Abstract
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 these methods by reviewing results published previously by various authors. We show that both methods are order optimal when the regularization parameter is chosen according to the discrepancy principle. The theory developed suggests situations in which the fractional methods yield approximate solutions of higher quality than Tikhonov regularization in standard form. Computed examples that illustrate the behavior of the methods are presented.
Funding source: Austrian Science Fund (FWF)
Award Identifier / Grant number: W1214-N15
Funding source: Austrian Science Fund (FWF)
Award Identifier / Grant number: T529-N18
Funding source: NSF
Award Identifier / Grant number: DMS-1115385
© 2015 by De Gruyter
