DOI HERE 00 Advance Access Publication Date: Day Month Year Paper Published by
Oxford University Press on behalf of the Institute of Mathematics and its
Applications. All rights reserved.

Barbara Kaltenbacher

[*]Corresponding author<EMAIL_ADDRESS>
0Year 0Year 0Year

#  Convergence rates under a range invariance condition with application to
electrical impedance tomography

Barbara Kaltenbacher  Department of Mathematics, University of Klagenfurt,
Universitätsstr. 65-67, 9020, , Klagenfurt

(2024; 2024; Date; Date; Date)

###### Abstract

This paper is devoted to proving convergence rates of variational and
iterative regularization methods under variational source conditions VSCs for
inverse problems whose linearization satisfies a range invariance condition.
In order to achieve this, often an appropriate relaxation of the problem needs
to be found that is usually based on an augmentation of the set of unknowns
and leads to a particularly structured reformulation of the inverse problem.
We analyze three approaches that make use of this structure, namely a
variational and a Newton type scheme, whose convergence without rates has
already been established in [17]; additionally we propose a split minimization
approach that can be show to satisfy the same rates results.
The range invariance condition has been verified for several coefficient
identification problems for partial differential equations from boundary
observations as relevant in a variety of tomographic imaging modalities. Our
motivation particularly comes from the by now classical inverse problem of
electrical impedance tomography EIT and we study both the original formulation
by a diffusion type equation and its reformulation as a Schrödinger equation.
For both of them we find relaxations that can be proven to satisfy the range
invariance condition. Combining results on VSCs from [28] with the abstract
framework for the three approaches mentioned above, we arrive at convergence
rates results for the variational, split minimization and Newton type method
in EIT.

###### keywords:

range invariance condition; convergence rates; iterative regularization;
variational regularization; electrical impedance tomography.

##  1 Introduction

Consider an inverse problem either in its all-at-once formulation

$\left.\begin{array}[]{ll}A(q,u)=0&\mbox{(model equation)}\\\
Bu=y&\mbox{(observation equation)}\end{array}\right\\}\quad\Leftrightarrow:\
\mathbb{F}(q,u)=(0,y)^{T}$ (1.1)

or in its reduced formulation, with a parameter-to-state operator
$S:\mathcal{D}(\mathbb{F})\to V$ that is implicitly defined by the first
equation in

$A(q,S(q))=0\mbox{ and }B(S(q))=y\quad\Leftrightarrow:\ \mathbf{F}(q)=y.$
(1.2)

Here $A:\mathcal{D}(S)(\subseteq Q)\times V\to W^{*}$ is the (possibly
nonlinear) model operator, $B:V\to Z$ the (linear) observation operator and
$Q$, $V$, $W^{*}$, $Z$ are Banach spaces. Often $A$ represents the weak form
of a partial differential equation (with boundary and – if applicable –
initial conditions) and $W$ is the dual of a normed space. More generally,
consider

$F(x)=y$ (1.3)

with an operator $F:\mathcal{D}(F)(\subseteq X)\to Y$, mapping between Banach
spaces $X$ and $Y$, given noisy data $y^{\delta}$ with

$\|y-y^{\delta}\|_{Y}\leq\delta$ (1.4)

and denoting by $x^{\dagger}$ an exact solution. This comprises both all-at-
once (1.1) with $x=(q,u)$, $F=\mathbb{F}$ and reduced (1.2) with $x=q$,
$F=\mathbf{F}$ formulations. Since (1.3) is typically ill-posed,
regularization needs to be applied, cf., e.g., [2, 9, 21, 13, 27]. We will
here consider variational and iterative approaches for this purpose.

### Range invariance and other nonlinearity conditions

In order to prove convergence for regularization methods, conditions on the
nonlinearity of the forward operator $F$ are needed. One of the most commonly
used of these is the tangential cone condition

$\forall x,\tilde{x}\in U\,:\
\|F(x)-F(\tilde{x})-F^{\prime}(x)(x-\tilde{x})\|_{Y}\leq
c_{\textup{tc}}\|F(x)-F(\tilde{x})\|_{Y}$ (1.5)

in a neighborhood $U$ of the exact solution. This condition (or closely
related ones; cf. the introduction of [17] for a somewhat more detailed
overview) allows to show convergence of iterative regularization methods [11,
8, 18] as well as local convexity of the Tikhonov functional in variational
regularization [7]. The control by the residual provided by (1.5) allows to
use the discrepancy principle as an a posteriori rule for the regularization
parameter choice, which makes these methods - in particular the iterative ones
- very convenient to implement. However, the major drawback of (1.5) is that
it could so far only be verified in case of full observations or in a
discretized (thus not ill-posed in the sense of instability) setting.

An alternative condition that avoids these limitations is range invariance of
the linearized forward operator, cf., e.g., [4, 5, 8, 15, 14, 16, 26], which
we here (like in [17]) impose in a differential formulation:

$\exists x_{0}\in U\,,\ K\in L(\widetilde{X},Y)\,\forall x\in U\,\exists
r(x)\in\widetilde{X}\,:\ F(x)-F(x_{0})=Kr(x).$ (1.6)

This condition has been verified for a number of inverse problems with
boundary (or actually arbitrary) observations, see, e.g., [17, 19, 20].
However, as has been demonstrated there and we will also see in Examples 1 and
2 below, establishing (1.6) sometimes requires extension of the original
parameter space $\check{X}$ to some larger space $X$, that is, a relaxation of
the original inverse problem, and this may lead to a loss of unique
identifiability. Thus we add a penalty term that in the limit restricts
reconstructions to the original parameter space, where they can (more likely)
be shown to be unique. We do so by means of a penalty functional
$\mathcal{P}:X\to[0,\infty]$ such that $\mathcal{P}(x^{\dagger})=0$ for the
exact solution $x^{\dagger}$, more concretely
$\mathcal{P}(x^{\dagger})=\|P(x)\|_{X}^{p}$ for some operator $P$ (typically a
projection operator) and some $p\in[1,\infty)$.

The inverse problem (1.3) can therefore (locally in $U$) be rewritten as a
system

$\displaystyle K\hat{r}=y-F(x_{0})$ $\displaystyle\textup{ in }Y$ (1.7)
$\displaystyle r(x)=\hat{r}$ $\displaystyle\textup{ in }\widetilde{X}$
$\displaystyle P(x)=0$ $\displaystyle\textup{ in }X$

for the unknowns $(\hat{r},x)\in\widetilde{X}\times U$, $U\subseteq X$; see
also [8, Remark 2.2].

### Structure exploiting reconstruction approaches

A natural way of applying variational regularization to (1.7) is

$\displaystyle(\hat{r}_{\alpha,\beta}^{\delta},x_{\alpha,\beta}^{\delta})\in\mbox{argmin}_{(\hat{r},x)\in\widetilde{X}\times
U}J_{\alpha,\beta}^{\delta}(\hat{r},x)$ (1.8) $\displaystyle\text{where
}J_{\alpha,\beta}^{\delta}(\hat{r},x):=\underbrace{\|K\hat{r}+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha\mathcal{R}(\hat{r})}_{=:J_{\alpha}(\hat{r})}+\underbrace{\beta\mathcal{Q}(r(x),\hat{r})^{p}+\|P(x)\|_{X}^{p}}_{=:J_{\beta}(\hat{r},x)}$

with some $p\in[1,\infty)$, some regularization functional
$\mathcal{R}:\widetilde{X}\to[0,\infty]$, and some distance functional
$\mathcal{Q}:\widetilde{X}^{2}\to[0,\infty]$ such that
$\mathcal{Q}(\hat{r}_{1},\hat{r}_{2})=0$ implies $\hat{r}_{1}=\hat{r}_{2}$.
Note that a minimizer of (1.8) can be computed more easily than for the
original Tikhonov functional

$\check{J}_{\alpha}^{\delta}(\check{x})=\|Kr(\check{x})+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha\check{\mathcal{R}}(\check{x}),$
(1.9)

since if $\mathcal{R}$ is convex, so is $J_{\alpha}$ (in case of $p=2$ and
$\mathcal{R}$ being defined by a squared norm, $J_{\alpha}$ is even quadratic)
and for $\mathcal{Q}$ convex with respect to its first component,
$J_{\beta}(\hat{r},\cdot)$ is locally uniformly convex.

An efficient way to solve a structured minimization problem like (1.8) is to
alternatingly solve minimization problems for the two additive parts for a
subset of the variables only and iterate this procedure, see, e.g. [1, 3, 10,
25, 22]. Since in our case $J_{\alpha}$ only depends on $\hat{r}$, this leads
to the following (non-iterative) one-step split minimization procedure.

$\displaystyle\hat{r}_{sm}^{\delta}\in\mbox{argmin}_{\hat{r}\in\widetilde{X}}J_{\alpha}(\hat{r})\text{
where
}J_{\alpha}(\hat{r})=\|K\hat{r}+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha\mathcal{R}(\hat{r})$
(1.10) $\displaystyle x_{sm}^{\delta}\in\mbox{argmin}_{x\in
U}J_{\beta}(\hat{r},x)\text{ where
}J_{\beta}(\hat{r},x)=\beta\mathcal{Q}(r(x),\hat{r}_{sm})^{p}+\|P(x)\|_{X}^{p}.$

To define a regularized (frozen) Newton method, we locally linearize $r$ in
(1.8), assuming that it is Gâteaux differentiable.

$\displaystyle(\hat{r}_{n+1}^{\delta},x_{n+1}^{\delta})\in\mbox{argmin}_{(\hat{r},x)\in\widetilde{X}\times
U}J_{n}^{\delta}(\hat{r},x)$ (1.11) $\displaystyle\mbox{where
}J_{n}^{\delta}(\hat{r},x):=\|K\hat{r}+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha_{n}\mathcal{R}(\hat{r})$
$\displaystyle\hskip
113.81102pt+\beta_{n}\mathcal{Q}(r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x-x_{n}^{\delta}),\hat{r})^{p}+\|P(x)\|^{p}$

with some $p\in[1,\infty)$, $\alpha_{n},\beta_{n}>0$ and functionals
$\mathcal{Q}$, $\mathcal{R}$ as above.

In [17], convergence without rates has been proven for (1.8), (1.11) (but see
Proposition 12 for the current, slightly different setting). In this paper we
aim to establish convergence rates under additional regularity assumptions on
the exact solution $x^{\dagger}$ that can be cast into so-called variational
source conditions. Moreover, we will analyze convergence of the split
minimization method (1.10) under these conditions.

### Some illustrative examples

######  Example 1.

Identify $c(x)$ (that is, $q=c$ in (1.1) and (1.2)) in the elliptic PDE

$-\Delta u+c\,u=0\text{ in }\Omega$ (1.12)

on a smooth bounded domain $\Omega\subseteq\mathbb{R}^{d}$, $d\in\\{2,3\\}$,
from the Neumann-to-Dirichlet N-t-D map $\Lambda\in
L(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$; that is,
$\check{\mathbf{F}}(c):=(\text{tr}^{D}_{\partial\Omega}u_{n})_{n\in\mathbb{N}}$
where $u_{n}$ solves

$-\Delta u_{n}+c\,u_{n}=0\ \text{ in
}\Omega,\quad\partial_{\nu}u_{n}=\varphi_{n}\ \text{ on
}\partial\Omega,\quad\int_{\partial\Omega}u_{n}\,d\Gamma=0$ (1.13)

for a basis of boundary currents $\varphi_{n}\in L^{2}(\partial\Omega)$. Here
$\text{tr}^{D}_{\partial\Omega}:H^{1}(\Omega)\to H^{1/2}(\partial\Omega)$
denotes the Dirichlet trace operator; later on we will also use the Neumann
trace $\text{tr}^{N}_{\partial\Omega}:H^{s}(\Omega)\to
H^{s-3/2}(\partial\Omega)$, $s\in\mathbb{R}$.

Note that the Hilbert-Schmidt norm of $\Lambda$ when considered as an operator
from $L^{2}(\partial\Omega)$ into itself [28, Section 6.1.2] is equivalent to
the $l^{2}(\mathbb{N};L^{2}(\partial\Omega))$ norm of $\check{\mathbf{F}}(c)$,
when ortho-normalizing the basis functions $\varphi_{n}$ in
$L^{2}(\partial\Omega)$.

To establish (1.6), we extend the parameter space 111 the space $Q$ yet to be
specified, cf. Example 3 below by substituting $c\in Q$ with a sequence of
potentials $\vec{c}=(c_{j})_{j\in\mathbb{N}}\in\ell^{\infty}(Q)$ and on the
other hand penalizing $\|P\vec{c}\|$, where

$P\vec{c}=\Bigl{(}c_{j}-\bigl{(}\sum_{k\in\mathbb{N}}w_{k}\bigr{)}^{-1}\,\sum_{k\in\mathbb{N}}w_{k}c_{k}\Bigr{)}_{j\in\mathbb{N}}.$
(1.14)

for some $\vec{w}\in\ell^{1}(\mathbb{R}^{+})$ (e.g.,
$\vec{w}:=(j^{-2})_{j\in\mathbb{N}}$). The $\ell^{\infty}$ norm is used to
allow for the true solution – the constant sequence with value $c$ – to be
contained in this space. The projection $I-P$ on the subspace of constant
sequences is Lipschitz continuous, cf. (3.6) below. Correspondigly, we re-
define the forward operator

$\mathbf{F}(\vec{c})_{n}:=(\text{tr}^{D}_{\partial\Omega}u_{n})_{n\in\mathbb{N}}\text{
where $u_{n}=S(\vec{c})_{n}$ solves }\left\\{\begin{array}[]{rll}-\Delta
u_{n}+c_{n}\,u_{n}&=0&\text{ in }\Omega\\\
\partial_{\nu}u_{n}&=\varphi_{n}&\text{ on }\partial\Omega\\\
\int_{\partial\Omega}u_{n}\,d\Gamma&=0.\end{array}\right.$ (1.15)

Indeed, it is readily checked that by setting

$r(\vec{c})_{j}:=(c_{j}-c_{0,j})\frac{u_{j}}{u_{0,j}},\quad
K:=F^{\prime}(\vec{c}_{0})$ (1.16)

where $u_{j}=S(\vec{c})_{j}$, $u_{0j}=S(\vec{c}_{0})_{j}$, we formally obtain
(1.6). The problem of dividing by zero in (1.16) in view of the last condition
in (1.15) can be avoided by using an all-at-once formulation, in which we
define the forward operator by

$\mathbb{F}(\vec{c},\vec{u})_{n}:=\left(\begin{array}[]{c}v\mapsto\int_{\Omega}(\nabla
u_{n}\cdot\nabla
v+c_{n}\,u_{n}\,v)\,dx-\int_{\partial\Omega}\varphi_{n}\,v\,d\Gamma\\\
\int_{\partial\Omega}u_{n}\,d\Gamma\\\
\text{tr}^{D}_{\partial\Omega}u_{n}\end{array}\right)$ (1.17)

Also in this setting, the function defined by (1.16) can be used to formally
verify (1.6), but $u_{0,n}$ does not need to be a PDE solution and in
particular does not need to have vanishing average over the boundary any more.

######  Example 2.

Identify $\sigma(x)$ (that is, $q=\sigma$ in (1.1) and (1.2)) in

$-\nabla\cdot(\sigma\nabla u)=0\textup{ in }\Omega$ (1.18)

on a smooth bounded domain $\Omega\subseteq\mathbb{R}^{d}$, $d\in\\{2,3\\}$,
from the N-t-D map $\Lambda\in
L(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$;
that is,
$\check{\mathbf{F}}(\sigma)=(\text{tr}^{D}_{\partial\Omega}u_{n})_{n\in\mathbb{N}}$
where $u_{n}$ solves

$-\nabla\cdot(\sigma\nabla u_{n})=0\ \text{ in
}\Omega,\quad\partial_{\nu}u_{n}=\varphi_{n},\ \text{ on
}\partial\Omega,\quad\int_{\partial\Omega}u_{n}\,d\Gamma=0$

for a basis of boundary currents $\varphi_{n}\in L^{2}(\partial\Omega)$. This
is a prototype version of the well-known electrical impedance tomography EIT
problem going back to Calderón [6], and extensively studied in the inverse
problems literature, see, e.g., [23, Chapter 12] or [28, Chapter 6] for
further references.

Here, in order to establish (1.6), we do the extension on the observations, by
introducing artificial boundary voltages $\vec{z}=(z_{n})_{n\in\mathbb{N}}$,
thus extending the forward operator to

$\mathbf{F}(\sigma,\vec{z})=(\text{tr}^{D}_{\partial\Omega}u_{n}+z_{n})_{n\in\mathbb{N}}\text{
where $u_{n}=S(\vec{c})_{n}$ solves
}\left\\{\begin{array}[]{rll}-\nabla\cdot(\sigma\nabla u_{n})&=0&\text{ in
}\Omega\\\ \partial_{\nu}u_{n}&=\varphi_{n}&\text{ on }\partial\Omega\\\
\int_{\partial\Omega}u_{n}\,d\Gamma&=0.\end{array}\right.$ (1.19)

and penalizing $\|P(\sigma,\vec{z})\|$, where

$P(\sigma,\vec{z})=\vec{z},$ (1.20)

This formally satisfies (1.6) with

$\displaystyle r(\sigma,\vec{z})_{j}:=(\sigma-\sigma_{0},\
z_{j}-z_{0,j}+\text{tr}^{D}_{\partial\Omega}v_{j}),\quad
K(\underline{d\sigma},\underline{d\vec{z}}):=\check{\mathbf{F}}^{\prime}(\sigma_{0})\underline{d\sigma}+\underline{d\vec{z}},$
(1.21) $\displaystyle\text{ where $v_{j}$ solves
}\left\\{\begin{array}[]{rll}-\nabla\cdot(\sigma\nabla
v_{j})&=\nabla\cdot\bigl{(}(\sigma-\sigma_{0})\nabla(u_{j}-u_{0,j})\bigr{)}&\text{
in }\Omega\\\ \partial_{\nu}v_{j}&=0&\text{ on }\partial\Omega\\\
\int_{\partial\Omega}v_{j}\,d\Gamma&=0.\end{array}\right.$

The relaxation we have done here is by far less extensive than the one from
Example 1: In view of the bijectivity of the operator mapping $\sigma$ to the
sequence of boundary voltage data corresponding to currents
$(\varphi_{n})_{n\in\mathbb{N}}$ it can be viewed as equivalent to adding a
second conductivity function as an artificial parameter.

Note that this extension follows a general concept of achieving (1.6) by
extension in data space, that we will briefly discuss in Section 2.4.

A well-known relation to Example 1 that has often been used for uniqueness and
stability proofs and also for establishing variational source conditions
(1.28) in [28] is given by the identity

$c=\Phi(\sigma)=\frac{\Delta\sqrt{\sigma}}{\sqrt{\sigma}}.$ (1.22)

that allows to substitute $u$ solving (1.18) by $\tilde{u}:=\sqrt{\sigma}\,u$
solving (1.12). Vice versa, since $\sigma$ is assumed to take the known (often
also constant) background value $\sigma_{bg}$ outside a ball of radius $\rho$
contained in $\Omega$

$\sigma(x)=\sigma_{bg}(x)\quad
x\in\mathbb{R}^{d}\setminus\mathcal{B_{\rho}}(0),\quad\mathcal{B}_{\rho+\epsilon}(0)\subseteq\Omega$
(1.23)

for some $\rho$, $\epsilon>0$, we can also uniquely determine $\sigma$ from
$c$ by means of the elliptic boundary value problem

$-\Delta\sqrt{\sigma}+c\sqrt{\sigma}=0\ \text{ in
}\mathcal{B_{\rho}}(0),\quad\sqrt{\sigma}=\sqrt{\sigma_{bg}}\text{ on
}\partial\mathcal{B_{\rho}}(0).$ (1.24)

The precise function space setting for both examples will be provided in
Section 3 below, where due to the relations (1.22), (1.24) we have also
subsumed Example 1 under “application to EIT”.

### Notation and some convex analysis tools

With the dual space $X^{*}$ of some normed space $X$, we denote the dual
pairing by $\langle x^{*},x\rangle_{X^{*},X}$ for $x^{*}\in X^{*}$, $x\in X$.
For some proper convex functional $f$ on $X$, the subdifferential $\partial
f(x)=\\{\xi\in X^{*}\,:\,\langle\xi,\tilde{x}-x\rangle_{X^{*},X}\leq
f(\tilde{x})-f(x)\text{ for all }\tilde{x}\in\mathrm{dom}(f)\\}$ is known to
be nonempty.
We will use the Bregman distance:

$\Delta^{f}_{\xi_{2}}(x_{1},x_{2})=f(x_{1})-f(x_{2})-\langle
x_{2},x_{1}-x_{2}\rangle_{X^{*},X}\quad\text{for some selection
}\xi_{2}\in\partial f(x_{2}).$ (1.25)

The convex conjugate of a real function $f:\mathbb{R}\to\mathbb{R}$ is defined
by

$f^{*}(s)=\sup_{t\in\mathbb{R}}(s\,t-f(t)).$ (1.26)

We denote the (smallest) Lipschitz constant of a Lipschitz continuous map $f$
by $L_{f}$.

### Variational source conditions

In order to prove rates, we will make use of a variational source condition
VSC in the original formulation (before extension of the parameter space)

$\check{F}(\check{x})=y\text{ with
}\check{F}:\mathcal{D}(\check{F})(\subseteq\check{X})\to Y,\
\check{F}(\check{x})=F(\check{x})=Kr(\check{x})+F(x_{0})$ (1.27)

being defined on a subspace $\check{X}$ of $X$ such that $P|_{\check{X}}=0$
and endowed with a proper regularization functional
$\check{\mathcal{R}}:\check{X}\to[0,\infty]$. The VSC reads as follows

$\displaystyle\exists\check{\xi}\in\partial\check{\mathcal{R}}(x^{\dagger})\subseteq\check{X}^{*}\,\forall\check{x}\in\mathcal{L}_{\overline{R}}^{\check{\mathcal{R}}}:\quad-\langle\check{\xi},\check{x}-x^{\dagger}\rangle\leq
b\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}(\check{x},x^{\dagger})+\psi(\|\check{F}(\check{x})-\check{F}(x^{\dagger}))\|^{p})$
(1.28)

for some $b\in(0,1)$ (with
$\mathcal{L}_{\overline{R}}^{\check{\mathcal{R}}}=\\{\check{x}\in\check{X}\,:\,\check{\mathcal{R}}(\check{x})\leq\overline{R}\\}$
for some $\overline{R}\geq\check{\mathcal{R}}(x^{\dagger})$). We use VSCs in
this original resticted setting because for some important inverse problems
such as EIT, they are readily available from the literature, see for example
[28]. Here $\psi$ is supposed to be an index function, that is,

$\psi:[0,\infty)\to[0,\infty)\text{ is continuous and monotonically increasing
with }\psi(0)=0,$ (1.29)

in particular a low rate one

$\psi(t)\geq c_{\psi}t,\ t>0.$ (1.30)

Additionally, we extend $\psi$ by $-\infty$ for $t<0$ and assume that

$\displaystyle-\psi\text{ is convex, }(-\psi)^{*}\text{ is an index function
and }t\mapsto t\,(-\psi)^{*}(-\tfrac{1}{t})\text{ is strictly monotone}$
(1.31)

The most commonly used classes of such index functions are Hölder type and
logarithm type, respectively:

$(a)\ \psi(t)=t^{\mu},\qquad(b)\ \psi(t)=(-\log(\min\\{t,t_{0}\\}))^{-\nu}$
(1.32)

for some $\mu\in(0,1]$, $\nu>0$, $t_{0}>0$, cf, e.g., [28, Definition 2.2 and
(2.3a), (2.3b)]. The convex conjugates of $-\psi$ are given by [28, (2.25 a),
(2.25 b)]

$(a)\ (-\psi)^{*}(-\tfrac{1}{t})=c_{\mu}t^{\mu/(1-\mu)},\qquad(b)\
\psi(-\tfrac{1}{t})=(-\log(\min\\{t,t_{0}\\}))^{-\nu}(1+o(1))$

with $c_{\mu}=\mu^{\mu/(1-\mu)}-\mu^{1/(1-\mu)}$ and satisfy (1.31).

Since we will partly use VSCs in a reduced setting for proving rates in an
all-at-once setting, it is important to note that they are equivalent for the
reduced and the all-at-once formulation in the following sense.

######  Lemma 1.

Condition (1.28) for $\check{F}=\mathbb{F}$ as in (1.1) implies (1.28) for
$\check{F}=\mathbf{F}$ as in (1.2) with the same regularization functional
$\check{\mathcal{R}}$ (depending only on $q$).
If for any $q\in U$, the operator $A(q,\cdot)$ has a uniformly Lipschitz
continous inverse and $\check{\mathcal{R}}$ only depends on $q$, then
condition (1.28) for $\check{F}=\mathbf{F}$ as in (1.2) implies (1.28) for
$\check{F}=\mathbb{F}$ as in (1.1) with $u^{\dagger}=S(q^{\dagger})$, the same
regularization functional $\check{\mathcal{R}}$, and $\psi(t)$ replaced by
$\psi(\bar{C}t)$ with $\bar{C}=2^{p-1}\max\\{1,\|B\|^{p}L_{A^{-1}}^{p}\\}$.

Proof. The first assertion is trivial. The second one follows from
monotonicity of $\psi$ and the fact that for any fixed $q\in Q$ and any $u\in
V$

$\displaystyle\|\mathbf{F}(q)-\mathbf{F}(q^{\dagger})\|_{Y}^{p}=\|B(S(q)-S(q^{\dagger}))\|_{Y}^{p}\leq\Bigl{(}\|B(u-u^{\dagger})\|_{Y}+\|B\|\,\|u-S(q)\|\Bigr{)}^{p}$
$\displaystyle\leq\Bigl{(}\|B(u-u^{\dagger})\|_{Y}+\|B\|\,L_{A^{-1}}\|A(q,u)-A(q,S(q))\|\Bigr{)}^{p}$
$\displaystyle\leq
2^{p-1}\Bigl{(}\|B(u-u^{\dagger})\|_{Y}^{p}+\|B\|^{p}\,L_{A^{-1}}^{p}\|A(q,u)\|^{p}\Bigr{)}\leq\bar{C}\|\mathbb{F}(q,u)-\mathbb{F}(q^{\dagger},u^{\dagger})\|_{Y}^{p},$

since $A(q,S(q))=0$.

$\diamondsuit$

Since for fixed $q\in U$, the forward problem $A(q,\cdot)=0$ is typically
well-posed, Lipschitz continuity of the inverse of $A(q,\cdot)$ is a natural
condition to hold.
Note that also here we have the respective all-at-once and reduced
formulations before extension of the parameter space in mind in order to make
use of the results from [28, Section 5] for Example 1; see Example 3 below.

### Plan of the paper

Most of this paper is devoted to proving convergence rates of the methods
(1.8), (1.10), (1.11) in Section 2. In Section 3 we apply these results to the
EIT Examples 1, 2, thereby making use of results on variational source
conditions from [28].

##  2 Convergence rates

In this section we will prove convergence rates for the three regularization
schemes (1.8), (1.10), (1.11) in the relaxed formulation (1.7) under a VSC
(1.28) in the original formulation (1.27). To this end, throughout this
section we will assume that

$\displaystyle(I-P)(X)\subseteq\check{X}\,,\quad
P(x^{\dagger})=P(x_{0})=0,\text{ thus }\quad x^{\dagger},\ x_{0}\in\check{X}.$
(2.1)

This is satisfied, e.g, if $X$ is a Hilbert space,
$I-P=\text{Proj}_{\check{X}}$, (the orthogonal projection onto the subspace
$\check{X}$).

The key steps for deriving convergence rates for Tikhonov regularization (1.9)
in the original formulation (1.27) are easily explained in case $\delta=0$.
Firstly, minimality yields
$\check{J}_{\alpha}^{\delta}(\check{x}_{\alpha}^{\delta})\leq\check{J}_{\alpha}^{\delta}(x^{\dagger})$
and thus by (1.4), (1.25), (1.28)

$\|\check{F}(\check{x}_{\alpha}^{\delta})-\check{F}(x^{\dagger})\|^{p}+(1-b)\alpha\Delta^{\\!\\!\check{\mathcal{R}}}(\check{x}_{\alpha}^{\delta},x^{\dagger})\leq\psi(\|\check{F}(\check{x}_{\alpha}^{\delta})-\check{F}(x^{\dagger})\|^{p}),$

that is, an estimate in terms of the residual
$\|\check{F}(\check{x}_{\alpha}^{\delta})-\check{F}(x^{\dagger})\|^{p}$ and
the error
$\Delta^{\\!\\!\check{\mathcal{R}}}(\check{x}_{\alpha}^{\delta},x^{\dagger})$.
This can be resolved for the two single quantities by means of the following
Lemma, that can be found in several recent publications on convergence rates,
e.g., as part of [28, Theorem 2.20 (a)], and whose proof we provide in the
appendix, for the sake of completeness.

######  Lemma 2.

For any $\alpha$, $c_{1}$, $c_{2}$, $C_{3}$, $d$, err, $\textrm{res}\
\in[0,\infty)$ the relation

$c_{1}\textrm{res}+c_{2}\alpha\,\textrm{err}\leq
d+\alpha\,\psi(C_{3}\textrm{res})$ (2.2)

implies the estimates

$\textrm{err}\leq\frac{1}{c_{2}}\Bigl{(}\frac{d}{\alpha}+(-\psi)^{*}\Bigl{(}-\frac{\tilde{c}_{1}}{\alpha}\Bigr{)}\Bigr{)},\quad\textrm{res}\leq\frac{2}{c_{1}}\Bigl{(}d+\alpha(-\psi)^{*}\Bigl{(}-\frac{\tilde{c}_{1}}{2\alpha}\Bigr{)}\Bigr{)},$
(2.3)

where $(-\psi)^{*}(z)=\sup_{s\in\mathbb{R}}(z\,s-(-\psi)(s))$ is the convex
conjugate of $-\psi$ and $\tilde{c}_{1}=\tfrac{c_{1}}{C_{3}}$.

Combined with a proper choice of $\alpha=\alpha(\delta)$ this also yields
convergence rates with noisy data. In the relaxed setting (1.7) and for
methods (1.8), (1.10), (1.11), the estimates resulting from minimality become
substantially more involved and have to take into account and control the
interplay among the three components in (1.7).

###  2.1 Variational regularization

Existence of a minimizer of (1.8) can be established under certain coercivity,
lower semicontinuity and closedness conditions on $J_{\alpha,\beta}^{\delta}$
and $\widetilde{X}\times U$, cf. [17]. However, we can avoid them (as well as
exact computation of a minimizer) by defining
$(\hat{r}_{\alpha,\beta,\eta}^{\delta},x_{\alpha,\beta,\eta}^{\delta})\in\widetilde{X}\times
U$ to be any pair satisfying

$\forall(\hat{r},x)\in\widetilde{X}\times U\,:\
J_{\alpha,\beta}^{\delta}(\hat{r}_{\alpha,\beta,\eta}^{\delta},x_{\alpha,\beta,\eta}^{\delta})\leq
J_{\alpha,\beta}^{\delta}(\hat{r},x)+\eta$ (2.4)

for some positive tolerance $\eta>0$ that is chosen in dependence of the noise
level $\delta$, cf., e.g., [24] as is of course also the regularization
parameter $\alpha=\alpha(\delta)>0$. For the latter purpose, as opposed to the
case of a tangential cone condition (1.5), where the discrepancy principle is
a natural a posteriori choice, we here remain with an a priori choice:

$\eta\leq
C_{\eta}\delta^{p},\quad\underline{\tau}\delta^{p}\leq\varphi^{-1}(\alpha):=\alpha(-\psi)^{*}\left(-\frac{1}{2\bar{C}\alpha}\right)\leq\overline{\tau}\delta^{p}$
(2.5)

cf. (1.31) with some sufficiently large $\bar{C}>0$ (cf. (2.15), (2.29) below)
and some fixed safety factors $0<\underline{\tau}<\overline{\tau}$. The
parameter $\beta$ will be chosen as a fixed constant independent of $\delta$
in most of what follows (except for Proposition 12 where the choice is more
general but includes constant $\beta$) and $C_{\eta}>0$ is a fixed constant as
well.

We first of all aim to prove that under a variational source condition (1.28)
the rates

$\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})),x^{\dagger})=O(\widetilde{\psi}(\delta^{p})),\quad\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|_{Y}=O(\delta).$
(2.6)

hold with $\widetilde{\psi}$ related to $\psi$ (ideally
$\widetilde{\psi}=\psi$), cf. Theorem 3 below. Later on, from this and further
estimates, we will derive convergence rates on
$x_{\alpha,\beta,\eta}^{\delta}$, cf. Corollary 4.

To be able to make use of the VSC (1.28), we will require that the
regularization functional used in (1.8) is related to the functional appearing
in (1.28) by

$\mathcal{R}(\hat{r}):=\check{\mathcal{R}}((I-P)(r^{-1}(\hat{r})))$ (2.7)

and that the ties between $x$ and $\hat{r}$ imposed by the $\mathcal{Q}$ term
in $J_{\alpha,\beta}^{\delta}$ are strong enough to control the out-
of-$\check{X}$ parts

$\forall\hat{r}_{1},\hat{r}_{2}\in X\,:\
\|P(r^{-1}(\hat{r}_{1}))-P(r^{-1}(\hat{r}_{2}))\|_{X}\leq
C_{Q}\mathcal{Q}(\hat{r}_{1},\hat{r}_{2}).$ (2.8)

The latter holds, e.g., if $r:U\mapsto r(U)$ is bijective with Lipschitz
continuous inverse,
$\mathcal{Q}(\hat{r}_{1},\hat{r}_{2})=\|\hat{r}_{1}-\hat{r}_{2}\|_{X}$, and
$P$ is Lipschitz continuous, for example, the projection operator on a Hilbert
space.

######  Theorem 3.

Under the assumption (2.1), let $F$ satisfy (1.6) with Lipschitz continuous
$r$ and let the variational source condition (1.28) hold with $\psi$
satisfying (1.29), (1.30), (1.31). Moreover, assume that $\mathcal{R}$,
$\mathcal{Q}$ are chosen according to (2.7), (2.8).

Then for any family of data $(y^{\delta})_{\delta\in(0,\bar{\delta}]}$
satisfying (1.4) and regularization parameters $\alpha=\alpha(\delta)$,
$\eta=\eta(\delta)$ satisfying (2.5), any corresponding family of regularized
reconstructions according to (2.4) satisfies the convergence rate (2.6) with
(2.16), as well as

$\|P(x_{\alpha,\beta,\eta}^{\delta})\|=O(\delta),\quad\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta})=O(\delta).$
(2.9)

Proof. As usual in the analysis of variational regularization methods, we
start with a minimality estimate. Comparing
$J_{\alpha,\beta}^{\delta}(\hat{r}_{\alpha,\beta,\eta}^{\delta},x_{\alpha,\beta,\eta}^{\delta})$
with $J_{\alpha,\beta}^{\delta}(r(x^{\dagger}),x^{\dagger})$, using
$P(x^{\dagger})=0$ and

$\|K\hat{r}_{\alpha,\beta,\eta}^{\delta}+F(x_{0})-y^{\delta}\|_{Y}^{p}=\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))+y-y^{\delta}\|_{Y}^{p}\geq
2^{1-p}\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}-\delta^{p}$
(2.10)

we obtain

$\displaystyle
2^{1-p}\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}+\alpha\,(\mathcal{R}(\hat{r}_{\alpha,\beta,\eta}^{\delta})-\mathcal{R}(r(x^{\dagger})))$
(2.11)
$\displaystyle+\beta\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta})^{p}+\|P(x_{\alpha,\beta,\eta}^{\delta})\|^{p}\leq(2+C_{\eta})\delta^{p}.$

Now, using the VSC (1.28) with
$\check{x}=(I-P)r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})$ together with
(1.25), (2.7) we conclude

$\displaystyle\mathcal{R}(\hat{r}_{\alpha,\beta,\eta}^{\delta})-\mathcal{R}(r(x^{\dagger}))=\check{\mathcal{R}}((I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})))-\check{\mathcal{R}}(x^{\dagger})$
$\displaystyle=\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})),x^{\dagger})+\langle\check{\xi},(I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}))-x^{\dagger}\rangle$

where

$\displaystyle-\langle\check{\xi},(I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}))-x^{\dagger}\rangle$
$\displaystyle\leq
b\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}),x^{\dagger})+\psi\left(\|K\Bigl{(}r\bigl{(}(I-P)r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})\bigr{)}-r(x^{\dagger})\bigr{)}\|^{p}\right).$

Here, using (2.1), (2.8), and the monotonicity of $\psi$, we can further
estimate

$\displaystyle\psi\Bigl{(}\|K\bigl{(}r\bigl{(}(I-P)r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})\bigr{)}-r(x^{\dagger})\bigr{)}\|^{p}\Bigr{)}$
(2.12)
$\displaystyle\leq\psi\Bigl{(}2^{p-1}\Bigl{(}\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|^{p}+\|K\|^{p}L_{r}^{p}\|Pr^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})\|^{p}\Bigr{)}\Bigr{)}$
$\displaystyle\leq\psi\Bigl{(}2^{p-1}\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|^{p}+2^{2p-2}\|K\|^{p}L_{r}^{p}\Bigl{(}\|P(x_{\alpha,\beta,\eta}^{\delta})\|^{p}+C_{Q}^{p}\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta})^{p}\Bigr{)}\Bigr{)}.$

Altogether we obtain

$\displaystyle
2^{1-p}\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}+\beta\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta})^{p}+\|P(x_{\alpha,\beta,\eta}^{\delta})\|^{p}$
(2.13) $\displaystyle\hskip
85.35826pt+\alpha\,(1-b)\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}),x^{\dagger})$
$\displaystyle\leq\psi\Bigl{(}2^{p-1}\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|^{p}+2^{2p-2}\|K\|^{p}L_{r}^{p}\Bigl{(}C_{Q}^{p}\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta})^{p}+\|P(x_{\alpha,\beta,\eta}^{\delta})\|^{p}\Bigr{)}\Bigr{)}$
$\displaystyle\qquad+(2+C_{\eta})\delta^{p}.$

Abbreviating

err
$\displaystyle:=(1-b)\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})),x^{\dagger}),$
res
$\displaystyle:=2^{1-p}\|K\Bigl{(}\hat{r}_{\alpha,\beta,\eta}^{\delta})-r(x^{\dagger})\Bigr{)}\|^{p}+\beta\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta})^{p}+\|P(x_{\alpha,\beta,\eta}^{\delta})\|^{p},$

we obtain that

$\textrm{res}+\alpha\,\textrm{err}\leq(2+C_{\eta})\delta^{p}+\alpha\,\psi(\bar{C}\textrm{res})$
(2.14)

with

$\bar{C}:=2^{2p-2}\max\\{1,\|K\|^{p}L_{r}^{p},\tfrac{\|K\|^{p}L_{r}^{p}C_{Q}^{p}}{\beta}\\}.$
(2.15)

Now an application of Lemma 2 to (2.14) with the regularization parameter
choice (2.5) and (1.31) implies the rate

$\displaystyle\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta})),x^{\dagger})\leq\bar{\bar{C}}(-\psi)^{*}\Bigl{(}-\frac{1}{\bar{C}\alpha}\Bigr{)}\leq\bar{\bar{C}}(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\varphi(\overline{\tau}\delta^{p})}\Bigr{)}$
$\displaystyle\|K(\hat{r}_{\alpha,\beta,\eta}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}\leq
2^{p-1}\Bigl{(}(2+C_{\eta})\delta^{p}+\alpha(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\varphi(\overline{\tau}\delta^{p})}\Bigr{)}\Bigr{)}\leq
2^{p-1}(2+C_{\eta}+\overline{\tau})\delta^{p}$

with
$\bar{\bar{C}}=\frac{1}{1-b}\Bigl{(}\frac{2+C_{\eta}}{\underline{\tau}}+1\Bigr{)}$;
that is, (2.6) with

$\widetilde{\psi}(d)=(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\varphi(\overline{\tau}d)}\Bigr{)},\qquad\varphi^{-1}(t)=t\,(-\psi)^{*}(-\frac{1}{2\bar{C}t}),$
(2.16)

cf. (1.31).

$\diamondsuit$

######  Remark 1.

Note that the ideal choice of $\alpha=\alpha(\delta)$ minimizing
$\frac{d}{\alpha}+(-\psi)^{*}\Bigl{(}-\frac{1}{\bar{C}\alpha}\Bigr{)}$ with
$d=(2+C_{\eta})\delta^{p}$ yields the best possible rate cf. [28, Chapter IV]
and the references therein.

err
$\displaystyle\leq\inf_{\alpha}\Bigl{(}\frac{d}{\alpha}+(-\psi)^{*}\Bigl{(}-\frac{1}{\bar{C}\alpha}\Bigr{)}\Bigr{)}=\inf_{\alpha}\Bigl{(}\left(-\bar{C}d\right)\left(-\frac{1}{\bar{C}\alpha}\right)+(-\psi)^{*}\Bigl{(}-\frac{1}{\bar{C}\alpha}\Bigr{)}\Bigr{)}$
$\displaystyle=-\sup_{s}\Bigl{(}\left(\bar{C}d\right)\,s-(-\psi)^{*}(s)\Bigr{)}=-(-\psi)^{**}\left(\bar{C}d\right)=\psi\left(\bar{C}d\right).$

On the other hand, in the logarithmic case (1.32) (b), it is readily checked
that the simple choice $\alpha(\delta)\sim\delta^{\tilde{p}}$ with
$\tilde{p}\in(0,p]$ yields the same rate.

A convergence rate of $x_{\alpha,\beta,\eta}^{\delta}$ to $x^{\dagger}$ can be
obtained by combining the $\Delta^{\\!\\!\check{\mathcal{R}}}_{\xi}$,
$\mathcal{Q}$ and $P$ terms. To this end, consider the more particular setting
that $\check{\mathcal{R}}$ is defined by a norm on an $s$-convex space $X$ and
correspondingly $\mathcal{Q}$ is defined by the norm on $\widetilde{X}$

$\check{\mathcal{R}}(\check{x}):=\|\check{x}\|_{\check{X}}^{s}\,\quad\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}(\check{x},x^{\dagger})\geq
c_{s}\|\check{x}-x^{\dagger}\|_{\check{X}}^{s}\,\quad\mathcal{Q}(\hat{r}_{1},\hat{r}_{2}):=\|\hat{r}_{1}-\hat{r}_{2}\|_{\widetilde{X}}$
(2.17)

for all $\hat{r}_{1},\hat{r}_{2}\in\widetilde{X}$, $\check{x}\in\check{X}$.
Here $s\geq 2$, cf. e.g. [27, Example 2.47].

######  Corollary 4.

Under the assumption (2.1) with an $s$-convex space $\check{X}$, let $F$
satisfy (1.6) with Lipschitz continuous $r$, $r^{-1}$, $P$, and let the
variational source condition (1.28) hold with $\psi$ satisfying (1.29),
(1.30), (1.31). Moreover, assume that $\mathcal{R}$, $\mathcal{Q}$ are chosen
according to (2.7), (2.17).

Then

$\|x_{\alpha,\beta,\eta}^{\delta}-x^{\dagger}\|_{X}^{s}=O(\widetilde{\psi}(\delta^{p})).$

Proof. By the triangle inequality we have

$\displaystyle\|x_{\alpha,\beta,\eta}^{\delta}-x^{\dagger}\|_{X}$
$\displaystyle\leq\|P(x_{\alpha,\beta,\eta}^{\delta})\|_{X}+\|(I-P)(x_{\alpha,\beta,\eta}^{\delta})-(I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}))\|_{X}+\|(I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}))-x^{\dagger}\|_{X}$
$\displaystyle\leq O(\delta)+O(\widetilde{\psi}(\delta^{p}))^{1/s},$

where we have estimated

$\|(I-P)(x_{\alpha,\beta,\eta}^{\delta})-(I-P)(r^{-1}(\hat{r}_{\alpha,\beta,\eta}^{\delta}))\|_{X}\leq
L_{I-P}L_{r^{-1}}\mathcal{Q}(r(x_{\alpha,\beta,\eta}^{\delta}),\hat{r}_{\alpha,\beta,\eta}^{\delta}).$

and used (2.9).

$\diamondsuit$

For the sake of completeness and to show that these results are compatible
with convergence without rates in case a source condition is missing or
unknown, we provide a result (whose proof is very similar to the one of [17,
Theorem 2.3]) in the setting under consideration here in the appendix, see
Proposition 12 there.

###  2.2 Split minimization

As in Section 2.1, in both subproblems of (1.10) we minimize only up to a
tolerance $\eta\leq C_{\eta}\delta^{p}$,

$\displaystyle\forall\hat{r}\in\widetilde{X}\,:\
J_{\alpha}^{\delta}(\hat{r}_{sm}^{\delta})\leq
J_{\alpha}^{\delta}(\hat{r})+\eta$ (2.18) $\displaystyle\forall x\in U\,:\
J_{\beta}^{\delta}(\hat{r}_{sm}^{\delta},x_{sm}^{\delta})\leq
J_{\beta}^{\delta}(\hat{r}_{sm}^{\delta},x)+\eta$

and compare with the value at $\hat{r}=r(x^{\dagger})$ and at $x=x^{\dagger}$
to obtain

$\displaystyle\|K\hat{r}_{sm}^{\delta}+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha\mathcal{R}(\hat{r}_{sm}^{\delta})\leq\alpha\mathcal{R}(r(x^{\dagger}))+(1+C_{\eta})\delta^{p}$
(2.19)
$\displaystyle\beta\mathcal{Q}(r(x_{sm}),\hat{r}_{sm})^{p}+\|P(x_{sm})\|_{X}^{p}\leq\beta\mathcal{Q}(r(x^{\dagger}),\hat{r}_{sm})^{p}+C_{\eta}\delta^{p}$

Note that the additional right hand side term
$\beta\mathcal{Q}(r(x^{\dagger}),\hat{r}_{sm})^{p}$ is due to suboptimality of
split as compared to joint minimization and needs a dedicated estimate. For
this reason, instead of (2.7) we set

$\displaystyle\mathcal{R}(\hat{r}):=\check{\mathcal{R}}((I-P)(r^{-1}(\hat{r})))+\mathcal{R}_{P}(P(r^{-1}(\hat{r})))$
(2.20) $\displaystyle\text{ with
}\mathcal{R}_{P}(P(r^{-1}(\hat{r})))\geq\gamma\psi(C_{P}\|P(r^{-1}(\hat{r}))\|^{p})$

for sufficiently large constants $\gamma$, $C_{P}>0$

$\gamma>C_{\psi},\quad C_{P}\geq 2^{p-1}\|K\|^{p}L_{r}^{p}$ (2.21)

and make an assumption that allows to control
$\beta\mathcal{Q}(r(x^{\dagger}),\hat{r}_{sm})^{p}$ by means of the
regularzation term

$\forall\hat{r}\in\widetilde{X}:\quad\mathcal{Q}(r(x^{\dagger}),\hat{r})^{p}\leq\tilde{C}_{Q}\Bigl{(}\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}),x^{\dagger})+\psi\Bigl{(}C_{P}\|P(r^{-1}(\hat{r}))\|^{p}\Bigr{)}.$
(2.22)

We also use an alternative estimate of the $\psi$ term as compared to (2.12)
that is obtained by replacing the assumption (2.8) by

$\forall a,b\in\mathbb{R}^{+}:\quad\psi(a+b)\leq C_{\psi}(\psi(a)+\psi(b))$
(2.23)

for some $C_{\psi}>0$ (which is satisfied for all relevant cases of $\psi$,
see the appendix).

######  Theorem 5.

Under the assumption (2.1), let $F$ satisfy (1.6) with Lipschitz continuous
$r$ and let the variational source condition (1.28) hold with $\psi$
satisfying (1.29), (1.30), (1.31), (2.23). Moreover, assume that
$\mathcal{R}$, $\mathcal{Q}$ are chosen according to (2.20), (2.21), (2.22).

Then for any family of data $(y^{\delta})_{\delta\in(0,\bar{\delta}]}$
satisfying (1.4) and regularization parameters $\alpha=\alpha(\delta)$,
$\eta=\eta(\delta)$ satisfying (2.5), a corresponding family of regularized
reconstructions according to (2.4) or (2.18) satisfies the convergence rate
(2.6) with (2.16) as well as

$\displaystyle\|P(x_{sm})\|^{p}=O(\widetilde{\psi}(\delta^{p})),\quad\mathcal{Q}(r(x_{sm}^{\delta}),\hat{r}_{sm}^{\delta})^{p}=O(\widetilde{\psi}(\delta^{p}))$
(2.24)
$\displaystyle\|P(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}^{p}=O(\psi^{-1}(\widetilde{\psi}(\delta^{p}))).$

Proof. Assumption (2.23) allows to bound the $\psi$ term arising from setting
$\check{x}:=(I-P)r^{-1}(\hat{r}_{sm}^{\delta})$ in the VSC (1.28) as follows

$\displaystyle\psi\Bigl{(}\|K(r\bigl{(}(I-P)r^{-1}(\hat{r}_{sm}^{\delta})\bigr{)}-r(x^{\dagger}))\|^{p}\Bigr{)}$
(2.25) $\displaystyle\leq
C_{\psi}\Bigl{(}\psi\Bigl{(}2^{p-1}\|K(\hat{r}_{sm}^{\delta}-r(x^{\dagger}))\|^{p}\Bigr{)}+\psi\Bigl{(}2^{p-1}\|K\|^{p}L_{r}^{p}\|Pr^{-1}(\hat{r}_{sm}^{\delta})\|^{p}\Bigr{)}\Bigr{)}.$

From (2.19), analogously to (2.13), using (2.10), (2.20), we obtain

$\displaystyle
2^{1-p}\|K(\hat{r}_{sm}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}+\alpha\,(1-b)\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{sm}^{\delta}),x^{\dagger})$
(2.26) $\displaystyle\hskip
85.35826pt+\alpha\,\Bigl{(}\gamma\psi\Bigl{(}C_{P}\|P(r^{-1}(\hat{r}_{sm}))\|^{p}\Bigr{)}-C_{\psi}\psi\Bigl{(}2^{p-1}\|K\|^{p}L_{r}^{p}\|Pr^{-1}(\hat{r}_{sm}^{\delta})\|^{p}\Bigr{)}$
$\displaystyle\hskip 85.35826pt\leq\alpha
C_{\psi}\psi\Bigl{(}2^{p-1}\Bigl{(}\|K(\hat{r}_{sm}^{\delta}-r(x^{\dagger}))\|^{p}\Bigr{)}+(2+C_{\eta})\delta^{p}$
$\displaystyle\beta\mathcal{Q}(r(x_{sm}),\hat{r}_{sm})^{p}+\|P(x_{sm})\|_{X}^{p}\leq\beta\mathcal{Q}(r(x^{\dagger}),\hat{r}_{sm})^{p}+C_{\eta}\delta^{p},$

(where the second estimate is just a repetition of the second estimate in
(2.19) for the convenience of the reader). This suggests to choose $\gamma$,
$C_{P}$ such that (2.21) holds and motivates the assumption (2.22) on
$\mathcal{Q}$.

With this, an additive combination of both estimates in (2.26) (after
multiplying the second one with $\frac{\alpha}{\beta}\underline{c}$, where
$\underline{c}=\frac{\min\\{1-b,\gamma-C_{\psi}\\}}{2\tilde{C}_{Q}}$) yields

$\displaystyle
2^{1-p}\|K(\hat{r}_{sm}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}+\alpha\,\tfrac{\min\\{1-b,\gamma-
C_{\psi}\\}}{2}\Bigl{(}\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{sm}^{\delta}),x^{\dagger})+\psi\bigl{(}C_{P}\|P(r^{-1}(\hat{r}_{sm}))\|^{p}\bigr{)}\Bigr{)}$
(2.27) $\displaystyle\hskip
85.35826pt+\alpha\underline{c}\Bigl{(}\mathcal{Q}(r(x_{sm}),\hat{r}_{sm})^{p}+\tfrac{1}{\beta}\|P(x_{sm})\|_{X}^{p}\Bigr{)}$
$\displaystyle\hskip 85.35826pt\leq
C_{\psi}\psi\Bigl{(}2^{p-1}\Bigl{(}\|K(\hat{r}_{sm}^{\delta}-r(x^{\dagger}))\|^{p}\Bigr{)}+(2+C_{\eta}+\tfrac{\alpha}{\beta}\underline{c}C_{\eta})\delta^{p}.$

Setting

err $\displaystyle:=\tfrac{\min\\{1-b,\gamma-
C_{\psi}\\}}{2C_{\psi}}\Bigl{(}\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{sm}^{\delta}),x^{\dagger})+\psi\bigl{(}C_{P}\|P(r^{-1}(\hat{r}_{sm}))\|^{p}\bigr{)}\Bigr{)}$
$\displaystyle\hskip
85.35826pt+\tfrac{\underline{c}}{C_{\psi}}\Bigl{(}\mathcal{Q}(r(x_{sm}),\hat{r}_{sm})^{p}+\tfrac{1}{\beta}\|P(x_{sm})\|_{X}^{p}\Bigr{)},$
res
$\displaystyle:=\tfrac{2^{1-p}}{C_{\psi}}\|K(\hat{r}_{sm}^{\delta})-r(x^{\dagger})\Bigr{)}\|^{p}$

we can rewrite this as

$\textrm{res}+\alpha\,\textrm{err}\leq\tfrac{2+C_{\eta}+\tfrac{\alpha}{\beta}\underline{c}C_{\eta}}{C_{\psi}}\delta^{p}+\alpha\,\psi(\bar{C}\textrm{res})$
(2.28)

with

$\bar{C}:=2^{2p-2}.$ (2.29)

Estimate (2.19) (with the right hand side in the second line replaced by
$(1+C_{\eta})\delta^{p}$) also holds true for the original simultaneous
minimization version (1.8), so that we can draw the same conclusions on it.

Applying Lemma 2 to (2.28), analogously to Theorem 3 we obtain the assertion.

$\diamondsuit$

To obtain a convergence rate of $x_{sm}^{\delta}$ to $x^{\dagger}$ we again
assume that $\check{\mathcal{R}}$ is defined by a norm on an $s$-convex space
$X$, but in order to be compatible with (2.22), make a slightly different
assumption on $\mathcal{Q}$ as compared to (2.17), namely:

$\displaystyle\check{\mathcal{R}}(x):=\|x\|_{X}^{s}\,\quad\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}(x,x^{\dagger})\geq
c_{s}\|x-x^{\dagger}\|_{X}^{s}\,$ (2.30)
$\displaystyle\overline{C}_{Q}\mathcal{Q}(\hat{r}_{1},\hat{r}_{2})^{p}\geq\max\Bigl{\\{}\|(I-P)r^{-1}(\hat{r}_{2})-(I-P)r^{-1}(\hat{r}_{1})\|_{X}^{s},$
$\displaystyle\hskip
113.81102ptc_{\psi}C_{P}\|P(r^{-1}(\hat{r}_{2}))-P(r^{-1}(\hat{r}_{1}))\|^{p}\Bigr{\\}},$

for all $\hat{r}_{1},\hat{r}_{2}\in\widetilde{X}$, $x\in X$, with $c_{\psi}$
as in (1.31).

######  Corollary 6.

Under the assumption (2.1) with an $s$-convex space $\check{X}$, let $F$
satisfy (1.6) with Lipschitz continuous $r$, $r^{-1}$, and let the variational
source condition (1.28) hold with $\psi$ satisfying (1.29), (1.30), (1.31),
(2.23). Moreover, assume that $\mathcal{R}$, $\mathcal{Q}$ are chosen
according to (2.20), (2.21), (2.22), (2.30).

Then

$\|x_{sm}^{\delta}-x^{\dagger}\|_{X}^{s}=O(\widetilde{\psi}(\delta^{p})).$

Proof. By the triangle inequality we have

$\displaystyle\|x_{sm}^{\delta}-x^{\dagger}\|_{X}$
$\displaystyle\leq\|x_{sm}^{\delta}-r^{-1}(\hat{r}_{sm}^{\delta})\|_{X}+\|r^{-1}(\hat{r}_{sm}^{\delta})-x^{\dagger}\|_{X}$
$\displaystyle\leq\|(I-P)(x_{sm}^{\delta})-(I-P)(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}+\|P(x_{sm}^{\delta})-P(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}$
$\displaystyle+\|(I-P)(r^{-1}(\hat{r}_{sm}^{\delta}))-x^{\dagger}\|_{X}+\|P(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}$
$\displaystyle\leq
O(\widetilde{\psi}(\delta^{p})^{1/s})+O(\widetilde{\psi}(\delta^{p})^{1/p})+O(\psi^{-1}(\widetilde{\psi}(\delta^{p})\bigr{)}^{1/p})$

where we have estimated (cf. (2.24))

$\displaystyle\|(I-P)(x_{sm}^{\delta})-(I-P)(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}\leq\overline{C}_{Q}^{1/s}\mathcal{Q}(r(x_{sm}^{\delta}),\hat{r}_{sm}^{\delta})^{p/s}=O(\widetilde{\psi}(\delta^{p})^{1/s})$
$\displaystyle\|P(x_{sm}^{\delta})-P(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}\leq\left(\tfrac{\overline{C}_{Q}}{C_{P}c_{\psi}}\right)^{1/p}\mathcal{Q}(r(x_{sm}^{\delta}),\hat{r}_{sm}^{\delta})=O(\widetilde{\psi}(\delta^{p})^{1/p})$
$\displaystyle\|(I-P)(r^{-1}(\hat{r}_{sm}^{\delta}))-x^{\dagger}\|_{X}\leq
c_{s}^{-1/s}\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{sm}^{\delta})),x^{\dagger})^{1/s}=O(\widetilde{\psi}(\delta^{p})^{1/s})$
$\displaystyle\|P(r^{-1}(\hat{r}_{sm}^{\delta}))\|_{X}=O(\psi^{-1}(\widetilde{\psi}(\delta^{p}))^{1/p}).$

$\diamondsuit$

######  Remark 2.

Since $\psi$ (or at least its decay rate) needs to be known to enable
construction of a regularization functional satisfying (2.20), a convergence
result without source condition like Proposition 12 would not make sense here.

###  2.3 Frozen Newton

A minimizer to (1.11) exists if $U$ is weakly closed (e.g., $U$ is closed and
convex) and $\mathcal{P}$ is weakly lower semicontinuous, but as in (2.4),
(2.18) we can avoid imposing conditions for existence of a minimizer and
computing it exactly by adding a tolerance $\eta_{n}>0$ that tends to zero as
$n\to\infty$, thus defining the Newton step by

$\forall(\hat{r},x)\in\widetilde{X}\times U\,:\
J_{n}^{\delta}(\hat{r}_{n+1}^{\delta},x_{n+1}^{\delta})\leq
J_{n}^{\delta}(\hat{r},x)+\eta_{n}$ (2.31)

More precisely, we will choose the tolerance, the regularization parameters
and the stopping index according to

$\forall n\in\mathbb{N}:\quad\alpha_{n+1}<\alpha_{n},\quad\eta_{n}\leq
C_{\eta}\varphi^{-1}(\alpha_{n}),\qquad
n_{*}=\min\\{n\in\mathbb{N}\,:\,\alpha_{n}\leq\varphi(\tau\delta^{p})\\}.$
(2.32)

for fixed $C_{\eta}$, $\tau>0$, where we additionally assume the growth
condition

$\forall
n\in\mathbb{N}:\quad(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\alpha_{n}}\Bigr{)}\leq
C_{(-\psi)^{*}}(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\alpha_{n+1}}\Bigr{)}$
(2.33)

to hold for some $C_{(-\psi)^{*}}>1$. Moreover, we set

$\beta_{n}=c_{\beta\alpha}\alpha_{n}\text{ with
}0<c_{\beta\alpha}\leq\tfrac{\min\\{1-b,\gamma-C_{\psi}\\}}{4\tilde{C}_{Q}},$
(2.34)

define $\mathcal{R}$ according to (2.20) with (2.21) and assume $\psi$ to
satisfy (2.23).

A further smoothness assumtion on $r$ will have to be formulated in terms of
$\mathcal{Q}$ so that we can control the Taylor remainder estimates that will
appear in the analysis. In order not cloud the procedure too much with
technicalities we here restrict ourselves to the simple norm-based choice

$\mathcal{Q}(\hat{r}_{1},\hat{r}_{2}):=\|\hat{r}_{1}-\hat{r}_{2}\|_{\widetilde{X}}$
(2.35)

that automatically satisfies (2.22), (2.30), and that we have already
encountered earlier cf. (2.17). The assumption we make on $r$

$\forall x_{1},x_{2}\in
U:\quad\|r(x_{1})-r(x_{2})-r^{\prime}(x_{2})(x_{1}-x_{2})\|\leq
c_{tcr}\|r(x_{1})-r(x_{2})\|$ (2.36)

with $c_{tcr}>0$ small enough

$c_{tcr}<2^{2-2p}\,\max\left\\{1,\,C_{(-\psi)^{*}}(1+2^{1-p})+1\right\\}^{-1}$
(2.37)

resembles the tangential cone condition but now is naturally satisfied, given
that $r$ is a local homeomorphism and we have constrained the minimization to
a sufficiently small neighborhood $U$ of $x^{\dagger}$.

######  Theorem 7.

Under the assumption (2.1) with a $p$-convex space $\check{X}$, let $F$
satisfy (1.6) with Lipschitz continuous $r$ satisfying (2.36), (2.37), and let
the variational source condition (1.28) hold with $\psi$ satisfying (1.29),
(1.30), (1.31), (2.23). Moreover, assume that $\mathcal{R}$, $\mathcal{Q}$ are
chosen according to (2.20), (2.21), (2.35).

Then for any family of data $(y^{\delta})_{\delta\in(0,\bar{\delta}]}$
satisfying (1.4) and regularization parameters $\alpha_{n}$, $\eta_{n}$,
$n_{*}$ satisfying (2.32), (2.33), a corresponding family of regularized
reconstructions according to (2.31) satisfies the convergence rate

$\displaystyle\|\hat{r}_{n_{*}}-r(x^{\dagger})\|=O(\widetilde{\psi}(\delta^{p}))^{1/p},\qquad\|r(x_{n_{*}}^{\delta})-r(x^{\dagger})\|=O(\widetilde{\psi}(\delta^{p}))^{1/p},$
(2.38)
$\displaystyle\|K\Bigl{(}\hat{r}_{n_{*}}^{\delta})-r(x^{\dagger})\Bigr{)}\|_{Y}=O(\delta),\qquad\|P(x_{n_{*}}^{\delta})\|=O(\delta)$

with $\tilde{\psi}$ as in (2.16).

Proof. As in the analysis of (2.4), and (2.18) we start with a minimality
estimate. Comparing $J_{n}^{\delta}(\hat{r}_{n+1}^{\delta},x_{n+1}^{\delta})$
with $J_{n}^{\delta}(r(x^{\dagger}),x^{\dagger})$, using $P(x^{\dagger})=0$
and an analogon to (2.10), we obtain the estimate (cf. (2.11) with
linearization in the first argument of $\mathcal{Q}$)

$\displaystyle
2^{1-p}\|K(\hat{r}_{n+1}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}+\alpha_{n}\,\Bigl{(}\mathcal{R}(\hat{r}_{n+1}^{\delta})-\mathcal{R}(r(x^{\dagger}))\Bigr{)}$
$\displaystyle\qquad+\beta_{n}\mathcal{Q}(r(x_{n+1}^{\delta}),\hat{r}_{n+1}^{\delta})^{p}+\|P(x_{n+1}^{\delta})\|^{p}$
$\displaystyle\leq
2\delta^{p}+\eta_{n}+\beta_{n}\mathcal{Q}(r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x^{\dagger}-x_{n}^{\delta}),r(x^{\dagger}))^{p}$
$\displaystyle\qquad+\beta_{n}\max\Bigl{\\{}0,\mathcal{Q}(r(x_{n+1}^{\delta}),\hat{r}_{n+1}^{\delta})-\mathcal{Q}(r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x_{n}^{\delta}),\hat{r}_{n+1}^{\delta})\Bigr{\\}}^{p}.$

Making use of (2.20), (2.35), (2.36) as well as the variational source
condition (1.28) together with (2.23) and

$\|r(x_{n+1}^{\delta})-\hat{r}_{n+1}^{\delta}\|^{p}\geq
2^{1-p}\|r(x_{n+1}^{\delta})-r(x^{\dagger})\|^{p}-\|\hat{r}_{n+1}^{\delta}-r(x^{\dagger})\|^{p}$
$\|r(x_{n+1}^{\delta})-r(x_{n}^{\delta})\|^{p}\leq
2^{p-1}\|r(x_{n+1}^{\delta})-r(x^{\dagger})\|^{p}+2^{p-1}\|r(x_{n}^{\delta})-r(x^{\dagger})\|^{p}$

we arrive at

$\displaystyle 2^{1-p}\|K(\hat{r}_{n+1}^{\delta}-r(x^{\dagger}))\|_{Y}^{p}$
(2.39) $\displaystyle\qquad+\alpha_{n}\,\tfrac{\min\\{1-b,\gamma-
C_{\psi}\\}}{2}\Bigl{(}\Delta^{\\!\\!\check{\mathcal{R}}}_{\check{\xi}}((I-P)r^{-1}(\hat{r}_{n+1}^{\delta}),x^{\dagger})+\psi\bigl{(}C_{P}\|P(r^{-1}(\hat{r}_{n+1}))\|^{p}\bigr{)}\Bigr{)}$
$\displaystyle\qquad+\beta_{n}(2^{1-p}-2^{p-1}c_{tcr})\|r(x_{n+1}^{\delta})-r(x^{\dagger})\|^{p}+\|P(x_{n+1})\|^{p}$
$\displaystyle\leq\beta_{n}(1+2^{p-1})c_{tcr}\|r(x_{n}^{\delta})-r(x^{\dagger})\|^{p}+\beta_{n}\|\hat{r}_{n+1}-r(x^{\dagger})\|^{p}$
$\displaystyle\qquad+\alpha_{n}C_{\psi}\psi\Bigl{(}2^{p-1}\Bigl{(}\|K(\hat{r}_{n+1}^{\delta}-r(x^{\dagger}))\|^{p}\Bigr{)}+2\delta^{p}+\eta_{n}.$

Here under condition (2.22) we can control
$\beta_{n}\|\hat{r}_{n+1}-r(x^{\dagger})\|^{p}$ by means of the second term on
the left hand side, provided $\beta_{n}$ is chosen according to (2.34).
Setting

$\displaystyle\textrm{err}_{n+1}$
$\displaystyle:=\tfrac{c_{\beta\alpha}}{C_{\psi}}\Bigl{(}\|\hat{r}_{n+1}-r(x^{\dagger})\|^{p}+(2^{1-p}-2^{p-1}c_{tcr})\|r(x_{n+1}^{\delta})-r(x^{\dagger})\|^{p}\Bigr{)},$
(2.40) $\displaystyle\textrm{res}_{n+1}$
$\displaystyle:=2^{1-p}\|K\Bigl{(}\hat{r}_{n+1}^{\delta})-r(x^{\dagger})\Bigr{)}\|^{p}+\|P(x_{n+1}^{\delta})\|^{p}$
$\displaystyle d_{n}$
$\displaystyle:=\tfrac{c_{\beta\alpha}}{C_{\psi}}\alpha_{n}(1+2^{p-1})c_{tcr}\|r(x_{n}^{\delta})-r(x^{\dagger})\|^{p}+\frac{2\delta^{p}+\eta_{n}}{C_{\psi}}$

and $\bar{C}$ as in (2.29) we can rewrite this as

$\textrm{res}_{n+1}+\alpha_{n}\,\textrm{err}_{n+1}\leq
d_{n}+\alpha_{n}\,\psi(\bar{C}\textrm{res}_{n+1}),$

thus by Lemma 2

$\textrm{err}_{n+1}\leq\frac{d_{n}}{\alpha_{n}}+(-\psi)^{*}\Bigl{(}-\frac{1}{\bar{C}\alpha_{n}}\Bigr{)},\quad\textrm{res}_{n+1}\leq
2\Bigl{(}d_{n}+\alpha_{n}(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\alpha_{n}}\Bigr{)}\Bigr{)}.$
(2.41)

As a consequence we get the recursion

$\displaystyle\textrm{err}_{n+1}\leq
q\,\textrm{err}_{n}+\frac{2\delta^{p}+\eta_{n}}{C_{\psi}\,\alpha_{n}}+(-\psi)^{*}\Bigl{(}-\frac{1}{\bar{C}\alpha_{n}}\Bigr{)}\leq
q\,\textrm{err}_{n}+\hat{C}\,(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\alpha_{n}}\Bigr{)}$

with

$q:=\frac{1+2^{p-1}}{2^{1-p}-2^{p-1}c_{tcr}}c_{tcr}\in(0,1/C_{(-\psi)^{*}})\subseteq(0,1),$

which holds under the smallness condition (2.37) on $c_{tcr}$, and
$\hat{C}=\frac{2}{\tau C_{\psi}}+\frac{C_{\eta}}{C_{\psi}}+1$ for $n<n_{*}$ by
(2.32). Resolving this recursion and using (2.32), (2.33), we obtain

$\displaystyle\textrm{err}_{n_{*}}\leq
q^{n_{*}}\textrm{err}_{0}+\hat{C}\sum_{n=0}^{n_{*}-1}q^{n_{*}-n-1}\,(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\alpha_{n}}\Bigr{)}$
$\displaystyle\leq\Bigl{(}(qC_{(-\psi)^{*}})^{n_{*}}\frac{\textrm{err}_{0}}{(-\psi)^{*}\left(-\frac{1}{2\bar{C}\alpha_{0}}\right)}+\frac{\hat{C}}{q(1-qC_{(-\psi)^{*}})}\Bigr{)}\,(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\alpha_{n_{*}}}\Bigr{)}$
$\displaystyle\leq\tilde{C}\,(-\psi)^{*}\Bigl{(}-\frac{1}{2\bar{C}\varphi(\tau\delta^{p})}\Bigr{)}$

with
$\tilde{C}=\frac{\textrm{err}_{0}}{(-\psi)^{*}\left(-\frac{1}{2\bar{C}\alpha_{0}}\right)}+\frac{C}{q(1-qC_{(-\psi)^{*}})}$.

This together with the second estimate in (2.41) and the definition (2.40)
yields the convergence rates.

$\diamondsuit$

###  2.4 A canonical relaxation leading to range invariance

A general way of extending an arbitrary inverse problem

$\check{F}(\check{x})=y$

with $\check{F}:\mathcal{D}(\check{F})(\subseteq\check{X})\to Y$ to an
operator equation (1.3) satisfying the range invariance condition (1.6) is by
defining

$X=\check{X}\times Y,\quad x=(\check{x},z),\quad
F(x)=\check{F}(\check{x})+z,\quad P(x)=z$

Indeed, it is readily checked that (1.6) formally holds with

$r(x)=\left(\begin{array}[]{c}\check{x}-\check{x}_{0}\\\
z-z_{0}+\bigl{(}\check{F}(\check{x})-\check{F}(\check{x}_{0})-\check{F}^{\prime}(\check{x}_{0})(\check{x}-\check{x}_{0})\bigr{)}\end{array}\right),\qquad
K\underline{dx}=\check{F}^{\prime}(\check{x}_{0})\underline{d\check{x}}+\underline{dz}.$

A sufficient condition for the requirements of the convergence results in
Section 2, namely local bijectivity of $r$ and Lipschitz continuity of $r$ and
its inverse, is

$\|r(x_{1})-r(x_{2})-(x_{1}-x_{2})\|\leq c_{r}\|(x_{1}-x_{2})\|,\qquad
x_{1},x_{2}\in U\times Y$

for some $c_{r}\in(0,1)$, which in the above setting amounts to a Taylor
remainder estimate of the form

$\|\check{F}(\check{x}_{1})-\check{F}(\check{x}_{2})-\check{F}^{\prime}(\check{x}_{0})(\check{x}_{1}-\check{x}_{2})\|_{Y}\leq
c_{r}\|\check{x}_{1}-\check{x}_{2}\|\qquad\check{x}_{1},\check{x}_{2}\in U.$
(2.42)

Likewise, the additional condition (2.36) required for proving convergence
rates for (2.31) reads as

$\|\check{F}(\check{x}_{1})-\check{F}(\check{x}_{2})-\check{F}^{\prime}(\check{x}_{2})(\check{x}_{1}-\check{x}_{2})\|_{Y}\leq
c_{tcr}\|\check{x}_{1}-\check{x}_{2}\|\qquad\check{x}_{1},\check{x}_{2}\in U.$
(2.43)

The methods analyzed above in this setting read as follows.

  * •

variational regularization (1.8):

$\displaystyle(\hat{r}_{\check{x},\alpha,\beta}^{\delta},\hat{r}_{z,\alpha,\beta}^{\delta},\check{x}_{\alpha,\beta}^{\delta},z_{\alpha,\beta}^{\delta})\in\mbox{argmin}_{(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)\in\check{X}\times
Y\times U\times
Y}J_{\alpha,\beta}^{\delta}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)$
(2.44) $\displaystyle\text{where
}J_{\alpha,\beta}^{\delta}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)=J_{\alpha}(\hat{r}_{\check{x}},\hat{r}_{z})+J_{\beta}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)$
$\displaystyle
J_{\alpha}(\hat{r}_{\check{x}},\hat{r}_{z})=\|\check{F}^{\prime}(\check{x}_{0})\hat{r}_{\check{x}}+\check{F}(\check{x}_{0})+\hat{r}_{z}+z_{0}-y^{\delta}\|_{Y}^{p}+\alpha\check{\mathcal{R}}(\hat{r}_{\check{x}})$
$\displaystyle
J_{\beta}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)=\beta\Bigl{(}\mathcal{Q}_{\check{x}}(\check{x}-\check{x}_{0},\hat{r}_{\check{x}})^{p}$
$\displaystyle\hskip
85.35826pt+\mathcal{Q}_{z}(z-z_{0}+\bigl{(}\check{F}(\check{x})-\check{F}(\check{x}_{0})-\check{F}^{\prime}(\check{x}_{0})(\check{x}-\check{x}_{0})\bigr{)},\hat{r}_{z})^{p}\Bigr{)}$
$\displaystyle\hskip 85.35826pt+\|z\|_{X}^{p}$

  * •

split minimization (1.10):

$\displaystyle(\hat{r}_{\check{x},sm}^{\delta},\hat{r}_{z,sm}^{\delta})\in\mbox{argmin}_{(\hat{r}_{\check{x}},\hat{r}_{z})\in\widetilde{X}}J_{\alpha}(\hat{r}_{\check{x}},\hat{r}_{z})$
(2.45)
$\displaystyle(\check{x}_{sm}^{\delta},z_{sm}^{\delta})\in\mbox{argmin}_{(\check{x},z)\in
U\times
Y}J_{\beta}(\hat{r}_{\check{x},sm}^{\delta},\hat{r}_{z,sm}^{\delta},\check{x},z)$

  * •

frozen Newton (1.11):

$\displaystyle(\hat{r}_{\check{x},n+1}^{\delta},\hat{r}_{z,n+1}^{\delta},\check{x}_{n+1}^{\delta},z_{n+1}^{\delta})\in\mbox{argmin}_{(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)\in\check{X}\times
Y\times U\times Y}J_{n}^{\delta}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)$
(2.46) $\displaystyle\text{where
}J_{n}^{\delta}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z):=J_{\alpha_{n}}(\hat{r}_{\check{x}},\hat{r}_{z})+J_{\beta_{n},n}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)$
$\displaystyle
J_{\beta,n}(\hat{r}_{\check{x}},\hat{r}_{z},\check{x},z)=\beta\Bigl{(}\mathcal{Q}_{\check{x}}(\check{x}-\check{x}_{0},\hat{r}_{\check{x}})^{p}$
$\displaystyle\hskip
85.35826pt+\mathcal{Q}_{z}(z-z_{0}+\bigl{(}(\check{F}^{\prime}(\check{x}_{n})-\check{F}(\check{x}_{0}))(\check{x}-\check{x}_{n})+y_{n}\bigr{)},\hat{r}_{z})^{p}\Bigr{)}$
$\displaystyle\hskip 85.35826pt+\|z\|_{X}^{p}$ $\displaystyle
y_{n}=\check{F}(\check{x}_{n})-\check{F}(\check{x}_{0})-\check{F}^{\prime}(\check{x}_{0})(\check{x}_{n}-\check{x}_{0})$

##  3 Application to EIT

######  Example 3 (Example 1 revisited).

We adopt a convenient function space setting for this problem from [28,
Section 6.2] and show that formal range invariance (1.6) with (1.16) can be
made rigorous in this framework. This is done in the all-at-once formulation,
since the latter allows to avoid division by zero in (1.16) by a proper choice
of $u_{0}$. A logarithmic variational source condition for the reduced
formulation is taken from [28], cf. Theorem 8 below, and extended to the all-
at-once setting by means of Lemma 1.

The inverse problem can be written in reduced form as

$\check{\mathbf{F}}(c)=y$ (3.1)

or equivalently in all-at-once form as

$\check{\mathbb{F}}(c,u):=\left(\begin{array}[]{c}\check{A}(c,u)\\\
Bu\end{array}\right)=\left(\begin{array}[]{c}0\\\ y\end{array}\right)$ (3.2)

(note that we are using the $\check{\cdot}$ notation for the formulation
before extension) or equivalently, after extension of the parameter space and
penalization, as

$\displaystyle\mathbb{F}(\vec{c},u):=\left(\begin{array}[]{c}A(\vec{c},u)\\\
Bu\end{array}\right)=\left(\begin{array}[]{c}0\\\ y\end{array}\right)$ (3.3)
$\displaystyle P(\vec{c},u)=0$

cf. (1.1) with

$\displaystyle\check{\mathbf{F}}:\mathcal{D}(\check{\mathbf{F}})(\subseteq\check{Q})\to
Z,\quad\check{A}:\mathcal{D}(\check{A})(\subseteq\check{X})\to(0,\vec{\varphi},0)+W^{*},$
(3.4) $\displaystyle A:\mathcal{D}(A)(\subseteq
X)\to(0,\vec{\varphi},0)+W^{*},\quad B:V\to Z$
$\displaystyle\mathcal{D}(\check{\mathbf{F}})=\Phi(\mathcal{D}),\quad\mathcal{D}(\check{A})=\mathcal{D}(\check{\mathbf{F}})\times
V,\quad\mathcal{D}(A)=\ell^{\infty}(\mathcal{D}(\check{\mathbf{F}}))\times V$
$\displaystyle\text{ where }\mathcal{D}=\\{\sigma\in
H^{2+s_{1}}(\Omega)\,:\,0<\underline{\sigma}\leq\sigma(x)\leq\overline{\sigma}<\infty\
x\in\Omega\text{ and \eqref{sigmabg} holds}\\},$
$\displaystyle\check{Q}=H^{s_{1}}(\Omega),\quad
Q=\ell^{\infty}(\check{Q}),\quad V=\ell^{2}(H^{2}(\Omega)),\quad
W^{*}=\ell^{2}(L^{2}(\Omega))\times\ell^{2}(H^{1/2}(\partial\Omega))\times\ell^{2}(\mathbb{R}),\quad$
$\displaystyle\check{X}=\check{Q}\times V,\quad X=Q\times V,\quad
Z=\ell^{2}(L^{2}(\partial\Omega)),\quad Y:=W^{*}\times Z.$

Note that taking $Z=\ell^{2}(L^{2}(\Omega))$ together with the
$L^{2}(\partial\Omega)$ normalization of the currents $\varphi_{n}$
corresponds to taking the Hilbert-Schmidt norm of the N-t-D operator. Also
note that regardless of the use of $\ell^{\infty}$ in the definition of $X$,
the original space $\check{X}$ is still $s$-convex with $s=2$. The operators
are defined by

$\displaystyle A(\vec{c},u)=\left(\begin{array}[]{c}-\Delta
u_{n}+c_{n}u_{n}\\\ \text{tr}^{N}_{\partial\Omega}u_{n}-\varphi_{n}\\\
\int_{\partial\Omega}u_{n}\,d\Gamma\end{array}\right)_{n\in\mathbb{N}}\\!\\!\\!\\!\\!\\!\\!\\!,\quad\check{A}(c,u)=A(\vec{c},u)\text{
with $c_{n}\equiv c$},$ (3.5) $\displaystyle
Bu=\left(\text{tr}^{D}_{\partial\Omega}u_{n}\right)_{n\in\mathbb{N}}\quad
P(\vec{c},u)=\Bigl{(}c_{n}-\bigl{(}\sum_{k\in\mathbb{N}}w_{k}\bigr{)}^{-1}\,\sum_{k\in\mathbb{N}}w_{k}c_{k}\Bigr{)}_{n\in\mathbb{N}},$
$\displaystyle\check{\mathbf{F}}(c)=\mathbf{F}(c):=(\text{tr}^{D}_{\partial\Omega}u_{n})_{n\in\mathbb{N}}\text{
where $u_{n}=S(c)_{n}$ and }\check{A}(c,S(c))=0$
$\displaystyle\mathbf{F}(\vec{c}):=(\text{tr}^{D}_{\partial\Omega}u_{n})_{n\in\mathbb{N}}\text{
where $u_{n}=\vec{S}(\vec{c})_{n}$ and }A(\vec{c},\vec{S}(\vec{c}))=0$

with the Neumann trace operator $\text{tr}^{N}_{\partial\Omega}$, Note that
slightly deviating from [28], in order to avoid division by zero in (1.16), we
use all of $H^{2}(\Omega)$ rather than incorporating the vanishing boundary
average constraint into the state space. Moreover, we increase regularity as
compared to the $H^{1}(\Omega)$ setting from [28] as this is possible with
$s_{1}\geq 0$ (note that even $s_{1}\in(1/2,3/2)$ is assumed in Theorem 8) and
will be needed in the rigorous verification of (1.6).

Indeed, the mapping property $A(\vec{c},u):\mathcal{D}(A)\to W^{*}$ holds true
due to boundedness of $\text{tr}^{N}_{\partial\Omega}:H^{2}(\Omega)\to
H^{1/2}(\partial\Omega)$, as well as continuity of the multiplication operator
$M:H^{s_{1}}(\Omega)\times H^{2}(\Omega)\mapsto L^{2}(\Omega)$, $(c,u)\mapsto
c\,u$ for $s_{1}\geq 0$, that allows to estimate

$\displaystyle\|A(\vec{c},u)+(0,\vec{\varphi},0)\|_{W^{*}}$
$\displaystyle\leq\Bigl{(}\sum_{n\in\mathbb{N}}\bigl{(}1+C_{H^{2},L^{\infty}}^{\Omega}\|c_{n}\|_{L^{2}(\Omega)}+\|\text{tr}^{N}_{\partial\Omega}\|_{H^{2}(\Omega),H^{1/2}(\partial\Omega)}+\|\text{tr}^{D}_{\partial\Omega}\|_{H^{2}(\Omega),L^{1}(\partial\Omega)}\bigr{)}^{2}\|u_{n}\|_{H^{2}(\Omega)}^{2}\Bigr{)}^{1/2}$
$\displaystyle\leq\tilde{C}(1+\|\vec{c}\|_{\ell^{\infty}(L^{2}(\Omega))}\|u\|_{\ell^{2}(H^{2}(\Omega))}.$

On the other hand, for any fixed $c\in\mathcal{D}(\check{\mathbf{F}})$, as a
consequence of the Lax-Milgram Lemma and elliptic regularity, the operator
$\check{A}(c,u)$ is Lipschitz continuously invertible, as needed for the
application of Lemma 1. More precisely, for any
$(\vec{f},\vec{\phi},\vec{m})\in W^{*}$ we have that
$(\vec{f},\vec{\phi},\vec{m})=\check{A}(c,u_{1})-\check{A}(c,u_{2})$ iff for
all $n\in\mathbb{N}$, $\tilde{u}_{n}:=u_{1,n}-u_{2,n}-m_{n}/|\partial\Omega|$
solves

$-\Delta\tilde{u}_{n}+c\,\tilde{u}_{n}=f_{n}-c_{n}\tfrac{m_{n}}{|\partial\Omega|}\
\text{ in }\Omega,\quad\partial_{\nu}\tilde{u}_{n}=\phi_{n},\ \text{ on
}\partial\Omega,\quad\int_{\partial\Omega}\tilde{u}_{n}\,d\Gamma=0$

and therefore

$\displaystyle\|u_{1}-u_{2}\|_{\ell^{2}(H^{2}(\Omega))}=\Bigl{(}\sum_{n\in\mathbb{N}}\|\tilde{u}_{n}+\tfrac{m_{n}}{|\partial\Omega|}\|_{H^{2}(\Omega)}^{2}\Bigr{)}^{1/2}$
$\displaystyle\leq\Bigl{(}\sum_{n\in\mathbb{N}}\Bigl{(}\tilde{C}\bigl{(}\|f_{n}-c_{n}\tfrac{m_{n}}{|\partial\Omega|}\|_{L^{2}(\Omega)}+\|\phi_{n}\|_{H^{1/2}(\partial\Omega)}\bigr{)}+\|\tfrac{m_{n}}{|\partial\Omega|}\|_{H^{2}(\Omega)}\Bigr{)}^{2}\Bigr{)}^{1/2}$
$\displaystyle\leq\tilde{C}\|\vec{f}\|_{\ell^{2}(L^{2}(\Omega))}+\tfrac{\tilde{C}}{|\partial\Omega|}\|\vec{c}\|_{\ell^{\infty}(L^{2}(\Omega))}\|\vec{m}\|_{\ell^{2}(\mathbb{R})}+\tilde{C}\|\vec{\phi}\|_{\ell^{2}(H^{1/2}(\partial\Omega))}+\tfrac{\sqrt{|\Omega|}}{|\partial\Omega|}\|\vec{m}\|_{\ell^{2}(\mathbb{R})}$
$\displaystyle\leq\tilde{\tilde{C}}\|\check{A}(c,u_{1})-\check{A}(c,u_{2})\|_{W^{*}}$

Also Lipschitz continuity of $(I-P):X\to X$ and therefore of $P:X\to X$ is
easily verified:

$\|(I-P)(\vec{c}_{1},u_{1})-(I-P)(\vec{c}_{2},u_{2})\|_{X}\leq\tfrac{1}{\|\vec{w}\|_{\ell^{1}(\mathbb{R})}}\sup_{j\in\mathbb{N}}\sum_{k\in\mathbb{N}}w_{k}\|c_{1,k}-c_{2,k}\|_{\check{Q}}\leq\|\vec{c}_{1}-\vec{c}_{2}\|_{\ell^{\infty}(\check{Q})}.$
(3.6)

We make use of the following result from [28] that we quote in a
particularized way for the convenience of the reader.

######  Theorem 8.

([28, Theorem 6.10, 6.12 with $p=2$]) Let
$\Omega=\mathcal{B}_{1}(0)\subseteq\mathbb{R}^{d}$, $d=3$, $\sigma_{bg}\equiv
1$. Set $\check{R}(c):=\tfrac{1}{2}\|c\|_{H^{s_{1}}(\Omega)}^{2}$ for some
$s_{1}\in(1/2,3/2)$ and assume that $c^{\dagger}\in B^{s_{2}}_{2,\infty}$ for
some $s_{2}\in(s_{1},2s_{1}+1/2)$. Then there exist constants $C_{1}$,
$C_{2}$, $t_{0}>0$ such that (1.28) holds for $\check{F}=\check{\mathbf{F}}$
with
$\psi(t)=C_{1}(-\log(\min\\{t,t_{0}\\}+C_{2}))^{-2(s_{2}-s_{1})\frac{2s_{1}+1}{2s_{2}+1}}$
and $b=3/4$.

This follows directly from an application of [28, Theorem 6.12] with (in the
notation from there) $p=q=r=2$, $s_{0}=-1/2$ to the stability result [28,
Theorem 6.10].

In order to conclude a VSC for the all-at-once forward operator
$\check{\mathbb{F}}$, we apply Lemma 1. The purpose of using the all-at-once
formulation is to allow for a bound

$(\tfrac{1}{u_{0,j}})_{j\in\mathbb{N}}\in\ell^{\infty}(H^{d/2+\epsilon}(\Omega))$
(3.7)

for some $\epsilon\in(0,2-d/2]$, that will be required for a rigorous
verification of the range invariance condition (1.6).

Using this in the convergence results of Sections 2.1, 2.2, 2.3, we obtain
convergence rates for the inverse protential problem.

######  Corollary 9.

Under the conditions of Theorem 8, with (3.7), $p=2$,
$\mathcal{Q}(\hat{r}_{1},\hat{r}_{2})=\|\hat{r}_{1}-\hat{r}_{2}\|_{X}$,
$\alpha\sim\delta^{\tilde{p}}$, $\tilde{p}\in(0,2]$, we obtain the convergence
rate

$\|\vec{c}^{\;\delta}-c^{\dagger}\|_{\ell^{\infty}(H^{s_{1}}(\Omega))}+\|u^{\delta}-S(c^{\dagger})\|_{\ell^{2}(H^{2}(\Omega)}=O((-\log(\delta^{2}+C_{2}))^{-2(s_{2}-s_{1})\frac{2s_{1}+1}{(2s_{2}+1)p}}$

  1. 1.

for $(\vec{c}^{\;\delta},u^{\delta})=x_{\alpha,\beta,\eta}^{\delta}$ defined
by (1.8) with (2.7), (2.5);

  2. 2.

for $(\vec{c}^{\;\delta},u^{\delta})=x_{sm}^{\delta}$ defined by (1.10) with
(2.20), (2.21), (2.5);

  3. 3.

for $(\vec{c}^{\;\delta},u^{\delta})=x_{n_{*}}^{\delta}$ defined by (1.11)
with (2.20), (2.21), (2.32).

Proof. Invertibility of $r$ as defined in (1.16), that is, in the all-at-once
setting used here,

$r(\vec{c},u)_{j}:=\left(\begin{array}[]{c}(c_{j}-c_{0,j})(1+\tfrac{u_{j}-u_{0,j}}{u_{0,j}})\\\
u_{j}-u_{0,j}\end{array}\right),\quad
r^{-1}(\hat{r}_{c},\hat{r}_{u})=\left(\begin{array}[]{c}\hat{r}_{c,j}(1+\tfrac{\hat{r}_{u,j}}{u_{0,j}})^{-1}+c_{0,j}\\\
\hat{r}_{u}+u_{0,j}\end{array}\right),$ (3.8)

as well as Lipschitz continuity of $r$ and $r^{-1}$ follow from the estimate

$\displaystyle\|r(\vec{c}_{1},u_{1})-r(\vec{c}_{2},u_{2})-\bigl{(}(\vec{c}_{1},u_{1})-(\vec{c}_{2},u_{2})\bigr{)}\|_{X}$
(3.9)
$\displaystyle=\|((c_{1,j}-c_{0,j})\tfrac{u_{1,j}-u_{0,j}}{u_{0,j}}-(c_{2,j}-c_{0,j})\tfrac{u_{2,j}-u_{0,j}}{u_{0,j}})_{j\in\mathbb{N}}\|_{\ell^{\infty}(H^{s_{1}}(\Omega))}$
$\displaystyle=\sup_{j\in\mathbb{N}}\|\tfrac{1}{u_{0,j}}\bigl{(}(c_{1,j}-c_{2,j})(u_{1,j}-u_{0,j})+(c_{2,j}-c_{0,j})(u_{1,j}-u_{2,j})\bigr{)}\|_{H^{s_{1}}(\Omega)}$
$\displaystyle\leq
C\sup_{j\in\mathbb{N}}\|\tfrac{1}{u_{0,j}}\|_{H^{d/2+\epsilon}(\Omega)}\Bigl{(}\|c_{1,j}-c_{2,j}\|_{H^{s_{1}}(\Omega)}\|u_{1,j}-u_{0,j}\|_{H^{2}(\Omega)}$
$\displaystyle\hskip
142.26378pt+\|c_{2,j}-c_{0,j}\|_{H^{s_{1}}(\Omega)}\|u_{1,j}-u_{2,j}\|_{H^{2}(\Omega)}\Bigr{)}$
$\displaystyle\leq\tilde{C}\|(\vec{c}_{1},u_{1})-(\vec{c}_{2},u_{2})\|_{X}\bigl{(}\|(\vec{c}_{1},u_{1})-(\vec{c}_{0},u_{0})\|_{X}+\|(\vec{c}_{2},u_{2})-(\vec{c}_{0},u_{0})\|_{X}\bigr{)}$
$\displaystyle\leq c_{r}\|(\vec{c}_{1},u_{1})-(\vec{c}_{2},u_{2})\|_{X}$

with $c_{r}\in(0,1)$ for $(\vec{c}_{i},u_{i})\in
U:=\mathcal{B}_{\rho_{0}}^{X}(\vec{c}_{0},u_{0})$, $i\in\\{1,2\\}$ with
$\rho_{0}$ small enough. Here we have used the assumption (3.7) and [12,
Theorem 6.1] (see also [28, Theorem B.22]) with $s_{0}=d/2+\epsilon$ and
$p_{0}=p_{1}=p=2$ to obtain, e.g,
$\|\tfrac{1}{u_{0,j}}(c_{1,j}-c_{2,j})(u_{1,j}-u_{0,j})\|_{H^{s_{1}}(\Omega)}\leq
C\|c_{1,j}-c_{2,j}\|_{H^{s_{1}}(\Omega)}\|\tfrac{1}{u_{0,j}}(u_{1,j}-u_{0,j})\|_{H^{d/2+\epsilon}(\Omega)}$
and then the fact that $H^{d/2+\epsilon}(\Omega)$ with the pointwise
multiplication is a Banach algebra as well as the embedding
$\ell^{2}(H^{2}(\Omega))\subseteq\ell^{\infty}(H^{2}(\Omega))$ of the sequence
spaces.

It only remains to verify the tangential cone condition for $r$, (2.36), that
appears in the convergence proof of (1.11)

$\displaystyle\|r(\vec{c}_{1},u_{1})-r(\vec{c}_{2},u_{2})-r^{\prime}(\vec{c}_{2},u_{2})\bigl{(}(\vec{c}_{1},u_{1})-(\vec{c}_{2},u_{2})\bigr{)}\|_{X}$
$\displaystyle=\sup_{j\in\mathbb{N}}\|(c_{1,j}-c_{0,j})\tfrac{u_{1,j}-u_{0,j}}{u_{0,j}}-(c_{2,j}-c_{0,j})\tfrac{u_{2,j}-u_{0,j}}{u_{0,j}}$
$\displaystyle\hskip
56.9055pt-(c_{1,j}-c_{2,j})\tfrac{u_{2,j}-u_{0,j}}{u_{0,j}}-(c_{2,j}-c_{0,j})\tfrac{u_{1,j}-u_{2,j}}{u_{0,j}}\|_{H^{s_{1}}(\Omega)}$
$\displaystyle=\sup_{j\in\mathbb{N}}\|\tfrac{1}{u_{0,j}}(c_{1,j}-c_{2,j})(u_{1,j}-u_{2,j})\|_{H^{s_{1}}(\Omega)}$
$\displaystyle\leq\tilde{C}\|(\vec{c}_{1},u_{1})-(\vec{c}_{2},u_{2})\|_{X}^{2}\leq
c_{tcr}\|r(\vec{c}_{1},u_{1})-r(\vec{c}_{2},u_{2})\|_{X}$

similarly to (3.9).

$\diamondsuit$

Alternatively, we could apply the general relaxation from Section 2.4, by
remaining with the reduced original formulation

$\displaystyle\check{\mathbf{F}}:\mathcal{D}(\check{\mathbf{F}})(\subseteq\check{Q})\to
Z,\quad\check{X}=\check{Q},\quad Y=Z=\ell^{2}(L^{2}(\partial\Omega)),$
$\displaystyle\check{\mathbf{F}}(c):=(\text{tr}^{D}_{\partial\Omega}u_{n})_{n\in\mathbb{N}}\text{
where $u_{n}=S(c)_{n}$ and }\check{A}(c,S(c))=0$

with $\mathcal{D}(\check{\mathbf{F}})$ as in (3.4), but defining the extended
operator by

$\mathbf{F}:\mathcal{D}(\check{\mathbf{F}})\times Z\to Z,\quad
X=\check{Q}\times Z,\quad\mathbf{F}(c,z):=\mathbf{F}(c)+z$

and

$\displaystyle r(c,z)_{j}:=(c-c_{0},\
z_{j}-z_{0,j}+\text{tr}^{D}_{\partial\Omega}v_{j})$ (3.10)
$\displaystyle\text{ where $v_{j}$ solves }\left\\{\begin{array}[]{rll}-\Delta
v_{j}+c\,v_{j}&=(c-c_{0})(S(c)_{j}-S(c_{0})_{j})&\text{ in }\Omega\\\
\partial_{\nu}v_{j}&=0&\text{ on }\partial\Omega\\\
\int_{\partial\Omega}v_{j}\,d\Gamma&=0.\end{array}\right.$

Since the functions $z_{j}$ are defined on the $d-1$ dimensional manifold
$\partial\Omega$, this is a more parsimonious relaxation in the sense of
augmentation of the set of unknowns. It also avoids a condition like (3.7) and
thus allows to stay in the reduced setting. However, this comes at the prize
of the more involved definition of $r$ according to (3.10) and its inverse

$\displaystyle r^{-1}(\hat{r}_{c},\hat{r}_{z})_{j}:=(c_{0}+\hat{r}_{c},\
z_{0,j}+\hat{r}_{z,j}-\text{tr}^{D}_{\partial\Omega}v_{j})$
$\displaystyle\text{ where $v_{j}$ solves }\left\\{\begin{array}[]{rll}-\Delta
v_{j}+(c_{0}+\hat{r}_{c})\,v_{j}&=\hat{r}_{c}(S(c_{0}+\hat{r}_{c})_{j}-S(c_{0})_{j})&\text{
in }\Omega\\\ \partial_{\nu}v_{j}&=0&\text{ on }\partial\Omega\\\
\int_{\partial\Omega}v_{j}\,d\Gamma&=0.\end{array}\right.$

as compared to (3.8).

######  Example 4 (Example 2 revisited).

The plan is similar to the one in Example 3 above: Also here we make use of
the function space setting detailed in [28] and use a variational source
condition from there, but stay in the reduced setting.

We thus write the inverse problem as

$\check{\mathbf{F}}(\sigma)=y$ (3.11)

or equivalently, after extension of the parameter space and penalization
according to Section 2.4, as

$\displaystyle\mathbf{F}(\sigma,z)=(0,y)^{T}$ (3.12) $\displaystyle
P(\sigma,z)=0$

cf. (1.2) with

$\displaystyle\check{\mathbf{F}}:\mathcal{D}(\check{\mathbf{F}})(\subseteq\check{X})\to
Y,\quad\mathbf{F}:\mathcal{D}(\check{\mathbf{F}})\times Z\to Y,$
$\displaystyle\mathcal{D}(\check{\mathbf{F}})=\\{\sigma\in H^{s_{1}}(\Omega)\
:\
0<\underline{\sigma}\leq\sigma(x)\leq\overline{\sigma}<\infty,\,x\in\Omega,\text{
and \eqref{sigmabg} holds}\\}$ $\displaystyle\check{X}=H^{s_{1}}(\Omega),\quad
X=\check{X}\times Z,\quad Y=Z=\ell^{2}(L^{2}(\partial\Omega))$

with $s_{1}>d/2$ (so that $\check{X}$ continuously embeds into
$L^{\infty}(\Omega)$) and (1.19), (1.20),
$\check{\mathbf{F}}(\sigma)=\mathbf{F}(\sigma,0)$. This satisfies (1.6) with
(1.21).

Since we employ the relaxation from Section 2.4, we also have to verify the
Taylor remainder estimate (2.42) and, in case of Newton (1.11) also condition
(2.43). We do so by observing that with $u_{k,j}=S(\sigma_{k})_{j}$,
$k\in\\{1,2,i\\}$, $i\in\\{0,2\\}$

$\displaystyle\check{\mathbf{F}}(\sigma_{1})-\check{\mathbf{F}}(\sigma_{2})-\check{\mathbf{F}}^{\prime}(\sigma_{i})=(\text{tr}^{D}_{\partial\Omega}\tilde{v}_{j})_{j\in\mathbb{N}}\quad\text{
where $\tilde{v}_{j}$ solves }$
$\displaystyle\left\\{\begin{array}[]{rll}-\nabla\cdot(\sigma_{i}\nabla\tilde{v}_{j})&=\nabla\cdot\bigl{(}(\sigma_{1}-\sigma_{i})\nabla(u_{1,j}-u_{2,j})+(\sigma_{1}-\sigma_{2})\nabla(u_{2,j}-u_{i,j})\bigr{)}&\text{
in }\Omega\\\ \partial_{\nu}\tilde{v}_{j}&=0&\text{ on }\partial\Omega\\\
\int_{\partial\Omega}\tilde{v}_{j}\,d\Gamma&=0.\end{array}\right.$

Thus with the Poincaré-Friedrichs type estimate

$\|v\|_{H^{1}(\Omega)}\leq C_{PF}\left(\|\nabla
v\|_{L^{2}(\Omega)}+\left|\int_{\partial\Omega}v\,d\Gamma\right|\right),\quad
v\in H^{1}(\Omega)$

applied to the weak formulation

$\displaystyle\int_{\Omega}\sigma_{i}\nabla\tilde{v}_{j}\cdot\nabla
w\,dx=-\int_{\Omega}\bigl{(}(\sigma_{1}-\sigma_{i})\nabla(u_{1,j}-u_{2,j})+(\sigma_{1}-\sigma_{2})\nabla(u_{2,j}-u_{i,j})\bigr{)}\cdot\nabla
w\,dx,$ $\displaystyle\text{for all }w\in H^{1}(\Omega)\text{ with
}\int_{\partial\Omega}w\,d\Gamma=0,$

setting $w=\tilde{v}_{j}$ and applying the Cauchy-Schwarz inequality, we
obtain

$\displaystyle\|\check{\mathbf{F}}(\sigma_{1})-\check{\mathbf{F}}(\sigma_{2})-\check{\mathbf{F}}^{\prime}(\sigma_{i})\|_{L^{2}(\partial\Omega)}$
$\displaystyle\leq\tfrac{1}{\underline{\sigma}}C_{PF}\|\text{tr}^{D}_{\partial\Omega}\|_{H^{1}(\Omega),L^{2}(\partial\Omega)}\|(\sigma_{1}-\sigma_{i})\nabla(u_{1,j}-u_{2,j})+(\sigma_{1}-\sigma_{2})\nabla(u_{2,j}-u_{i,j})\|_{L^{2}(\Omega)}$
$\displaystyle\leq\tfrac{1}{\underline{\sigma}}C_{PF}\|\text{tr}^{D}_{\partial\Omega}\|_{H^{1}(\Omega),L^{2}(\partial\Omega)}\Bigl{(}\|\sigma_{1}-\sigma_{i}\|_{L^{\infty}(\Omega)}\|u_{1,j}-u_{2,j}\|_{H^{1}(\Omega)}$
$\displaystyle\hskip
128.0374pt+\|\sigma_{1}-\sigma_{2}\|_{L^{\infty}(\Omega)}\|u_{2,j}-u_{i,j}\|_{H^{1}(\Omega)}\Bigr{)}$
$\displaystyle\leq\tilde{C}\Bigl{(}\|\sigma_{1}-\sigma_{i}\|_{\check{X}}+\|\sigma_{2}-\sigma_{i}\|_{\check{X}}\Bigr{)}\|\sigma_{1}-\sigma_{2}\|,$

due to Lipschitz continuity of $S:L^{\infty}(\Omega)\to H^{1}(\Omega)$.
Thus (2.42) and (2.43) hold on a sufficiently small neighborhood $U$ of
$\sigma^{\dagger}$.

######  Theorem 10.

([28, Theorem 6.15 with $p=2$]) Let
$\Omega=\mathcal{B}_{1}(0)\subseteq\mathbb{R}^{d}$, $d=3$, $\sigma_{bg}\equiv
1$. Set $\check{R}(\sigma):=\tfrac{1}{2}\|\sigma\|_{H^{s_{1}}(\Omega)}^{2}$
for some $s_{1}\in(5/2,7/2)$ and assume that $\sigma^{\dagger}\in
B^{s_{2}}_{2,\infty}$ for some $s_{2}\in(s_{1},2s_{1}-1)$. Then there exist
constants $C$, $t_{0}>0$ such that (1.28) holds with
$\psi(t)=C(-\log(\min\\{t,t_{0}\\}))^{-(s_{2}-s_{1})\frac{2s_{1}-3}{s_{1}-1}}$
and $b=3/4$.

Combining this with the convergence results of Sections 2.1, 2.2, 2.3, we
arrive at the following convergence rates result for EIT.

######  Corollary 11.

Under the conditions of Theorem 10, with $p=2$,
$\mathcal{Q}(\hat{r}_{1},\hat{r}_{2})=\|\hat{r}_{1}-\hat{r}_{2}\|_{X}$,
$\alpha\sim\delta^{\tilde{p}}$, $\tilde{p}\in(0,2]$, we obtain the convergence
rate

$\|\sigma^{\delta}-\sigma^{\dagger}\|_{H^{s_{1}}(\Omega)}+\|z^{\delta}\|_{\ell^{2}(L^{2}(\partial\Omega))}=O((-\log(\delta^{p}+C_{2}))^{-(s_{2}-s_{1})\frac{2s_{1}-3}{(s_{1}-1)p}}$

  1. 1.

for $(\sigma^{\delta},z^{\delta})=x_{\alpha,\beta,\eta}^{\delta}$ defined by
(2.4) with (2.7), (2.5);

  2. 2.

for $(\sigma^{\delta},z^{\delta})=x_{sm}^{\delta}$ defined by (2.18) with
(2.20), (2.21), (2.5);

  3. 3.

for $(\sigma^{\delta},z^{\delta})=x_{n_{*}}^{\delta}$ defined by (2.31) with
(2.20), (2.21), (2.32).

##  Appendix A Appendix

###  A.1 Proof of Lemma 2.2

Proof. Splitting the res term into a convex combination
$\textrm{res}=\lambda\textrm{res}+(1-\lambda)\textrm{res}$ we obtain from
(2.2)

$\lambda c_{1}\textrm{res}+c_{2}\alpha\,\textrm{err}\leq
d+\alpha\Bigl{(}-\frac{(1-\lambda)c_{1}}{C_{3}\alpha}\,C_{3}\textrm{res}-(-\psi)(C_{3}\textrm{res})\Bigr{)}\leq
d+\alpha(-\psi)^{*}\Bigl{(}-\frac{(1-\lambda)c_{1}}{C_{3}\alpha}\Bigr{)}$

Setting $\lambda=0$ yields the estimate on err and setting
$\lambda=\frac{1}{2}$ the one on res.

$\diamondsuit$

###  A.2 Convergence of (1.8) without VSC

under the following conditions.

######  Assumption 1.

$r$ is injective and there exist tolopogies $\widetilde{\mathcal{T}}$,
$\widetilde{\widetilde{\mathcal{T}}}$, on $\widetilde{X}$ and $\mathcal{T}$ on
$X$ such that

  1. (i)

sublevel sets of $\mathcal{R}$ are $\widetilde{\mathcal{T}}$ compact;

  2. (ii)

For any two sequences $(\hat{r}^{1}_{k})_{k\in\mathbb{N}}$,
$(\hat{r}^{2}_{k})_{k\in\mathbb{N}}$

$\mathcal{Q}(\hat{r}^{1}_{k},\hat{r}^{2}_{k})\to 0\text{ and
}\hat{r}^{2}_{k}\stackrel{{\scriptstyle\widetilde{\mathcal{T}}}}{{\longrightarrow}}\hat{r}^{2}\
\Rightarrow\ \text{ there exists a subsequence
}\hat{r}^{1}_{k_{\ell}}\stackrel{{\scriptstyle\widetilde{\widetilde{\mathcal{T}}}}}{{\longrightarrow}}\hat{r}^{2};$

  3. (iii)

$r^{-1}:r(U)\to U$ is $\widetilde{\widetilde{\mathcal{T}}}-\mathcal{T}$
continuous;

  4. (iv)

$\|K\cdot\|_{Y}$ is $\widetilde{\mathcal{T}}$ lower semicontinuous;

  5. (v)

$\mathcal{P}$ is $\mathcal{T}$ lower semicontinuous.

Moreover, the regularization and tolerance parameters $\alpha=\alpha(\delta)$,
$\beta=\beta(\delta)$, $\eta(\delta)$ are assumed to satisfy

$\alpha(\delta)\to 0,\qquad\frac{\delta^{p}+\eta(\delta)}{\alpha(\delta)}\to
0,\qquad\frac{\delta^{p}+\eta(\delta)}{\beta(\delta)}\to 0,\qquad\text{ as
}\delta\to 0.$ (A.1)

The a priori choice (2.5) is compatible with (A.1), and so is the particular
one in the logarithmic case indicated in Remark 1 when slightly restricted to
$\alpha(\delta)\sim\delta^{\tilde{p}}$ with $\tilde{p}\in(0,p)$.

######  Proposition 12.

Under condition (1.6) and Assumption 1, the approximations
$x_{\alpha,\beta,\eta}^{\delta}$ defined by (2.4) with (A.1) converge
$\mathcal{T}$ subseqentially to a solution of (1.7), that is, every
subsequence of
$(\hat{r}_{\alpha,\beta,\eta}^{\delta},x_{\alpha,\beta,\eta}^{\delta})_{\delta>0}$
has a $\widetilde{\mathcal{T}}\times\mathcal{T}$ convergent subsequence and
the limit $(\hat{r},x)$ of every $\widetilde{\mathcal{T}}\times\mathcal{T}$
convergent subsequence solves (1.7). If the $\mathcal{R}$ minimizing solution
$(\hat{r}^{\dagger},x^{\dagger})$ of (1.7)222i.e.,
$(\hat{r}^{\dagger},x^{\dagger})$ solving (1.7) such that
$\forall(\hat{r},x)\mbox{ solving \eqref{FP}}\,:\
\mathcal{R}(\hat{r}^{\dagger},x^{\dagger})\leq\mathcal{R}(\hat{r},x)$ is
unique and then
$\hat{r}^{\delta}\stackrel{{\scriptstyle\widetilde{\mathcal{T}}}}{{\longrightarrow}}r(x^{\dagger})$
and
$x^{\delta}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}x^{\dagger}$.

Proof. For an arbitrary sequence $\delta_{n}\to 0$, abbreviating
$(\hat{r}_{n},x_{n}):=(\hat{r}_{\alpha_{n}\beta_{n}\eta_{n}}^{\delta_{n}},x_{\alpha_{n}\beta_{n}\eta_{n}}^{\delta_{n}}$,
from (2.11) (with $(2+C_{\eta})\delta^{p}$ replaced by the more general bound
$2\delta^{p}+\eta(\delta)$ and (A.1) we obtain

$\displaystyle(a)$
$\displaystyle\limsup_{n\to\infty}\mathcal{R}(\hat{r}_{n})\leq\mathcal{R}(r(x^{\dagger}))$
$\displaystyle(b)$
$\displaystyle\limsup_{n\to\infty}\mathcal{Q}(r(x_{n}),\hat{r}_{n})\ =0$
$\displaystyle(c)$
$\displaystyle\limsup_{n\to\infty}\|K(\hat{r}_{n}-r(x^{\dagger}))\|_{Y}\ =0$
$\displaystyle(d)$ $\displaystyle\limsup_{n\to\infty}\|P(x_{n})\|=0.$

From (i) and (a) we conclude existence of a subsequence $\hat{r}_{n_{k}}$ an
and element $\hat{r}\in\widetilde{X}$ with
$\mathcal{R}(\hat{r})\leq\mathcal{R}(r(x^{\dagger}))$ such that
$\hat{r}_{n_{k}}\stackrel{{\scriptstyle\widetilde{\mathcal{T}}}}{{\to}}\hat{r}$.
This together with (b), (ii) yields existence of a subsequence such that
$r(x_{n_{k_{\ell}}})\stackrel{{\scriptstyle\tilde{\tilde{\mathcal{T}}}}}{{\to}}\hat{r}$.
Now by (iii) we obtain
$x_{n_{k_{\ell}}}\stackrel{{\scriptstyle\mathcal{T}}}{{\to}}x$ and
$r(x)=\hat{r}$. Finally, (c), (iv) imply $K(r(x))=K(r(x^{\dagger}))$ and (d),
(v) imply $P(x)=0$.

$\diamondsuit$

###  A.3 Verification of (2.23) for Hölder or logarithmic source conditions
(1.32)

######  Lemma 13.

The functions defined in (1.32) satisfy

$\psi(a+b)\leq\psi(a)+\psi(b),\text{ for all }a,b>0$

Proof. In the Hölder case, with $m:=1/\mu$, $x=(b/a)^{\mu}$ the claim is
equivalent to

$\phi(x):=\frac{1+x^{m}}{(1+x)^{m}}\leq 1,\text{ for all }x>0$

which in case of an integer $m$ immediately follows from the binomial theorem,
otherwise from the fact that $\lim_{x\searrow 0}\phi(x)=1$,
$\lim_{x\nearrow\infty}\phi(x)=1$,
$\phi^{\prime}(x)=(1+x)^{-2m}(mx^{m-1}(1+x)^{m}-(1+x^{m})m(1+x)^{m-1}=m(1+x)^{-m-1}(x^{m-1}-1)=0$
iff $x=1$ and $\phi(1)=2^{1-m}\leq 1$, since $m=1/\mu\geq 1$.

In the logarithmic case for simplicity of exposition we just consider $\nu=1$,
$t_{0}=1/e$. With $\alpha:=(-\log a)^{-1}$, $\beta:=(-\log b)^{-1}$, the claim
is equivalent to

$\phi(\beta):=\frac{-\log\left(e^{-1/\alpha}+e^{-1/\beta}\right)}{\alpha+\beta}\leq
1,\text{ for all }\beta>0$

and any fixed $\alpha>0$. It is readily checked that $\lim_{\beta\searrow
0}\phi(\beta)=1$, $\lim_{\beta\nearrow\infty}\phi(\beta)=0$ and
$\phi^{\prime}(\beta_{*})=(\alpha+\beta_{*})^{-2}\Bigl{(}-\left(e^{-1/\alpha}+e^{-1/\beta_{*}}\right)^{-1}e^{-1/\beta_{*}}\tfrac{\alpha+\beta_{*}}{\beta_{*}^{2}}+\log\left(e^{-1/\alpha}+e^{-1/\beta_{*}}\right)\Bigr{)}=0$
iff
$\log\left(e^{-1/\alpha}+e^{-1/\beta_{*}}\right)=\left(e^{-1/\alpha}+e^{-1/\beta_{*}}\right)^{-1}e^{-1/\beta_{*}}\tfrac{\alpha+\beta_{*}}{\beta_{*}^{2}}\geq
0$ which inserted into $\phi$ yields
$\phi(\beta_{*})=-\left(e^{-1/\alpha}+e^{-1/\beta_{*}}\right)^{-1}e^{-1/\beta_{*}}\,\beta_{*}^{-2}=-\frac{b_{*}}{a+b_{*}}(-\log
b_{*})^{-2}\leq 0$.

$\diamondsuit$

## References

  * [1] A. Auslender.  Méthodes numériques pour la décomposition et la minimisation de fonctions non différentiables.  Numer. Math., 18:213–223, 1971/72.
  * [2] A. B. Bakushinsky and M. Y. Kokurin.  Iterative Methods for Approximate Solution of Inverse Problems, volume 577 of Mathematics and Its Applications.  Springer, Dordrecht, 2004.
  * [3] Dimitri P. Bertsekas and John N. Tsitsiklis.  Parallel and distributed computation: numerical methods.  Athena Scientific, Belmont, MA, 2014.  Originally published by Prentice-Hall, Inc. in 1989. Includes corrections (1997).
  * [4] Barbara Blaschke-Kaltenbacher.  Some Newton Type Methods for the Regularization of Nonlinear Ill–posed Problems.  PhD thesis, University of Linz, 1996.
  * [5] Martin Burger and Barbara Kaltenbacher.  Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems.  SIAM J.Numer.Anal., 44:153–182, 2006.
  * [6] Alberto P Calderón.  On an inverse boundary value problem.  Seminar on Numerical Analysis and its Applications to Continuum Physics, 25:65–73, 1980.
  * [7] G. Chavent and K. Kunisch.  On weakly nonlinear inverse problems.  SIAM J. Appl. Math., 56(2):542–572, 1996.
  * [8] P. Deuflhard, H. W. Engl, and O. Scherzer.  A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions.  Inverse Problems, 14:1081–1106, 1998.
  * [9] Heinz W. Engl, Martin Hanke, and Andreas Neubauer.  Regularization of Inverse Problems.  Kluwer, Dordrecht, 1996.
  * [10] Luigi Grippo and Marco Sciandrone.  Globally convergent block-coordinate techniques for unconstrained optimization.  Optim. Methods Softw., 10(4):587–637, 1999.
  * [11] Martin Hanke, Andreas Neubauer, and Otmar Scherzer.  A convergence analysis of the Landweber iteration for nonlinear ill-posed problems.  Numer. Math., 72(1):21–37, 1995.
  * [12] Jon Johnsen.  Pointwise multiplication of besov and triebel-lizorkin spaces.  Mathematische Nachrichten, 175(1):85–133, 1995.
  * [13] B. Kaltenbacher, A. Neubauer, and O. Scherzer.  Iterative Regularization Methods for Nonlinear Problems.  de Gruyter, Berlin, New York, 2008.  Radon Series on Computational and Applied Mathematics.
  * [14] B. Kaltenbacher, A. Neubauer, and O. Scherzer.  Iterative Regularization Methods for Nonlinear Problems.  de Gruyter, Berlin, New York, 2008.  Radon Series on Computational and Applied Mathematics.
  * [15] Barbara Kaltenbacher.  Some Newton type methods for the regularization of nonlinear ill–posed problems.  Inverse Problems, 13:729–753, 1997.
  * [16] Barbara Kaltenbacher.  On Broyden’s method for nonlinear ill–posed problems.  Numerical Functional Analysis and Optimization, 19:807–833, 1998\.
  * [17] Barbara Kaltenbacher.  Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance.  IMA Journal of Numerical Analysis, page drad044, 07 2023.  see also arXiv:2209.12596 [math.NA].
  * [18] Barbara Kaltenbacher and Mario Previatti de Souza.  Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space.  Numerische Mathematik, 140:449–478, 2018.  arxiv:1707.07589 [math.NA].
  * [19] Barbara Kaltenbacher and Teresa Rauscher.  Simultaneous reconstruction of sound speed and nonlinearity parameter in a paraxial model of vibro-acoustography in frequency domain.  Computational Methods in Applied Mathematics, 2023.  to appear; see also arXiv:2303.17236 [math.NA].
  * [20] Barbara Kaltenbacher and William Rundell.  On the simultanenous reconstruction of two space dependent coefficients in acoustic nonlinearity parameter tomography.  Inverse Problems, 39:105001, 2023.  see also arXiv:2210.08063 [math.NA].
  * [21] Andreas Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems.  Applied Mathematical Sciences. Springer International Publishing, 3rd ed. edition, 2021.
  * [22] Z. Q. Luo and P. Tseng.  On the convergence of the coordinate descent method for convex differentiable minimization.  J. Optim. Theory Appl., 72(1):7–35, 1992.
  * [23] Jennifer L. Mueller and Samuli Siltanen.  Linear and Nonlinear Inverse Problems with Practical Applications.  Society for Industrial and Applied Mathematics, Philadelphia, PA, 2012\.
  * [24] Andreas Neubauer and Otmar Scherzer.  Finite-dimensional approximation of Tikhonov regularized solutions of nonlinear ill-posed problems.  Numer. Funct. Anal. Optim., 11(1-2):85–99, 1990.
  * [25] J. M. Ortega and W. C. Rheinboldt.  Iterative solution of nonlinear equations in several variables.  Academic Press, New York-London, 1970.
  * [26] O. Scherzer, B. Hofmann, and Z. Nashed.  Gauss–newton method for solving linear inverse problems with neural network coders.  Sampl. Theory Signal Process. Data Anal., 21(2), 2023.
  * [27] Thomas Schuster, Barbara Kaltenbacher, Bernd Hofmann, and Kamil S. Kazimierski.  Regularization Methods in Banach Spaces.  Walter de Gruyter, Berlin, 2012.
  * [28] Frederic Weidling.  Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs.  Universitätsverlag Göttingen, Göttingen, 2019.