(1)
where






We will assume attainability of the exact data y in a ball
, i.e., the equation F(x) = y is solvable in
. The element x 0 is an initial guess which may incorporate a-priori knowledge of an exact solution.


The actually available data y δ will in practice usually be contaminated with noise for which we here use a deterministic model, i.e.,

where the noise level δ is assumed to be known. For a convergence analysis with stochastic noise, see the references in section “Further Literature on Gauss–Newton Type Methods”.

(2)
2 Preliminaries
Conditions on F
For the proofs of well-definedness and local convergence of the iterative methods considered here we need several conditions on the operator F. Basically, we inductively show that the iterates remain in a neighborhood of the initial guess. Hence, to guarantee applicability of the forward operator to these iterates, we assume that

for some ρ > 0.

(3)
Moreover, we need that F is continuously Fréchet-differentiable, that
is uniformly bounded with respect to
, and that problem (1) is properly scaled, i.e., certain parameters occurring in the iterative methods have to be chosen appropriately in dependence of this uniform bound.


The assumption that F ′ is Lipschitz continuous,

that is often used to show convergence of iterative methods for well-posed problems, implies that

However, this Taylor remainder estimate is too weak for the ill-posed situation unless the solution is sufficiently smooth (see, e.g., case (ii) in Theorem 9 below). An assumption on F that can often be found in the literature on nonlinear ill-posed problems is the tangential cone condition

which implies that
for all
. One can even prove (see [70, Proposition 2.1]).

(4)

(5)

(6)


Proposition 1.
Let 




(i)
Then for all 
and
for all
. Moreover,
where instead of
equality holds if
.







(ii)
If F(x) = y is solvable in
, then a unique x 0 -minimum-norm solution exists. It is characterized as the solution
of F(x) = y in
satisfying the condition





(7)
If F(x) = y is solvable in
but a condition like (6) is not satisfied, then at least existence (but no uniqueness) of an x 0-minimum-norm solution is guaranteed provided that F is weakly sequentially closed (see [36, Chapter 10]).
