Optimization tools for solving nonlinear ill-posed problems

Using the L- and alpha-curve, we consider how a nonlinear ill-posed Tikhonov regularized problem can be solved by a Gauss-Newton method. The solution to the problem is chosen from the point on the logarithmic L-curve that has maximal curvature, i.e. at the corner. The L-curve is used for analysing how the tradeoff between minimizing the solution norm versus the residual norm changes with the choice of the regularization parameter. The alpha-curve is used to analyse how the object function changes for different choices of the regularization parameter. In the numerical tests we solve an inverse problem in heat transfer where the L-curve solution is compared to the optimal solution and the solution given by the discrepancy principle.