Algorithms for using the nonlinear L-curve

Abstract

When using Tikhonov regularization for finite dimensional ill-posed problems there is a problem dependent choice of the regularization parameter. We present general tools for determining a proper regularization parameter that are based on the nonlinear L-curve and the associated (dual) a-curve. Given approximations of the solution of the Tikhonov problem we define upper and lower piecewise linear approximations of the L- and a-curve called shadow curves. These shadow curves are thouroughly investigated. Finally, we present ways to update the shadow curves and their use to identify good regularized solutions. AMS(MOS) subject classification: 62J05, 65U05 Key words: Nonlinear least squares, optimization, L-curve, regularization, Gauss-Newton method Contents 1 Introduction 1 2 Local results 3 3 Shadow curves 4 3.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 The polygon shadow curves . . . . . . . . . . . . . . . . . . . . . 4 3.3 The connection betwe..