An improved fixed-point algorithm for determining a Tikhonov regularization parameter

Abstract

We re-analyze a Tikhonov parameter choice rule devised by Regińska (1996 SIAM J. Sci. Comput. 3 740–49) and algorithmically realized through a fast fixed-point (FP) method by Bazán (2008 Inverse Problems 24 035001). The method determines a Tikhonov parameter associated with a point near the L-corner of the maximum curvature and at which the L-curve is locally convex. In practice, it works well when the L-curve presents an L-shaped form with distinctive vertical and horizontal parts, but failures may occur when there are several local convex corners. We derive a simple and computable condition which describes the regions where the L-curve is concave/convex, while providing insight into the choice of the regularization parameter through the L-curve method or FP. Based on this, we introduce variants of the FP algorithm capable of handling the parameter choice problem even in the case where the L-curve has several local corners. The theory is illustrated both graphically and numerically, and the performance of the variants on a difficult ill-posed problem is evaluated by comparing the results with those provided by the L-curve method, generalized cross-validation and the discrepancy principle.