An adaptive pruning algorithm for the discrete L-curve criterion

Abstract

We describe a robust and adaptive implementation of the L-curve criterion. The algorithm locates the corner of a discrete L-curve which is a log–log plot of corresponding residual norms and solution norms of regularized solutions from a method with a discrete regularization parameter (such as truncated SVD or regularizing CG iterations). Our algorithm needs no predefined parameters, and in order to capture the global features of the curve in an adaptive fashion, we use a sequence of pruned L-curves that correspond to considering the curves at different scales. We compare our new algorithm to existing algorithms and demonstrate its robustness by numerical examples

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

“…We also applied heuristic parameter choice rules when the regularization matrices (27) and (28) were used. The restricted Regińska rule performs well in this setting, but many parameter choice rules do not.…”

“…Hansen et al [27] proposed an alternative approach to determining the "vertex" of the "L" from the point set (6). They construct a sequence of pruned L-curves, removing an increasing number of points, and consider a list of candidate "vertices" produced by two different selection algorithms.…”

“…The "vertex" of this point set is obtained by applying the L-corner method [27] to the set (7). The index k of the vertex is the truncation index for the TSVD method.…”

Comparingparameter choice methods for regularization of ill-posed problems

“…Our experience (see also [84]) is that these algorithms have similar performance. Other algorithms are given in [73] and the references therein.…”

FGMRES for linear discrete ill-posed problems

“…To make iterative methods useful for the solution of linear discrete ill-posed problems, they have to be equipped with a rule for estimating the valuek. This can be done in a variety of ways depending on what is known about the error e. Popular methods for estimatingk include the discrepancy principle, the Lcurve, generalized cross validation, and extrapolation; see [4,14,15,22,23] and references therein.…”