We call these special values Tikhonov parameter values. The principal application we intend is to equations that describe chemical reactions, in the context of quasi...

Magnetic resonance images are affected by motion artefacts due to breathing and cardiac beating that occur during the acquisition. Methods for joint reconstruction of image and motion have been proposed recently. Such optimization problems are ill-conditioned, therefore regularization methods are required such as motion smoothness constraints using the Tikhonov method. However with Tikhonov met...

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth, i.e., the approximate solution may lack many details that the desired exact solution might possess. Two different approaches, both referred to as fractional Tikhonov methods have been introduced to remedy this shortcoming. This paper investigates the convergence properties of ...

A convergence rate is established for nonstationary iterated Tik-honov regularization, applied to ill-posed problems involving closed, densely deened linear operators, under general conditions on the iteration parameters. It is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.

In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov–Mann iteration in W-hyperbolic spaces. This is an extension to a nonlinear setting modified Mann defined recently by Boţ, Csetnek and Meier Hilbert Furthermore, show that Douglas–Rachford forward-backward algorithms with Tikhonov regularization terms are special cases, spaces, our iteration.

We consider a local regularization method for the solution of first-kind Volterra integral equations with convolution kernel. The local regularization is based on a splitting of the original Volterra operator into “local” and “global” parts, and a use of Tikhonov regularization to stabilize the inversion of the local operator only. The regularization parameters for the local procedure include t...