Inexact Newton Regularization Using Conjugate Gradients as Inner Iteration

In our papers [Inverse Problems, 15, 309-327,1999] and [Numer. Math., 88, 347-365, 2001] we proposed algorithm REGINN being an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. REGINN consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by solving the linearized system. The convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration, expecially Landweber iteration, ν-methods as well as Tikhonov–Phillips regularization. In the present paper we prove convergence rates for REGINN when the conjugate gradient method, which is nonlinear, serves as inner iteration. Thereby we add to a convergence analysis of Hanke who investigated REGINN furnished with the conjugate gradient method before [Numer. Funct. Anal. Optim., 18, 971-993, 1997]. By numerical experiments we illustrate that the conjugate gradient method outperforms the ν-method as inner iteration.