Lavrentiev-type regularization methods for Hermitian problems

Abstract

Abstract. Lavrentiev regularization is a popular approach to the solution of linear discrete illposed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solution of certain matrix nearness problems are described. Computed examples illustrate the competitiveness of modified fractional Lavrentiev-type methods.

Fractional Tikhonov regularization with a nonlinear penalty term

“…A recent review of the literature with new theoretical results and further computed examples are presented in [8]. Fractional Lavrentiev regularization for linear discrete ill-posed problems (1.1) with a square positive semidefinite matrix A is discussed in [12,21]. Iterated fractional Tikhonov regularization has recently been described by Bianchi et al [2].…”

On generalized iterated Tikhonov regularization with operator-dependent seminorms

“…In the last years, new types of Tikhonov-based regularization methods were studied in [18] and [15] under the name of Fractional or Weighted Tikhonov and in [17,19] in order to dampen the oversmoothing effect on the regularized solution of classic Tikhonov and to exploit the information carried by the spectrum of the operator. Special attention was devoted to Fractional Tikhonov regularization studied and extended in [9,1,15], while for Hermitian problems, the fractional approach was combined with Lavrentiev regularization; see [21,14].…”