^{1}

^{1}

^{1}

^{2}

^{2}

^{1}

^{2}

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

This article is concerned with the stable solution of
operator equations of the first kind in Banach spaces. More precisely, we aim
at computing a solution

The development of explicit solvers for operator
equations in Banach spaces is a current field of research which has great
importance since the Banach space setting allows for dealing with inverse
problems in a mathematical framework which is often better adjusted to the
requirements of a certain application. Alber [

The idea of this paper is to get a solver for (

If

In the present paper, we consider two generalizations
of (

Alber et al. presented in [

In the next section, we give the necessary theoretical
tools and apply them in Sections

Throughout the paper, let

Let

We introduce
some definitions and preliminary results about the geometry of Banach spaces,
which can be found in [

The functions

A Banach space

uniformly
convex if

smooth if for every

uniformly
smooth if

There is a tight connection between the modulus of
convexity and the modulus of smoothness. The Lindenstrauss duality formula
implies that

For

One can show [

If

Let

In

In the sequence spaces

We also refer the interested reader to [

The next theorem (see [

(1) Let

(2) Let

We remark that in a real Hilbert space these inequalities reduce to the well-known
polarization identity (

It turns out that due to the geometrical characteristics of Banach spaces other than
Hilbert spaces, it is often more appropriate to use Bregman distances instead of
conventional-norm-based functionals

Let

We summarize a few facts concerning Bregman distances
and their relationship to the norm in

Let

To shorten the proof in Chapter 3, we formulate and prove the following.

Let

We
have

By Theorem

This completes the proof.

This section deals with an iterative method for
minimizing functionals of Tikhonov type. In contrast to the algorithm described
in the next section, we iterate directly in the dual space

Due to simplicity, we restrict ourselves to the
Tikhonov functional

We show the convergence of this method in a constructive way. This will be done via the following steps.

We show the
inequality

We choose
admissible stepsizes

We suppose

We establish an
upper estimate for

We choose

Finally, we state the iterative minimization scheme.

(i) First, we
calculate the estimate for

Under our assumptions on

Therefore,

We have

Finally, we
arrive at the desired inequality

(ii) Next, we
choose admissible stepsizes. Assume that

We see that the choice

minimizes the right-hand side of (

If we assume

For

(iii) We know the
behavior of the Bregman distances, if

(iv) We choose

Next, we calculate an index

Hence, the opposite case is

By choosing

we get

Figure

Geometry of
the problem. The iterates

(v) We are now
in the same situation as described in (2). If we replace

If the sequence

The iterative method, defined by

_{1}), defines an
iterative minimization method for the Tikhonov functional

A similar construction can be carried out
for any

Let

The
sequence

_{1}), converges
strongly to the unique minimizer

(a) If the stopping
criterion

(b) Due to the
properties of _{1}) is strictly
convex and differentiable with continuous derivative

By the above remark it suffices to prove convergence in case

By (

We have analyzed two conceptionally quite different
nonlinear iterative methods for finding the minimizer of norm-based Tikhonov
functionals in Banach spaces. One is the steepest descent method, where the
iterations are directly carried out in the

As already pointed out in Section

In what follows

The next lemma allows us to estimate

Let

We at
first prove (

Let us prove (

The proof of Theorem

We fix

Suppose

The first author was supported by Deutsche Forschungsgemeinschaft, Grant no. MA 1657/15-1.