A Regularization Parameter in Discrete Ill-Posed Problems

Abstract

The Tikhonov regularization method for discrete ill-posed problems is considered. For the practical choice of the regularization parameter c, some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all a considered. This paper contains an analysis of the shape of this plot and gives a theoreticaljustification for choosing the regularization parameter so it is related to the "L-comer" ofthe plot considered in the logarithmic scale. Moreover, a new criterion for choosing c is introduced (independent of the shape of the plot) which gives a new interpretation of the "comer criterion" mentioned above. The existence of"L-comer" is discussed.

Old and new parameter choice rules for discrete ill-posed problems

“…The discrete set (6) then is replaced by a continuous set. Regińska [41] proposed a parameter choice rule for Tikhonov regularization and showed how it relates to the L-curve criterion for Tikhonov regularization. This rule easily can be adapted to the situation when the regularization parameter is discrete.…”

“…Nevertheless, there is a need to use heuristic parameter choice rules, because for many discrete ill-posed problems (1) that arise in applications in science and engineering no accurate bound for e is available. For this reason, many heuristic parameter choice rules have been proposed in the literature, including the Lcurve criterion, the residual L-curve criterion, generalized cross validation, error estimation methods based on extrapolation, the quasi-optimality criterion, and Regińska's method; see, e.g., [6,7,24,25,28,29,38,39,41,43] and references therein.…”

Comparingparameter choice methods for regularization of ill-posed problems

“…There are several variants of the method; here we use the version proposed by Reginska [122,41]. The parameter choice is…”

“…This defines the "corner point" as the point on the L-curve where the "tangent" has slope -1 [122]. The method can be extended by minimizing Ax [122], where τ is a tuning constant, but we will not consider this here.…”

“…The method can be extended by minimizing Ax [122], where τ is a tuning constant, but we will not consider this here.…”