We consider Tikhonov and sparsity-promoting regularization in Banach spaces for inverse scattering from penetrable anisotropic media. To this end, we equip an admissible set of material parameters with the L-topology and use Meyers’ gradient estimate for solutions of elliptic equations to analyze the dependence of scattered fields and their Fréchet derivatives on the material parameter. This al...

In this paper, we study and analyze the regularized weighted total least squares (RWTLS) formulation. Our regularization of the weighted total least squares problem is based on the Tikhonov regularization. Numerical examples are presented to demonstrate the effectiveness of the RWTLS method.

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore t...

We consider solution of linear ill-posed problem Au = f by Tikhonov method and by Lavrentiev method. For increasing the qualification and accuracy of these methods we use extrapolation, taking for the approximate solution linear combination of n ≥ 2 approximations of Tikhonov or Lavrentiev methods with different parameters and with proper coefficients. If the solution u∗ belongs to R((A A)) and...

We describe a novel method for reducing a pair of large matrices {A,B} to a pair of small matrices {H,K}. The method is an extension of Golub–Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a disc...

We consider joint Tikhonovand Lavrentiev-regularization of control problems with pointwise controland state-constraints. We derive error estimates for the error which is introduced by the Tikhonov regularization. With the help of this results we show, that if the solution of the unconstrained problem has no active constraints, the same holds for the Tikhonov-regularized solution if the regulari...

in this paper we propose a descent method for solving variational inequality problems where the underlying operator is nonsmooth, locally Lipschitz, and monotone over a closed, convex feasible set. The idea is to combine a descent method for variational inequality problems whose operators are nonsmooth, locally Lipschitz, and strongly monotone, with the Tikonov-Browder regularization technique....

Computation of perfusion parameters by deconvolution from contrast-enhanced time-resolved CT or MR perfusion data sets is an ill-conditioned problem. Thus, adequate regularization and determination of corresponding regularization parameters is required. We present a novel method for Tikhonov regularization for perfusion imaging to locally adapt parameters to the SNR level by using a regression ...