Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix and an error-contaminated data vector (right-hand side). This regularization method replaces the given problem by a penalized least-squares problem. It is well known that Tikhonov regularization in standard form may yield approximate solutions that are too smooth, i.e., the computed approximate solution may lack many details that the desired solution of the associated, but unavailable, error-free problem might possess. Fractional Tikhonov regularization methods have been introduced to remedy this shortcoming. However, the computed solution determined by fractional Tikhonov methods in standard form may display undesirable spurious oscillations. This paper proposes that fractional Tikhonov methods be equipped with a nonlinear penalty term, such as a TV-norm penalty term, to reduce unwanted oscillations. Numerical examples illustrate the benefits of this approach.

Serena Morigi, Fiorella Sgallari, Lothar Reichel (2017). Fractional Tikhonov regularization with a nonlinear penalty term. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 324, 142-154 [10.1016/j.cam.2017.04.017].

### Fractional Tikhonov regularization with a nonlinear penalty term

#### Abstract

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix and an error-contaminated data vector (right-hand side). This regularization method replaces the given problem by a penalized least-squares problem. It is well known that Tikhonov regularization in standard form may yield approximate solutions that are too smooth, i.e., the computed approximate solution may lack many details that the desired solution of the associated, but unavailable, error-free problem might possess. Fractional Tikhonov regularization methods have been introduced to remedy this shortcoming. However, the computed solution determined by fractional Tikhonov methods in standard form may display undesirable spurious oscillations. This paper proposes that fractional Tikhonov methods be equipped with a nonlinear penalty term, such as a TV-norm penalty term, to reduce unwanted oscillations. Numerical examples illustrate the benefits of this approach.
##### Scheda breve Scheda completa Scheda completa (DC)
2017
Serena Morigi, Fiorella Sgallari, Lothar Reichel (2017). Fractional Tikhonov regularization with a nonlinear penalty term. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 324, 142-154 [10.1016/j.cam.2017.04.017].
Serena Morigi; Fiorella Sgallari; Lothar Reichel
File in questo prodotto:
File
JCAM324-2017.pdf

accesso aperto

Tipo: Postprint
Licenza: Licenza per Accesso Aperto. Creative Commons Attribuzione - Non commerciale - Non opere derivate (CCBYNCND)
Dimensione 372.34 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/589673`