On the regularization of nonlinear ill-posed problems via inexact Newton iterations

Abstract

Inexact Newton methods for the stable solution of nonlinear ill-posed problems are considered. The corresponding inner scheme can be chosen to be any linear regularization with a sufficient modulus of convergence. The regularization property of these Newton-type algorithms is verified, that is, the iterates converge to a solution of the nonlinear problem with exact data when the noise level tends to zero. Moreover, convergence rates are given. Finally, implementation issues are discussed and the algorithm is applied to a parameter identification problem for an elliptic PDE. The numerical results reproduce nicely theoretical predictions and show the efficiency of the proposed method.

Regularization of nonlinear ill-posed problems by exponential integrators

“…Our numerical experiments (see Sect. 7) showed that the parameter selection by Rieder [16] yields good results.…”

“…The first example is taken from [16]. We aim at identifying the non-negative coefficient u = u(ξ, η) in the two-dimensional elliptic problem −Δy + uy = f in Ω y = g on ∂Ω (7.1) from the knowledge of the solution y = y(ξ, η) in Ω = (0, 1) 2 .…”

“…This problem fits into our abstract framework with X = Y = L 2 (Ω), see also [16]. For its numerical solution, the problem is discretized in space by standard finite differences on a regular, uniform grid with N internal points in each direction.…”

“…Other options for regularization are variants of inexact Newton methods [12,16,17]. A nice review is given in the recent book by Kaltenbacher et al [14].…”

Towards a general convergence theory for inexact Newton regularizations

“…This class has been introduced and named REGINN (REGularization based on INexact Newton iteration) by the second author [14]. Each of the REGINN-methods consists of two components, the outer Newton iteration and the inner scheme providing the increment by regularizing the local linearization.…”