Using the Nonlinear L-curve and its Dual

Abstract

The L-curve has been used on linear inverse problems for a long time. We generalize the L-curve to the nonlinear finite dimensional setting and introduce another most useful dual curve. The analytic and geometrical properties of these curves are derived together with a discussion on their use in algorithms. Key words: Nonlinear least squares, optimization, L-curve, regularization, Gauss-Newton method 1 Introduction Inverse problems appear in many different engineering applications. An inverse problem consists of a direct problem and some unknown function(s) or parameter(s). In many cases the solution does not depend continuously on the unknown quantities and the problem is ill-posed. A typical ill-posed problem is when the task is to determine these unkowns given measured, inexact, data. Given such an ill-posed problem it is a good idea to reformulate the original problem into a well-posed problem giving a solution that is not too large and with a small residual. 1.1 An example from..