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Barbara Kaltenbacher

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# Convergence guarantees for coefficient reconstruction in PDEs from boundary
measurements by variational and Newton type methods via range invariance

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

(2022; 2022; Date; Date; Date)

###### Abstract

A key observation underlying this paper is the fact that the range invariance
condition for convergence of regularization methods for nonlinear ill-posed
operator equations – such as coefficient identification in partial
differential equations PDEs from boundary observations – can often be achieved
by extending the searched for parameter in the sense of allowing it to depend
on additional variables. This clearly counteracts unique identifiability of
the parameter, though. The second key idea of this paper is now to restore the
original restricted dependency of the parameter by penalization. This is shown
to lead to convergence of variational (Tikhonov type) and iterative (Newton
type) regularization methods. We concretize the abstract convergence analysis
in a framework typical of parameter identification in PDEs in a reduced and an
all-at-once setting. This is further illustrated by three examples of
coefficient identification from boundary observations in elliptic and time
dependent PDEs.

###### keywords:

parameter identification in PDEs; iterative regularization; variational
regularization; range invariance condition.

##  1 Introduction

A proof of convergence of iterative methods for parameter identification
problems in partial differential equations PDEs from boundary measurements, as
relevant, e.g., in tomographic applications, has been a long-standing open
problem due to the fact that the convergence analysis of these methods can
only be carried out under restrictions on the nonlinearity of the forward
operator that could not be verified for such PDE coefficient identification
problems so far. Likewise, although at a first glance not burdened with such
restrictive assumptions on the forward operator, Tikhonov regularization
requires the computation of a global minimizer of a functional whose (local)
convexity can only be verified under similar restricions on the nonlinearity.

The goal of this paper is to revisit one of these conditions – range
invariance of the linearized forward operator – and show that an introduction
of additional/artificial degrees of freedom in the searched for parameter
111we will shortly call this “extension” often allows for its verification.
Since this counteracts unique identifiability of the parameter, we restore the
original restricted dependency of the parameter by a proper penalization
within the reconstruction method.

To set the stage and explain ideas, we consider an inverse problem either in
its all-at-once formulation

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

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

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

Here $A:Q\times V\to W^{*}$ is the model operator, $C$ the observation
operator and $Q$, $V$, $W^{*}$, $Y$ are Banach spaces. 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.

Due to the characteristic ill-posedness of these problems, regularization has
to be applied, cf., e.g., [7, 16, 43, 29, 55].

To guarantee well-definedness and convergence of these methods, additional
assumptions on the forward operator $F$ (that is, in the above settings,
$\mathbb{F}$ or $\mathbf{F}$) need to be imposed. These basically fall into
two categories:

In the first and most common are the tangential cone condition [54]

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

(in a neighborhood $U$ of the exact solution) for the convergence of Landweber
iteration [21] and Newton’s method [36] as well as the closely related weak
nonlinearity condition from [14] for local convexity of the Tikhonov
functional, whose global minimizer must be computed in Tikhonov
regularization. Also in this category is the Newton-Mysovskii condition

$\forall x,\tilde{x}\in U\,:\
\|(F^{\prime}(x)-F^{\prime}(\tilde{x}))F^{\prime}(x)^{\dagger}\|_{X}\leq
C_{\textup{NM}}\|x-\tilde{x}\|_{X},$ (1.6)

(where † denotes the generalized inverse), used and discussed in [15]. A
sufficient condition for these – and actually quite often used for their
verification – is invariance of the range of the adjoint of $F^{\prime}$

$\forall x,\tilde{x}\in
U\,:\quad\textup{rng}(F^{\prime}(x)^{\star})=\textup{rng}(F^{\prime}(\tilde{x})^{\star}).$
(1.7)

The second, less well-explored category of nonlinearity conditions is related
to range invariance of $F^{\prime}$ itself

$\forall x,\tilde{x}\in
U\,:\quad\textup{rng}(F^{\prime}(x))=\textup{rng}(F^{\prime}(\tilde{x})).$
(1.8)

It contains the affine covariant Lipschitz condition

$\forall x,\tilde{x}\in U\,:\
\|F^{\prime}(x)^{\dagger}(F^{\prime}(x)-F^{\prime}(\tilde{x}))\|_{X}\leq
C_{\textup{acL}}\|x-\tilde{x}\|_{X},$ (1.9)

(compare to (1.6)) which allows to prove convergence of Newton and quasi
Newton methods, cf., e.g., [9, 10, 15, 31, 30, 32].

Adjoint range invariance (1.7) (and alongside the tangential cone condition as
well as weak nonlinearity and the Newton–Mysovskii condition) can usually only
be proven to hold under a full observation assumption, that is, $C$ is just
the identity or some embedding operator, which excludes practically relevant
scenarios of identifying coefficients from boundary measurements. In contrast
to this, range invariance of $F^{\prime}$ itself (1.8) clearly follows from
range invariance of the derivative $S^{\prime}$ of $S$ in (1.2) and this is
actually the way it is usually verified. This condition is independent of the
observation $C$ and in particular also allows for boundary observations as
relevant in tomographic imaging or nondestructive testing. We will therefore
focus on range invariance of $F^{\prime}$ itself here.

By means of the Closed Graph Theorem, (1.8) can be shown to be equivalent to
boundedness and bounded invertibility of
$F^{\prime}(x)^{\dagger}F^{\prime}(\tilde{x})$ for all $x,\tilde{x}\in U$,
(see Lemma B.1 in Appendix B and compare to (1.9)) or, alternatively written,
to

$\forall x,\tilde{x}\in U\,\exists R(x,\tilde{x})\in L(X,X)\,:\
R(x,\tilde{x})^{-1}\in L(X,X)\mbox{ and
}F^{\prime}(x)=F^{\prime}(\tilde{x})R(x,\tilde{x}).$

In view of bounded invertibility of $R(x,\tilde{x})$, it suffices to fix
$\tilde{x}$ here to some $x_{0}\in U$

$\exists x_{0}\in U\,\forall x\in U\,\exists R(x)\in L(X,X)\,:\ R(x)^{-1}\in
L(X,X)\mbox{ and }F^{\prime}(x)=F^{\prime}(x_{0})R(x),$ (1.10)

where sufficiency can be seen by setting
$R(x,\tilde{x})=R(\tilde{x})^{-1}R(x)$. Here $U\subset\mathcal{D}(F)$ is a
neighborhood of $x^{\dagger}$ and $F^{\prime}$ the Gâteaux derivative, which
we assume to exist whenever using it.

######  Remark 1.

The validity of range invariance for arbitrary observation operator $C$ is
formally equivalent for all-at-once (1.1) and reduced (1.2) formulations,
provided $\tfrac{\partial A}{\partial u}(q,u)$ is an isomorphism for $(q,u)\in
U$ (as usually assumed in the reduced setting), see Lemma B.2 in Appendix B.
Note however, that much more freedom in the choice of function spaces is
allowed in the all-at-once case and when dropping the restrictions of
$\tfrac{\partial A}{\partial u}(q,u)$ to be an isomorphism and
$(q,u)=(q,S(q))$, $(q_{0},u_{0})=(q_{0},S(q_{0}))$. This becomes crucial when
establishing $R(x)\in L(X,X),\,R(x)^{-1}\in L(X,X)$.

###  1.1 Examples

For further motivation, we provide three classical examples of parameter
identification problems where range invariance of the linearization of the
forward operator can be verified. All the computations will be completely
formal; for details we refer to Appendix A. As can be clearly seen there, in
the all-at-once setting we have much more freedom in choosing the function
spaces as compared to the reduced case, where we are bound to PDE solution
theory in order to establish the parameter-to-state map $S$.

######  Example 1 (identification of a potential in an elliptic PDE).

(For the reduced case (1.2), see, e.g., [31, Example 3.2]). Consider
identification of the spatially varying potential $q\in L^{2}(\Omega)$ in the
elliptic boundary value problem

$\displaystyle-\Delta u+q(x)u$ $\displaystyle=f\mbox{ in }\Omega$ (1.11)
$\displaystyle\partial_{\nu}u$ $\displaystyle=h\mbox{ on }\partial\Omega$

with given $f$, $h$, from observations $y=Cu$ of $u$, for example measurements
at the boundary of the domain $\Omega$.

With an extension $\bar{h}$ of the boundary data according to (A.3), we can
write the inverse problem in a reduced form (1.2) as $\mathbf{F}(q)=y$, or in
an all-at-once form (1.1) as $\mathbb{F}(q,u)=(0,y)^{T}$ with the respective
reduced and all-at-once forward operators being defined by

$\displaystyle\mathbf{F}(q)=CS(q)=-C(-\Delta_{N}+q\cdot)^{-1}[f+\bar{h}],$
(1.12) $\displaystyle\mathbb{F}(q,u)=(-\Delta_{N}u+q\cdot
u-f-\bar{h},\,Cu)^{T}.$

where $\Delta_{N}$ denotes the Neumann Laplacian (A.2).

The linearizations of $\mathbf{F}$ and $\mathbb{F}$ are formally given by

$\displaystyle\mathbf{F}^{\prime}(q)\underline{dq}=-C(-\Delta_{N}+q\cdot)^{-1}\Bigl{[}\underline{dq}\cdot
S(q)\Bigr{]},$ (1.13)
$\displaystyle\mathbb{F}^{\prime}(q,u)(\underline{dq},\underline{du})=(-\Delta_{N}\underline{du}+q\cdot\underline{du}+\underline{dq}\cdot
u,\,C\underline{du})^{T},$

and can easily be verified to be Fréchet derivatives in the function space
setting given in (A.5), (A.6). For any fixed $q_{0}\in L^{2}(\Omega)$, setting

$\displaystyle\mathbf{R}(q)\underline{dq}$
$\displaystyle=\tfrac{1}{S(q_{0})}\cdot\bigl{(}(-\Delta_{N}+q_{0}\cdot)\,(-\Delta_{N}+q\cdot)^{-1}[\underline{dq}\cdot
S(q)]\bigr{)}$ (1.14)
$\displaystyle=\tfrac{1}{S(q_{0})}\cdot\bigl{(}\underline{dq}\cdot
S(q)+(q-q_{0})\cdot S^{\prime}(q)\underline{dq}\bigr{)}$
$\displaystyle\mathbb{R}(q,u)(\underline{dq},\underline{du})$
$\displaystyle=\Bigl{(}\tfrac{1}{u_{0}}\bigl{(}\underline{dq}\cdot
u+(q-q_{0})\cdot\underline{du}\bigr{)},\,\underline{du}\Bigr{)}$

we get

$\mathbf{F}^{\prime}(q)=\mathbf{F}^{\prime}(q_{0})\mathbf{R}(q),\quad\mathbb{F}^{\prime}(q,u)=\mathbb{F}^{\prime}(q_{0},u_{0})\mathbb{R}(q,u),$
(1.15)

where boundedness away from zero of the denominators can be guaranteed by
maximum principles for $S(q_{0})$ and simply by a proper choice for $u_{0}$.
One can also prove (see (A.7)) that

$\|\mathbf{R}(q)-\textup{id}\|_{Q\to Q}\leq
C\|q-q_{0}\|_{Q},\quad\|\mathbb{R}(q,u)-\textup{id}\|_{Q\times V\to Q\times
V}\leq C\|(q,u)-(q_{0},u_{0})\|_{Q\times V},$ (1.16)

so that in a sufficiently small neighborhood $U$ of $q_{0}$ (or of
$(q_{0},u_{0})$, respectively), the $R$ operators are bounded and boundedly
invertible, which implies range invariance of the linearizations

$\forall q\in U\,:\
\textup{rng}(\mathbf{F}^{\prime}(q))=\textup{rng}(\mathbf{F}^{\prime}(q_{0})),\quad\forall(q,u)\in
U\,:\
\textup{rng}(\mathbb{F}^{\prime}(q,u))=\textup{rng}(\mathbb{F}^{\prime}(q_{0},u_{0})),$

no matter which observation operator $C$ is contained in the definition of
$\mathbf{F}$, $\mathbb{F}$. Note that here, besides linearity and boundedness
$C\in L(V,Y)$, we do not make any assumptions on the observation operator $C$.
This example has also been shown to satisfy the tangential cone condition in
the reduced setting (1.2) [21], however only with full measurements, that is,
$C$ being the embedding operator $V\to L^{2}(\Omega)$.

Thus, range invariance is easy to verify here, even in case of restricted
measurements, as opposed to the tangential cone condition or adjoint range
invariance. Still, keeping also unique identifiability of $q$ in mind, the
operator $C$ should not be too restrictive either. For example by a
straightforward dimension count, uniqueness from a single boundary observation
cannot be expected to hold here and in fact the full Neumann-to-Dirichlet map
(see (4.1) below) assigning to any prescribed boundary flux $\partial_{\nu}u$
the corresponding Dirichlet data $\textup{tr}_{\partial\Omega}u$ can be shown
to yield uniqueness (see [28] for $d=2$, [45] for $d\geq 3$).

######  Example 2 (identification of a potential in a time-dependent PDE).

We add a time dimension by considering the transient version of Example 1

$\displaystyle D_{t}^{M}u-\Delta u+q(x)u$ $\displaystyle=f\mbox{ in
}\Omega\times(0,T)$ (1.17) $\displaystyle\partial_{\nu}u$
$\displaystyle=h\mbox{ on }\partial\Omega\times(0,T)$
$\displaystyle\partial_{t}^{m}u(x,0)$ $\displaystyle=u_{m}(x)\ x\in\Omega\quad
m\in\\{0,\ldots,m-1\\}$

with given $f$ and $h$. Here besides the parabolic case with
$D_{t}^{M}=\partial_{t}$ one can consider quite general transient models,
containing also wave and time fractional equations. In particular, we have
strongly damped wave equations with $M=2$ and
$D_{t}^{M}:=\partial_{tt}-b\Delta\partial_{t}$ for some $b>0$ in mind.

With an extension $\bar{h}$ of the initial and boundary data such that
$D_{t}^{M}\bar{h}-\Delta\bar{h}=f$ in $\Omega\times(0,T)$,
$\partial_{\nu}\bar{h}=h$ on $\partial\Omega\times(0,T)$,
$\partial_{t}^{m}\bar{h}(x,0)=u_{m}(x)$, $x\in\Omega$,
$m\in\\{0,\ldots,M-1\\}$ and $\hat{u}:=u-\bar{h}$, this becomes the abstract
ODE

$D_{t}^{M}\hat{u}(t)+(-\Delta_{N}+q\cdot)\hat{u}=-q\cdot\bar{h}\,,\quad\hat{u}(0)=0;$
(1.18)

the hat will be skipped in the following.

For a space- and possibly also time-dependent potential $\widetilde{q}$, the
operator $T(\widetilde{q})$ defined by
$(T(\widetilde{q})v)(t)=D_{t}^{M}v(t)-\Delta_{N}v(t)+\widetilde{q}(t)\cdot
v(t)$, can be shown to be bounded and boundedly invertible in appropriate
function spaces (cf. Appendix A). With this notation, we can write
$u=S(q)=-T(q)^{-1}[q\cdot\bar{h}]$ and this formula applies both for
stationary and time variable potentials $q$. This allows to define the reduced
and all-at-once forward operators

$\displaystyle\mathbf{F}(q)=CS(q)=-CT(q)^{-1}[q\cdot\bar{h}],$ (1.19)
$\displaystyle\mathbb{F}(q,u)=(u^{\prime}-\Delta_{N}u+q\cdot(u+\bar{h}),\,Cu)^{T}\,.$

Analogously to the above elliptic example (more or less just substituting
$(-\Delta_{N}+q\cdot)$ by $T(q)$) we arrive at range invariance (1.15) of the
linearized reduced and all-at-once forward operators with

$\displaystyle\mathbf{R}(q)\underline{dq}=\tfrac{1}{S(q_{0})+\bar{h}}\cdot\,T(q_{0})\,T(q)^{-1}[\underline{dq}\cdot(S(q)+\bar{h})]$
(1.20)
$\displaystyle\mathbb{R}(q,u)(\underline{dq},\underline{du})=\Bigl{(}\tfrac{1}{u_{0}+\bar{h}}\bigl{(}\underline{dq}\cdot(u+\bar{h})+(q-q_{0})\cdot\underline{du}\bigr{)},\,\underline{du}\Bigr{)}.$

Also smallness of the difference between the $R$ operators and the identity
can be established similarly to the elliptic example.

Note that even if $q$ depends on $x$ only, due to multiplication with the time
dependent function $\tfrac{1}{S(q_{0})+\bar{h}}$ or
$\tfrac{1}{u_{0}+\bar{h}}$, the result after application of one of the $R$
operators in (1.20) will in general be a space and time dependent function.
Thus we are forced to use a time dependent parameter space, e.g.
$L^{2}(0,T;L^{2}(\Omega))$.

In this example, uniqueness of $q(x)$ from a single time trace on the boundary
or final time measurements of $u$

$C=\textup{tr}_{\Gamma\times(0,T)}\ \textup{ or }\
C=\textup{tr}_{\Omega\times\\{T\\}}$

can be shown [27, 48, 52]. However, as we have just seen, establishing range
invariance requires extension of the parameter space, which leads to a loss of
uniqueness. Therefore, we will introduce a restriction procedure by
penalization, that allows to show convergence to the unique solution of the
original inverse problem (that is, with the original smaller parameter space).
In this example, this can be done, e.g., by means of the functional
$\mathcal{P}(q):=\|Pq\|_{Z}^{2}$ with $Z=L^{2}(0,T;L^{2}(\Omega))$ and
$P(q)(t)=q(t)-\frac{1}{T}\int_{0}^{T}q(s)\,ds$, using the $L^{2}$ projection
of $q$ on the subspace of functions that are constant in time.

######  Example 3 (identification of a diffusion coefficient in an elliptic
PDE).

We now consider identification of the coefficient $a(x)$ in

$\displaystyle-\nabla\cdot(a(x)\nabla u)$ $\displaystyle=f\mbox{ in }\Omega$
(1.21) $\displaystyle a(x)\partial_{\nu}u$ $\displaystyle=h\mbox{ on
}\partial\Omega.$

with given $f$, $h$. In case $f=0$ and with the physical interpretation of $u$
being an electric potential, $a$ the conductivity, and the measurements
defined by all voltage-current pairs achievable by excitation at electrodes on
the boundary, this is the well-known and well-investigated Calderón problem
with application in electric impedance tomography EIT.

With general measurements defined by a linear operator $C$, this can be
written in a reduced (1.2) or in an all-at-once form (1.1) with $q=a$ and

$\displaystyle\mathbf{F}(a)=CS(a)=-C(-\Delta_{a,N})^{-1}[f+\bar{h}],$ (1.22)
$\displaystyle\mathbb{F}(a,u)=(-\Delta_{a,N}u-f-\bar{h},\,Cu)^{T},$

where

$\langle-\Delta_{a,N}u,v\rangle_{H^{1}(\Omega)^{*},H^{1}(\Omega)}:=\int_{\Omega}a\,\nabla
u\cdot\nabla v\,dx\quad\forall u,v\in H^{1}(\Omega)$ (1.23)

and $\bar{h}$ is defined by (A.3).

Here, verification of (1.8) (and likewise of (1.7)) only works out in one
space dimension. To see this, we consider the derivatives

$\displaystyle\mathbf{F}^{\prime}(a)\underline{da}=-C(-\Delta_{a,N})^{-1}\Bigl{[}\Delta_{\underline{da},N}S(a)\Bigr{]},$
$\displaystyle\mathbb{F}^{\prime}(a,u)(\underline{da},\underline{du})=(-\Delta_{a,N}\underline{du}-\Delta_{\underline{da},N}u,\,C\underline{du})^{T},$

and write the sufficient condition for (1.8) (which is also necessary if we
want it to hold for arbitrary $C$) in the reduced case as

$(-\Delta_{a,N})^{-1}\Bigl{[}\Delta_{\underline{da},N}S(a)\Bigr{]}=(-\Delta_{a_{0},N})^{-1}\Bigl{[}\Delta_{R(a)\underline{da},N}S(a_{0})\Bigr{]}\quad\forall\underline{da}\in
Q$

which is equivalent to

$\Delta_{R(a)\underline{da},N}S(a_{0})=(-\Delta_{a_{0},N})(-\Delta_{a,N})^{-1}\Bigl{[}\Delta_{\underline{da},N}S(a)\Bigr{]}\quad\forall\underline{da}\in
Q.$

This can be viewed as a transport equation

$\nabla r\cdot\nabla
S(a_{0})+r\Delta_{N}S(a_{0})=\Delta_{a-a_{0},N}S^{\prime}(a)\underline{da}+\nabla\cdot(\underline{da}\nabla
S(a))$ (1.24)

for $r=R(a)\underline{da}$. Since in general the directions of the gradients
$\nabla S(a_{0})$ and $\nabla S(a)$ vary in space and differ from each other,
an estimate of the $L^{p}$ norm of the $\underline{da}$ term will contain a
term of the form $\|\nabla\underline{da}\|_{L^{t}(\Omega)}\|\nabla
S(a)\|_{L^{s}(\Omega)}$ (with exponents $t,\,s$ chosen according to Hölder’s
inequality), whereas the left hand side of (1.24) only provides directional
derivatives of $r=R(a)\underline{da}$ in the direction of $\nabla S(a_{0})$
and thus not the full $W^{1,t}(\Omega)$ seminorm of $r$ in dimension $d>1$.

An analogous problem occurs in the all-at-once case. Thus we are limited to
the 1-d setting, where this loss of regularity does not occur. Indeed, it can
be easily checked that the choice
$R(a)\underline{da}:=-a_{0}\cdot\frac{(S^{\prime}(a)\underline{da})_{x}}{(S(a_{0}))_{x}}$
yields (1.8), provided $(S(a_{0}))_{x}$ is bounded away from zero.

An idealized formulation of the observations in EIT is given by the Neumann-
to-Dirichlet N-t-D map, defined analogously to (4.1) below; for the more
realistic complete electrode model, cf. [56]. As typical for tomographic
applications, these are boundary and not interior observations and so
verification of the tangential cone condition (1.5) or adjoint range
invariance (1.7) seems out of reach. We have seen that also range invariance
in the form (1.8) does not work out in higher than one space dimension. Still,
since (1.21) is related to (1.11) via the transform

$u\to\sqrt{a}u,\quad q=\frac{1}{4}\frac{|\nabla
a|^{2}}{a^{2}}-\frac{1}{2}\frac{\Delta a}{a},$ (1.25)

Example 1 shows that (1.8) indeed makes sense also in tomographic
applications. Clearly, the transform (1.25) requires $a$ to be sufficiently
smooth, which is often not the case in applications. Therefore it would be
desirable to cover the problem in its original form (1.21) that only requires
an $L^{\infty}$ coefficient $a$. In Example 6 we will show that this is in
fact possible in case of simultaneous identification of a diffusion and an
absorption coefficent.

The plan of this paper is as follows. In Section 2 we derive and analyze some
regularization methods making use of the structure of range invariance (1.10)
(or actually a more general condition in difference form (2.1). Doing so, we
will follow two paradigms: The first is variational, that is, by minimizing an
appropriately defined objective function as known from classical Tikhonov
regularization. The second is Newton’s method or frozen versions thereof,
which we regularize by stabilizing each Newton step and by early stopping.
Section 3 provides a general class of applications, typical of coefficient
identification in PDEs, where the abstract convergence conditions (in
particular (2.1)) can be verified in a reduced and an all-at-once setting,
respectively. Finally, in Section 4, we provide concrete examples of
coefficient identification in PDEs to which this framework applies.

### Notation

$L(X,Y)$ denotes the space of bounded linear operators between the normed
spaces $X$ and $Y$.
As already done above, we will denote the range of an operator $A:X\to Y$ by
$\textup{rng}(A)\subseteq Y$; its nullspace will be abbreviated by
$\textup{nsp}(A)\subseteq X$.
The dual of $X$ is denoted by $X^{*}$, the (Banach space) adjoint by $A^{*}\in
L(Y^{*},X^{*})$, and in case of $X$, $Y$ being Hilbert spaces, $A^{\star}\in
L(Y,X)$ denotes the Hilbert space adjoint.
A closed ball with radius $\rho>0$ and center $x_{0}$ is denoted by
$\mathcal{B}_{\rho}^{X}(x_{0})=\\{x\in X\,:\,\|x-x_{0}\|\leq\rho\\}\subseteq
X$.
We employ the notation $W^{s,p}(\Omega)$, $H^{s}=W^{s,2}(\Omega)$ for Sobolev
spaces over some domain $\Omega$ and $L^{p}_{\mu}(J;Z)$, $W^{s,p}_{\mu}(J;Z)$
for Bochner-Sobolev spaces of a variable in $J$ with values in the Banach
space $Z$, with respect to the measure $\mu$, cf., e.g., [51, Section 7.1].
In particular, we will use $H_{\diamondsuit}^{M}(0,T;L^{2}(\Omega))=\\{v\in
H^{M}(0,T;L^{2}(\Omega))\,:\,\partial_{t}^{m}v(0)=0,\
m\in\\{0,\ldots,M\\}\\}$.
The norm of the continuous embedding $X(\Omega)\to Y(\Omega)$ for function
spaces on a domain $\Omega$ is denoted by $C_{X,Y}^{\Omega}$.
For $\Gamma\subseteq\overline{\Omega}$ (the closure of a $d$-dimensional
domain $\Omega$), $\Gamma$ being a $d-1$ dimensional regular manifold (e.g.,
$\Gamma=\partial\Omega$) and $s\geq 0$, by
$\textup{tr}_{\Gamma}:H^{s}(\Omega)\to H^{s-1/2}(\Gamma)$ we denote the trace
operator.

##  2 Structure exploiting reconstruction methods and their convergence

We return to the general inverse problem (1.3) comprising reduced and all-at-
once formulations. Rather than working with (1.10) directly, we will base our
analysis on a differential range invariance condition:

$\exists x_{0}\in U\,,\ K\in L(\widetilde{X},Y)\,\forall x\in U\,\exists
r(x)\in\widetilde{X}\,:\ F(x)-F(x_{0})=Kr(x),$ (2.1)

where we assume $K\in L(\widetilde{X},Y)$ for some auxiliary space
$\widetilde{X}$. Analogously to Remark 1 one can show formal equivalence of
(2.1) for arbitrary observation operator $C$ for all-at-once (1.1) and reduced
(1.2) formulations, provided $\tfrac{\partial A}{\partial u}(q,u)$ is an
isomorphism for $(q,u)\in U$. But again the crucial difference in its
verifiability is made by the much wider choice of function spaces in the all-
at-once setting.

Condition (1.10) is sufficient for (2.1) with $\widetilde{X}=X$,
$K=F^{\prime}(x_{0})$,
$r(x)=\int_{0}^{1}R(x_{0}+\theta(x-x_{0}))(x-x_{0})\,d\theta$, provided $U$ is
convex. On the other hand, (2.1) implies

$\displaystyle\forall x,\tilde{x}\in U\,:\ F^{\prime}(x)(\tilde{x}-x)$
$\displaystyle=\lim_{\epsilon\to
0}\frac{1}{\epsilon}\Bigl{(}F(x+\epsilon(\tilde{x}-x))-F(x)\Bigr{)}$ (2.2)
$\displaystyle=K\lim_{\epsilon\to
0}\frac{1}{\epsilon}\Bigl{(}r(x+\epsilon(\tilde{x}-x))-r(x)\Bigr{)}=Kr^{\prime}(x)(\tilde{x}-x),$

where we have used continuity of $K$. Since $\tilde{x}$ and therewith the
direction $\tilde{x}-x$ is arbitrary this implies

$\forall x\in\mbox{int}(U)\,:\
F^{\prime}(x)=Kr^{\prime}(x)=F^{\prime}(x_{0})r^{\prime}(x_{0})^{-1}r^{\prime}(x),$

from which (1.10) follows provided $U$ is open, $r$ Gâteaux differentiable and
$r^{\prime}(x_{0})$ is boundedly invertible.

We will see in Section 4 that, e.g., in Example 1, condition (2.1) is
satisfied with easier to compute expressions as compared to (1.14), namely
$r(q)=(q-q_{0})\cdot\tfrac{S(q)}{S(q_{0})}$ for (1.2),
$r(q,u)=((q-q_{0})\cdot\tfrac{u}{u_{0}},\,u-u_{0})$ for (1.1) and likewise for
Example 2.

Besides being weaker than (1.10) by requiring less differentiability on the
forward operator $F$, condition (2.1) also allows for an immediate
computational approach by splitting the inverse problem into an ill-posed
linear and a well-posed nonlinear part as

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

see also [15, Remark 2.2].

Since establishing the range invariance condition (2.1) will sometimes require
extension of the parameter space and this may lead to a loss of unique
identifiability, we here 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 $x^{\dagger}$. The latter implies that $\mathcal{P}$ is proper;
typically, $\mathcal{P}$ will also be convex, although we do not explicitely
assume this in our analysis.

The problem to be solved can therefore be rewritten as a system

$\displaystyle K\hat{r}=y-F(x_{0})$ $\displaystyle\textup{ in }Y$ (2.3)
$\displaystyle r(x)=\hat{r}$ $\displaystyle\textup{ in }\widetilde{X}$
$\displaystyle\mathcal{P}(x)=0$ $\displaystyle\textup{ in }[0,\infty]$

for the unknowns $(\hat{r},x)\in\widetilde{X}\times U$, $U\subseteq X$.

We will consider four approaches for solving the system (2.3): A variational
and three Newton type methods, two of which will be described in Appendix C.

###  2.1 Variational reconstruction

We minimize a regularized combination of the misfit in the first two equations
in (2.3) and the penalty functional $\mathcal{P}:U(\subseteq X)\to[0,\infty]$

$\displaystyle(\hat{r}_{\alpha,\beta}^{\delta},x_{\alpha,\beta}^{\delta})\in\mbox{argmin}_{(\hat{r},x)\in\widetilde{X}\times
U}J_{\alpha,\beta}^{\delta}(\hat{r},x)$ (2.4) $\displaystyle\mbox{where
}J_{\alpha,\beta}^{\delta}(\hat{r},x):=\underbrace{\|K\hat{r}+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha\mathcal{R}(\hat{r})}_{=:J_{\alpha}(\hat{r})}+\underbrace{\beta\|r(x)-\hat{r}\|_{\widetilde{X}}^{b}+\mathcal{P}(x)}_{=:J_{\beta}(\hat{r},x)}$

with some $p,b\in[1,\infty)$ and some proper regularization functional
$\mathcal{R}:\widetilde{X}\to[0,\infty]$. In the particular case of exact
penalization, that is $b=1$ and $\beta$ sufficiently large (but finite) so
that $r(x)-\hat{r}=0$ is enforced, this reduces to Tikhonov regularization

$J_{\alpha}^{\delta\,\textup{Tikh}}(x):=\|F(x)-y^{\delta}\|_{Y}^{p}+\alpha\tilde{\mathcal{R}}(x)+\mathcal{P}(x)$
(2.5)

with $\tilde{\mathcal{R}}(x)=\mathcal{R}(r(x))$.

Existence of a (global) minimizer in (2.4) can be shown by the direct method
of calculus of variations, see, e.g., [24, Theorem 3.1] provided topologies
$\mathcal{T}$, $\tilde{\mathcal{T}}$, $\mathcal{T}_{Y}$ exists on $U$,
$\widetilde{X}$, $Y$ such that

######  Assumption 2.1.

  * •

sublevel sets of $J_{\alpha,\beta}^{\delta}$ are
$\tilde{\mathcal{T}}\times\mathcal{T}$ compact;

  * •

$K$ is $\tilde{\mathcal{T}}$-to-$\mathcal{T}_{Y}$ continuous and
$\|\cdot\|_{Y}$ is $\mathcal{T}_{Y}$ lower semicontinuous;

  * •

$r:U\to\widetilde{X}$ is $\mathcal{T}$ sequentially closed, that is,

$\forall(x_{n})_{n\in\mathbb{N}}\subseteq U\,\ \hat{r}\in\widetilde{X}:\
\Bigl{(}x_{n}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}x\mbox{
and
}r(x_{n})\stackrel{{\scriptstyle\widetilde{X}}}{{\longrightarrow}}\hat{r}\Bigr{)}\
\Rightarrow\ r\in U\mbox{ and }r(x)=\hat{r};$

  * •

$\mathcal{P}$ is $\mathcal{T}$ lower semicontinuous; $\mathcal{R}$ is
$\tilde{\mathcal{T}}$ lower semicontinuous;

These theoretical requirements as well as exact computation of a minimizer can
partially be avoided by just relying on existence of an infimum (which
trivially holds due to boundedness of the objective function from below by
zero) and considering inexact solution of the minimization problem according
to

$\forall(\hat{r},x)\in\widetilde{X}\times U\,:\
J_{\alpha,\beta}^{\delta}(\hat{r}_{\alpha,\beta,\eta}^{\delta},x_{\alpha,\beta,\eta}^{\delta})\leq
J_{\alpha,\beta}^{\delta}(\hat{r},x)+\eta$ (2.6)

with $\eta>0$ and $\eta=\eta(\delta)\to 0$ as $\delta\to 0$. Comparison with
$(r(x^{\dagger}),x^{\dagger})\in\widetilde{X}\times U$ and using (1.4) and
$\mathcal{P}(x^{\dagger})=0$ yields

$\displaystyle\|K\hat{r}_{\alpha,\beta,\eta}^{\delta}+F(x_{0})-y^{\delta}\|_{Y}^{p}+\alpha\mathcal{R}(\hat{r}_{\alpha,\beta,\eta}^{\delta})+\beta\|r(x)-\hat{r}_{\alpha,\beta,\eta}^{\delta}\|_{\widetilde{X}}^{b}+\mathcal{P}(x_{\alpha,\beta,\eta}^{\delta})$
$\displaystyle\hskip
199.16928pt\leq\delta^{p}+\alpha\mathcal{R}(r(x^{\dagger}))+\eta.$

Assuming that $\alpha=\alpha(\delta)$, $\beta=\beta(\delta)$,
$\eta=\eta(\delta)$ are chosen such that

$\frac{\delta^{p}}{\alpha}\to 0\,,\quad\frac{\eta}{\alpha}\to
0\,,\quad\frac{\delta^{p}}{\beta}\to 0\,,\quad\frac{\alpha}{\beta}\to
0\,,\quad\frac{\eta}{\beta}\to 0\,,\quad\alpha\to 0\,,\quad\eta\to
0\quad\mbox{ as }\delta\to 0,$ (2.7)

we therefore obtain for
$\hat{r}^{\delta}:=\hat{r}_{\alpha(\delta),\beta(\delta),\eta(\delta)}^{\delta}$,
$x^{\delta}:=x_{\alpha(\delta),\beta(\delta),\eta(\delta)}^{\delta}$,

$\displaystyle(a)$ $\displaystyle\limsup_{\delta\to
0}\mathcal{R}(\hat{r}^{\delta})\leq\mathcal{R}(r(x^{\dagger}))$
$\displaystyle(b)$ $\displaystyle\limsup_{\delta\to
0}\|r(x^{\delta})-\hat{r}^{\delta}\|_{\widetilde{X}}\ =0$ $\displaystyle(c)$
$\displaystyle\limsup_{\delta\to
0}\|K\hat{r}^{\delta}+F(x_{0})-y^{\delta}\|_{Y}\ =0$ $\displaystyle(d)$
$\displaystyle\limsup_{\delta\to 0}\mathcal{P}(x^{\delta})\ =0$

This yields convergence provided topologies $\mathcal{T}$,
$\tilde{\mathcal{T}}$, $\mathcal{T}_{Y}$ exists on $U$, $\widetilde{X}$, $Y$
such that for arbitrary sequences
$(\hat{r}_{n})_{n\in\mathbb{N}}\subseteq\widetilde{X}$

######  Assumption 2.2.

  * (i)

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

  * (ii)

$\hat{r}_{n}\stackrel{{\scriptstyle\widetilde{X}}}{{\longrightarrow}}0\
\Rightarrow\ \exists(\hat{r}_{n_{k}})_{k\in\mathbb{N}}:\
\hat{r}_{n_{k}}\stackrel{{\scriptstyle\tilde{\mathcal{T}}}}{{\longrightarrow}}0$;

  * (iii)

$K$ is $\tilde{\mathcal{T}}$-to-$\mathcal{T}_{Y}$ continuous and $\|\cdot\|$
is $\mathcal{T}_{Y}$ lower semicontinuous;

  * (iv)

$r(x_{n})\stackrel{{\scriptstyle\tilde{\mathcal{T}}}}{{\longrightarrow}}\hat{r}\
\Rightarrow\ \exists(x_{n_{k}})_{k\in\mathbb{N}}\subseteq U,\,x\in U:\
(x_{n_{k}}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}x$ and
$r(x)=\hat{r})$;

  * (v)

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

######  Theorem 1.

Under condition (2.1) and Assumption 2.2, the approximations $x^{\delta}$
defined by (2.6) with (2.7) converge $\mathcal{T}$ subseqentially to a
solution of (1.3) with $\mathcal{P}(x)=0$, (or equivalently, of (2.3)), that
is, every subsequence of $(\hat{r}^{\delta},x^{\delta})_{\delta>0}$ has a
$\tilde{\mathcal{T}}\times\mathcal{T}$ convergent subsequence and the limit
$(\hat{r},x)$ of every $\tilde{\mathcal{T}}\times\mathcal{T}$ convergent
subsequence solves (2.3). If the $\mathcal{R}$ minimizing solution
$(\hat{r}^{\dagger},x^{\dagger})$ of (2.3)222i.e.,
$(\hat{r}^{\dagger},x^{\dagger})$ solving (2.3) 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 $r$ is injective, then
$\hat{r}^{\delta}\stackrel{{\scriptstyle\tilde{\mathcal{T}}}}{{\longrightarrow}}r(x^{\dagger})$
and
$x^{\delta}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}x^{\dagger}$.

###### Proof.

(a) and (i) imply that $\hat{r}^{\delta}$ has a $\tilde{\mathcal{T}}$
convergent subsequence. For any such $\tilde{\mathcal{T}}$ convergent
subsequence
$\hat{r}^{\delta_{n}}\stackrel{{\scriptstyle\tilde{\mathcal{T}}}}{{\longrightarrow}}\hat{r}$,
by (b), (ii) we also have
$r(x^{\delta_{n}})\stackrel{{\scriptstyle\tilde{\mathcal{T}}}}{{\longrightarrow}}\hat{r}$,
which by (iv) implies existence of a sub-subsequence
$(x^{\delta_{n_{k}}})_{k\in\mathbb{N}}\subseteq U$ and an element $x\in U$
such that
$x^{\delta_{n_{k}}}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}x$
and $r(x)=\hat{r}$. From (c), (iii) we conclude that $Kr(x)=y-F(x)$ and from
(d), (v) $\mathcal{P}(x)=0$. In case of uniqueness, a subsequence-subsequence
argument together with injectivity of $r$ implies convergence of the whole
family $(\hat{r}^{\delta},x^{\delta})_{\delta\in(0,\bar{\delta})}$ as
$\delta\to 0$. ∎

######  Remark 2.

Computing a minimizer to (2.4) with $b>1$ is easier than finding a minimizer
to the classical Tikhonov functional (2.5), since in
$J_{\alpha,\beta}^{\delta}$ the first part $J_{\alpha}^{\delta}$ is convex due
to linearity of $K$ and so is usually the function $\mathcal{P}$. The
nonlinear term $\|r(x)-\hat{r}\|_{\widetilde{X}}^{b}$ is convex with respect
to $\hat{r}$ and also with respect to $x$ in a neighborhood of
$(\hat{r},x)=(r(x^{\dagger}),x^{\dagger})$,333uniformly so in case of a
uniformly convex norm $\|\cdot\|_{\widetilde{X}}$ and $b>1$ whereas the same
holds for $\|F(x)-y^{\delta}\|_{Y}^{p}$ only under some additional restriction
on the nonlinearity of the forward operator $F$ – the already mentioned weak
nonlinearity condition from [14], that is related to the tangential cone
condition (1.5). Thus, numerical approximation of a minimizer to (2.4) can,
e.g, be achieved by a Gauss-Seidel (also called alternating minimization or
coordinate descent) method based on alternatingly minimizing
$J_{\alpha}(\hat{r})$ with respect to $\hat{r}$ and then, for fixed $\hat{r}$,
minimizing $J_{\beta}(\hat{r},x)$ with respect to $x$, see, e.g., [5, 8, 18,
47, 44].

###  2.2 Iterative reconstruction

We exemplarily describe and analyze a simple regularized Newton method here
and provide an analysis of two further (somewhat more sophisticated) Newton
type methods in Appendix C.

####  2.2.1 A frozen Newton method

Based on the operator $K$ in (2.3) with $r(x)\approx x-x_{0}$ and (2.1) with
$\widetilde{X}=X$ we can define a frozen Newton iteration by

$x_{n+1}^{\delta}\in\mbox{argmin}_{x\in
U}\|K(x-x_{n}^{\delta})+F(x_{n}^{\delta})-y^{\delta}\|_{Y}^{p}+\alpha_{n}\mathcal{R}(x)+\mathcal{P}(x).$
(2.8)

We specify the setting to the case of Hilbert spaces $X$ and $Y$, assume
$p=2$, $\mathcal{R}(x)=\|x-x_{0}\|_{X}^{2}$ and $\mathcal{P}$ to be defined by
a linear operator $P\in L(X,Z)$ for another Hilbert space $Z$,
$\mathcal{P}(x)=\|Px\|_{Z}^{2}$. This allows to explicitely write the
minimizer as

$x_{n+1}^{\delta}=x_{n}^{\delta}+(K^{\star}K+P^{\star}P+\alpha_{n}\textup{id})^{-1}\Bigl{(}K^{\star}(y^{\delta}-F(x_{n}^{\delta}))-P^{\star}Px_{n}^{\delta}+\alpha_{n}(x_{0}-x_{n}^{\delta})\Bigr{)},$
(2.9)

where ⋆ denotes the Hilbert space adjoint. Using (2.1), which implies
$F(x_{n}^{\delta})=F(x_{0})+Kr(x_{n})=y+K(r(x_{n})-r(x^{\dagger}))$, and the
fact that $P^{\star}Px^{\dagger}=0$, we can express the recursion for the
error as

$\displaystyle x_{n+1}^{\delta}-x^{\dagger}=$
$\displaystyle(K^{\star}K+P^{\star}P+\alpha_{n}\textup{id})^{-1}$
$\displaystyle\Bigl{(}K^{\star}(y^{\delta}-y)+K^{\star}K\bigl{(}(r(x^{\dagger})-r(x_{n}^{\delta}))-(x^{\dagger}-x_{n}^{\delta})\bigr{)}+\alpha_{n}(x_{0}-x^{\dagger})\Bigr{)}.$

Assuming $r$ to be close to the identity (corresponding to the assumption
$\|R(x)-\textup{id}\|\leq c<1$ or $\|R(x)-\textup{id}\|\leq
C\|x-x^{\dagger}\|$ usually made in the context of (1.10), cf., e.g., [15, 32,
10])

$\exists c\in(0,1)\,\forall x\in U\,:\
\|(r(x^{\dagger})-r(x))-(x^{\dagger}-x)\|_{X}\leq c\|x^{\dagger}-x\|_{X}$
(2.10)

and using spectral calculus for the selfadjoint nonnegative linear operator
$A:=K^{\star}K+P^{\star}P$ we obtain

$\|x_{n+1}^{\delta}-x^{\dagger}\|_{X}\leq\frac{\delta}{\sqrt{\alpha_{n}}}+c\|x_{n}^{\delta}-x^{\dagger}\|_{X}+a_{n}(P)$
(2.11)

with

$a_{n}(P)=\alpha_{n}\|(K^{\star}K+P^{\star}P+\alpha_{n}\textup{id})^{-1}(x_{0}-x^{\dagger})\|_{X}\to
0\mbox{ as }n\to\infty$ (2.12)

provided
$x_{0}-x^{\dagger}\in(\textup{nsp}(K)\cap\textup{nsp}(P))^{\bot}\subseteq\textup{nsp}(A)^{\bot}$,
which follows from the implication

$(K^{\star}K+P^{\star}P)v=0\ \Rightarrow\ 0=\langle
v,(K^{\star}K+P^{\star}P)v\rangle_{X}=\|Kv\|_{Y}^{2}+\|Pv\|_{Z}^{2}.$

In (2.11) we have used the following estimates, whose proof can be found in
Appendix B.

######  Lemma 2.1.

For $K\in L(X,Y)$, $P\in L(X,Z)$ with $X,Y,Z$ Hilbert spaces satisfying

$\textup{(a) }\textup{nsp}(K)^{\bot}\subseteq\textup{nsp}(P)\textup{ or \ (b)
}P^{\star}P(K^{\star}K)^{1/2}=(K^{\star}K)^{1/2}P^{\star}P$ (2.13)

and any $\alpha>0$, the estimates

$\displaystyle\|(K^{\star}K+P^{\star}P+\alpha\textup{id})^{-1}K^{\star}K\|\leq
C$ (2.14)
$\displaystyle\|(K^{\star}K+P^{\star}P+\alpha\textup{id})^{-1}K^{\star}\|=\|K(K^{\star}K+P^{\star}P+\alpha\textup{id})^{-1}\|\leq\sqrt{\frac{C}{\alpha}},$

hold with $C=1$ in case (a) and $C=2$ in case (b).

######  Theorem 2.

Let $x_{0}\in U:=\mathcal{B}_{\rho}(x^{\dagger})$ for some $\rho>0$
sufficiently small and let
$x_{0}-x^{\dagger}\in\bigl{(}\textup{nsp}(K)\cap\textup{nsp}(P)\bigr{)}^{\bot}$
and (2.13) hold. Assume that $F$ satisfies (2.1) with $r$ Gâteaux
differentiable, satisfying (2.10). Let the stopping index
$n_{*}=n_{*}(\delta)$ be chosen such that

$n_{*}(\delta)\to
0,\quad\delta\sum_{j=0}^{n_{*}(\delta)-1}c^{j}\alpha_{n_{*}(\delta)-j-1}^{-1/2}\to
0\qquad\textup{ as }\delta\to 0$ (2.15)

with $c$ as in (2.10).

Then the iterates $(x_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are
well-defined by (2.9), remain in $\mathcal{B}_{\rho}(x^{\dagger})$ and
converge in $X$, $\|x_{n_{*}(\delta)}^{\delta}-x^{\dagger}\|_{X}\to 0$ as
$\delta\to 0$. In the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we
have $\|x_{n}-x^{\dagger}\|_{X}\to 0$ as $n\to\infty$.

###### Proof.

From estimate (2.11) we deduce the recursion

$\|x_{n+1}^{\delta}-x^{\dagger}\|\leq\delta\sum_{j=0}^{n}c^{j}\alpha_{n-j}^{-1/2}+c^{n+1}\|x_{0}^{\delta}-x^{\dagger}\|_{X}+\sum_{j=0}^{n}c^{j}a_{n-j},$

where by (2.12), $\sum_{j=0}^{n}c^{j}a_{n-j}\to 0$ as $n\to\infty$. The choice
of $n_{*}(\delta)$ according to (2.15) thus yields the assertions. ∎

##  3 A class of examples

We now discuss a framework in which range invariance (2.1) can be verified. It
will allow for considering multiple coefficients
$q=(q^{(0)},\ldots,q^{(I)})\in Q=\prod_{i=0}^{I}Q^{(i)}$,
$I\in\mathbb{N}_{0}$. This setting turns out to even enable verification of
(2.1) in the problem of identifying a diffusion coefficient in higher space
dimensions from boundary data (for which verification of range invariance and
also of the tangetial cone condition so far only succeeded in one space
dimension) as long as it is to be identified together with another
coefficient, see Example 6 below.

On the other hand, we also consider multiple equations in the sense of several
experiments carried out, which lead to states $u=(u^{j})_{j\in J}$ with
corresponding observations $C^{j}u^{j}$.

We thus study the following general model equation

$\displaystyle(D^{j}u^{j}+\sum_{i=0}^{I}B^{i,j}(u^{j})q^{(i)}-f^{j})_{j\in
J}=0$ (3.1) $\displaystyle\textup{where
}q=(q^{(i)})_{i\in\\{0,\ldots,I\\}},\quad u:=(u^{j})_{j\in J},$

where the operators $D=\textup{diag}((D^{j})_{j\in J})\in L(V_{J},W^{*}_{J})$,
$V_{J}\subseteq L^{2}_{\mu}(J,V)$, $W^{*}_{J}\subseteq L^{2}_{\mu}(J,W^{*})$,
$B^{i,j}(u^{j})\in L(Q^{(i)},W^{*})$, for any $u^{j}\in V$, $f^{j}\in W^{*}$
are given inhomogeneities, $\mu$ is a measure on $J$, and $L^{2}_{\mu}(J,V)$,
$L^{2}_{\mu}(J,W^{*})$ are the corresponding Bochner-Sobolev spaces. In here,
$D$ will typically be a differential operator, possibly equipped with
homogeneous boundary and/or initial conditions, and $B^{(i)}(u^{j})q^{(i)}$
are terms in which the unknown parameters $q^{(i)}$ appear as coefficients or
unknown source terms (the latter corresponding to $B^{(i)}(u^{j})$ being
constant in $u^{j}$). 444In principle, also the spaces $V$ and $W^{*}$ could
depend on $j$ but we do not consider this in order not overly complicate
notation.

Note that we restrict ourselves to model operators that are (affinely) linear
with respect to the coeffcients $q$, but not necessarily linear with respect
to the state $u$. We will see in Section 4 below that this comprises the
examples from the introduction as well as several others.

The observations are defined by the equations

$C^{j}u^{j}=y^{j},\quad C^{j}\in L(V,Y^{j}),\quad j\in J,$ (3.2)

with some arbitrary operators $C^{j}$; other than linearity and boundedness,
we will not impose further conditions on them in the following.

Clearly, more general choices also of the index set $\\{0,\ldots,I\\}$ for the
parameters are possible in principle. While they might be explored for
particular applications in future work, in this paper we stay with the setting
of finitely many parameters (each of which can be an infinite dimensional
quantity).

In order to achieve range invariance, we will – similarly to Example 2 –
introduce an artificial dependence of one of the parameters on $j$, that is,
for some fixed index $i_{0}\in\\{0,\ldots,I\\}$ (to be chosen such that the
corresponding operator $(B^{i_{0},j}(u_{0}^{j}))_{j\in J}$ is an isomorphism)
see (3.3)) below, we consider $q^{(i_{0})}=(q^{i_{0},j})_{j\in J}$. To recover
the original independence of $q^{(i_{0})}=(q^{i_{0}j})_{j\in J}$ on $j$ we use
the penalty term $\mathcal{P}(q):=\|Pq\|_{L^{2}_{\mu}(J;Q^{(i_{0})})}$, where
$P:L^{2}(J;Q^{(i_{0})})\to L^{2}(J;Q^{(i_{0})})$, $q\mapsto
q-\frac{1}{\int_{J}w\,d\mu}\int_{J}w\,q\,d\mu$ for some weight function $w>0$,
$w\in L^{1}_{\mu}(J)\cap L^{2}_{\mu}(J)$ (so that for any $v\in
L^{2}_{\mu}(J)$, $\frac{1}{\int_{J}w\,d\mu}\int_{J}w\,v\,d\mu$ is the
projection of $v$ on the constants in the weighted $L^{2}$ space with weight
$w$).

Correspondingly, one of the parameter spaces is lifted to be the Bochner space
$L^{2}_{\mu}(J;Q^{(i_{0})})$. To simplify notation, we continue using the same
symbol for the original parameter $q^{(i_{0})}\in Q^{(i_{0})}$ and its $j$
dependent extension $q^{(i_{0})}\in
L^{2}_{\mu}(J;Q^{(i_{0})})=:Q^{(i_{0})}_{J}$.

This covers some interesting cases such as time or frequency dependent PDEs
with $J=(0,T)$, $J=(\underline{\omega},\overline{\omega})$ or Neumann-to-
Dirichlet observations for elliptic PDEs with $J=\mathbb{N}$, see Examples 5,
6 below.

In order to verify the range invariance condition (2.1), throughout this
section we will make the assumption that the extended version of one of the
operators $B^{(i_{0})}(u_{0})$ is an isomorphism between $W^{*}_{J}$ and some
auxiliary space
$\widetilde{Q}^{(i_{0})}_{J}:=L^{2}_{\mu}(J;\widetilde{Q}^{(i_{0})})$:

$\displaystyle\exists i_{0}\in\\{0,\ldots,I\\}\,:$ $\displaystyle\
B_{0}:=B^{(i_{0})}(u_{0}):=(B^{i_{0},j}(u_{0}^{j}))_{j\in J}\textup{ bijective
and }$ (3.3) $\displaystyle B^{(i_{0})}(u_{0})\in
L(\widetilde{Q}^{(i_{0})}_{J},W^{*}_{J}),\quad B^{(i_{0})}(u_{0})^{-1}\in
L(W^{*}_{J},\widetilde{Q}^{(i_{0})}_{J});$

here $u_{0}:=(S(q_{0}))$ is the solution to (3.5) in the reduced case (1.2)
and arbitrary in the all-at-once case (1.1). In particular, (3.3) can always
be achieved by defining
$\widetilde{Q}^{(i_{0})}_{J}:=(B^{(i_{0})}(u_{0}))^{-1}(W^{*}_{J})$ as the
preimage of $W^{*}_{J}$ under the operator $B^{(i_{0})}(u_{0})$ with the norm

$\|\hat{r}^{(i_{0})}\|_{\widetilde{Q}^{(i_{0})}_{J}}:=\|B^{(i_{0})}(u_{0})\hat{r}^{(i_{0})}\|_{W^{*}_{J}}$
(3.4)

Indeed, this possibly weaker norm still enables boundedness of
$K=\mathbf{F}(q_{0})$ or $K=\mathbb{F}(q_{0},u_{0})$, respectively. By an
appropriate numbering of the unknowns $q^{(i)}$, without loss of generality we
assume $i_{0}=0$.

With the extended parameter $q^{(0)}=q^{(i_{0})}$, equation (3.1) becomes

$\displaystyle
A(q,u):=(D^{j}u^{j}+B^{0,j}(u^{j})q^{0,j}+\sum_{i=1}^{I}B^{i,j}(u^{j})q^{(i)}-f^{j}))_{j\in
J}$ (3.5)
$\displaystyle\qquad\quad=Du+B^{(0)}(u)q^{(0)}+\sum_{i=1}^{I}B^{(i)}(u)q^{(i)}-f=Du+B(u)q=0$

where

$\displaystyle B^{(0)}(u)q^{(0)}:=(B^{0,j}(u^{j})q^{0,j})_{j\in J},\quad
B^{(i)}(u)q^{(i)}:=(B^{i,j}(u^{j})q^{(i)})_{j\in J},\ i\in\\{1,\ldots,I\\}$
$\displaystyle q=(q^{(i)})_{i\in\\{0,\ldots,I\\}},\quad u:=(u^{j})_{j\in
J},\quad f:=(f^{j})_{j\in J}.$

The single experiment case is still covered by setting $J=\\{0\\}$,
$Q^{(i)}_{J}=Q^{(i)}$ and the single coefficient case by setting $I=0$.

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

We slightly extend Example 2 to the reconstruction of multiple potentials
$q^{(0)},\ldots q^{(I)}$ in

$D_{t}^{M}u-\Delta_{N}u+\sum_{i=0}^{I}q^{(i)}\cdot\Phi^{(i)}(u)=0,$

where $\Phi^{(1)}$,…, $\Phi^{(I)}$ are smooth real functions. This is clearly
covered by the framework of this section with

$\displaystyle j=t,\quad J=(0,T),\quad u^{j}=u(t)\ \in L^{2}(\Omega),$
$\displaystyle(Du)^{j}=D_{t}^{M}u(t)-\Delta_{N}u(t),\quad
B^{(i)}(u^{j})q^{(i)}=q^{(i)}\cdot\Phi^{(i)}(u)$ $\displaystyle J=(0,T),\quad
Q^{(0)}_{J}=L^{2}(0,T;L^{2}(\Omega)),\quad Q^{(i)}=L^{2}(\Omega),\
i\in\\{1,\ldots,I\\}$

provided there exists $c>0$ such that $|\Phi^{(0)}(u_{0})(x,t)|\geq c>0$ for
all $(x,t)\in\Omega\times(0,T)$ so that the multiplication operator
$B_{0}=B^{(0)}(u_{0}):Q^{(0)}_{J}\to W_{J}^{*}$,
$q^{(0)}\mapsto\Phi(u_{0})\cdot q^{(0)}$ is an isomorphism. This works out,
e.g., for the strongly damped wave equation example

$u_{tt}-\Delta_{N}u-b\Delta_{N}u_{t}+\sum_{i=0}^{I}q^{(i)}\cdot\Phi^{(i)}(u)=0$

with $b>0$ in the function space setting (cf. Appendix A)

$V=L^{\infty}(0,T;H^{2}(\Omega))\cap W^{1,\infty}(0,T;H^{1}(\Omega))\cap
H_{\diamondsuit}^{2}(0,T;L^{2}(\Omega)),\quad W^{*}=L^{2}(0,T;L^{2}(\Omega)).$

Identification of multiple coefficients that are of much more different
nature, namely a diffusion and an absorption coefficient, will be considered
in Example 6; see also [40] for another such two-coefficient identification
problem.

######  Remark 3.

Note that verification of condition (a) in (2.13) and
$x_{0}-x^{\dagger}\in(\textup{nsp}(K)\cap\textup{nsp}(P))^{\bot}$ in Theorem 2
(or the analogous conditions for the other Newton type methods) is related to
uniqueness proofs for the particular PDE and measurement setting and therefore
to be established on a case-by-case basis, see Section 4 below. We will
therefore in the corollaries of the current section just re-state (2.13) etc.
as assumptions.

In fact, we will use $K=\mathbf{F}(q_{0})$ in the reduced and
$K=\mathbb{F}(q_{0},u_{0})$ in the all-at-once setting, which implies
$r^{\prime}(q_{0})=\textup{Id}$ and in the examples of Section 4 show that

$\textup{nsp}(F^{\prime}(q_{0}))\cap\textup{nsp}(P)=\\{0\\}.$ (3.6)

With the canonical choice $u_{0}=S(q_{0})$, this also covers the all-at-once
setting due to the identity

$\textup{nsp}(\mathbb{F}(q_{0},S(q_{0})))=\\{(\underline{dq},-(D+[B^{\prime}(S(q_{0})\cdot]q_{0})^{-1}B(S(q_{0}))\underline{dq}\,:\,\underline{dq}\in\textup{nsp}(\mathbf{F}(q_{0}))\\}.$
(3.7)

###  3.1 Reduced formulation

We use (1.2) in the spaces

$X=Q_{J}=L^{2}_{\mu}(J,Q^{(0)})\times\prod_{i=1}^{I}Q^{(i)}=Q^{(0)}_{J}\times\prod_{i=1}^{I}Q^{(i)},\quad
Y=Y_{J}\subseteq\prod_{j\in J}Y^{j}\,.$

With

$\displaystyle q_{0}\in\mathcal{D}(\mathbf{F})\subseteq$ $\displaystyle\\{q\in
Q_{J}\,:\,u:=S(q)\in V_{J}\textup{ uniquely determined by }(D+B(\cdot)q)=f$
(3.8) $B$ Gâteaux differentiable at $S(q)$ and $\displaystyle\hskip
56.9055ptD+[B^{\prime}(S(q))\cdot]q:V_{J}\to W^{*}_{J}\mbox{ an isomorphism
}\\},$

we have

$\mathbf{F}(q)=CS(q)\mbox{ where }S(q)=(D+B(\cdot)q)^{-1}(f),$ (3.9)

and by implicit differentiation of the identity $D+B(S(q))q=f$,

$\mathbf{F}^{\prime}(q)=CS^{\prime}(q)\mbox{ where
}S^{\prime}(q)\underline{dq}=-(D+[B^{\prime}(S(q))\cdot]q)^{-1}B(S(q))\underline{dq}.$
(3.10)

Due to (3.8) and our assumption (3.3) of $B_{0}=B^{(i_{0})}(S(q_{0}))$ being
an isomorphism, we have

$\mathbf{F}^{\prime}(q_{0})\in L(\widetilde{Q}_{J},Y_{J})\ \textup{ for
}\widetilde{Q}_{J}=L^{2}_{\mu}(J,\widetilde{Q}^{(0)})\times\prod_{i=1}^{I}Q^{(i)}=\widetilde{Q}^{(0)}_{J}\times\prod_{i=1}^{I}Q^{(i)}.$

A sufficient condition for the range invariance (2.1) with
$K=\mathbf{F}^{\prime}(q_{0})$ to hold for an arbitrary observation operator
$C$ is

$\forall q\in Q_{J}\,\exists r(q)=(r^{0,j}(q))_{j\in
J},r^{(i)}(q))_{i\in\\{1,\ldots,I\\}})\in\widetilde{Q}_{J}\,:\
S(q)-S(q_{0})=S^{\prime}(q_{0})r(q).$ (3.11)

Due to the identities

$\displaystyle DS(q)+B(S(q))q=f$ (3.12) $\displaystyle
DS(\tilde{q})+B(S(\tilde{q}))\tilde{q}=f$ (3.13) $\displaystyle
D(S(q)-S(\tilde{q}))+[B(S(q))-B(S(\tilde{q}))]\tilde{q}+B(S(q))(q-\tilde{q})=0$
(3.14) $\displaystyle
D(S^{\prime}(\tilde{q})\underline{dq})+[B^{\prime}(S(\tilde{q}))S^{\prime}(\tilde{q})\underline{dq}]\tilde{q}+B(S(\tilde{q}))\underline{dq}=0,$
(3.15)

that imply

$\displaystyle Dv+B^{\prime}(S(\tilde{q}))v\tilde{q}$
$\displaystyle+[B(S(q))-B(S(\tilde{q}))-B^{\prime}(S(\tilde{q}))(S(q)-S(\tilde{q}))]\tilde{q}+B(S(q))(q-\tilde{q})-B(S(\tilde{q}))\underline{dq}=0$

for $v=S(q)-S(\tilde{q})-S^{\prime}(\tilde{q})\underline{dq}$ and any
$q,\tilde{q}\in\mathcal{D}(\mathbf{F})$, $\underline{dq}\in Q$, and the
assumed bijectivity of the operator
$D+[B^{\prime}(S(\tilde{q}))\cdot]\tilde{q}$, condition (3.11) is equivalent
to

$\displaystyle[B(S(q))-B(S(q_{0}))-B^{\prime}(S(q_{0}))(S(q)-S(q_{0}))]q_{0}$
$\displaystyle+[B(S(q))-B(S(q_{0}))](q-q_{0})-B(S(q_{0}))(r(q)-(q-q_{0}))=0\,.$

Under assumption (3.3), this can be achieved by setting
$r(q)=(r^{(0)}(q),\ldots,r^{(I)}(q))$ with

$\displaystyle r^{(0)}(q):=$ $\displaystyle
q^{(0)}-q^{(0)}_{0}+r^{(0)}_{\textup{diff}}(q,S(q);q_{0},S(q_{0}))$ (3.16)
$\displaystyle r^{(i)}(q):=$ $\displaystyle
r^{(i)}(q^{(i)})=q^{(i)}-q^{(i)}_{0},\quad i\in\\{1,\ldots,I\\}.$

with

$\displaystyle
r^{(0)}_{\textup{diff}}(q,u;q_{0},u_{0}):=B_{0}^{-1}\Bigl{(}[B(u)-B(u_{0})]q-[B^{\prime}(u_{0})(u-u_{0})]q_{0}\Bigr{)}$
(3.17)
$\displaystyle\quad=B_{0}^{-1}\Bigl{(}[B(u)-B(u_{0})-B^{\prime}(u_{0})(u-u_{0})]q_{0}+[B(u)-B(u_{0})](q-q_{0})\Bigr{)}.$

We now investigate the further convergence conditions for the methods in
Section 2.

A verification of Assumption 2.2 for the variational approach can be done in
the following setting

$\left.\begin{aligned} &Q^{(0)}_{J},Q^{(i)}\textup{ are Banach spaces with
}\tilde{\mathcal{T}}=\mathcal{T}:=\prod_{i=0}^{I}\mathcal{T}^{(i)}\textup{ the
norm topology on $Q_{J}$;}\\\ &Y_{J}\textup{ is reflexive or dual of a
separable space, }\mathcal{T}_{Y}\textup{ the weak(*) topology;}\\\ &q_{0}\in
U:=\mathcal{B}_{\sigma}(q^{\dagger})\subseteq\mathcal{D}(\mathbf{F})\textup{
with $\sigma>0$ sufficiently small;}\\\ &\textup{Sublevel sets of
$\mathcal{R}$ are compact in $Q$ (wrt the norm topology in $Q$);}\\\
&\textup{The functional $\mathcal{P}$ is $\mathcal{T}$ lower
semicontinuous;}\\\ &\textup{The operator $B$ is continuously differentiable
on }S(U)=\\{S(q)\,:\,q\in U\\}\\\ &\hskip 28.45274pt\textup{with uniformly
bounded dervative $B^{\prime}$ on $S(U)$; }\\\
&(D+[B^{\prime}(S(q))\cdot]q)^{-1}B(S(q))\in
L(\widetilde{Q},V),\end{aligned}\right\\}$ (3.18)

and based on Theorem 1 yields a convergence result for variational
regularization of (3.5) in a reduced setting.

######  Corollary 3.1 (variational, reduced).

Under conditions (3.2), (3.3), (3.18), $D\in L(V_{J},W^{*}_{J})$,
$\widetilde{Q}_{J}=Q_{J}$, the approximations $x^{\delta}=q^{\delta}$ defined
by (2.6), (3.9) with (2.7) converge norm subseqentially to a solution of
(1.3), (3.9) with $\mathcal{P}(q)=0$, If the $\mathcal{R}$ minimizing solution
$(\hat{r}^{\dagger},x^{\dagger})=(\hat{r}^{\dagger},q^{\dagger})$ of (1.3),
(3.9) with $\mathcal{P}(q)=0$ is unique, then
$\hat{r}^{\delta}\stackrel{{\scriptstyle\tilde{\mathcal{T}}}}{{\longrightarrow}}r(q^{\dagger})$
and $\|q^{\delta}-q^{\dagger}\|_{Q_{J}}\to 0$ as $\delta\to 0$.

###### Proof.

Items (i), (ii), (iii), (v) in Assumption 2.2 directly follow from (3.2),
(3.3), (3.18),
$q_{0}\in\mathcal{B}_{\sigma}(q^{\dagger})\subseteq\mathcal{D}(\mathbf{F})$,
and $K=CS^{\prime}(q_{0})$ with $S^{\prime}(q_{0})$ as in (3.10).

In order to verify item (iv), we assume that

$\hat{r}^{(i)}_{n}:=r^{(i)}(q_{n})\stackrel{{\scriptstyle\tilde{\mathcal{T}}^{(i)}}}{{\longrightarrow}}\hat{r}^{(i)},\
i\in\\{0,\ldots,I\\}$ (3.19)

and have to show existence of a subsequence $(q_{n_{k}})_{k\in\mathbb{N}}$
such that
$q^{(i)}_{n_{k}}\stackrel{{\scriptstyle\mathcal{T}^{(i)}}}{{\longrightarrow}}q^{(i)}_{*}$
and $r^{(i)}(q_{*})=\hat{r}^{(i)})$. This is trivial for
$i\in\\{1,\ldots,I\\}$ by the definition (3.16) of $r$ (here it would even
suffice to use the weak topologies for both $\hat{r}$ and $q$) which implies

$q^{(i)}_{n}=r^{(i)}(q_{n})+q^{(i)}_{0}\stackrel{{\scriptstyle\mathcal{T}^{(i)}}}{{\longrightarrow}}\hat{r}^{(i)}+q^{(i)}_{0}=:q^{(i)}_{*}\quad
i\in\\{1,\ldots,I\\}.$ (3.20)

For $i=0$, we define $q^{(0)}_{n}$ implicitly by

$\displaystyle G(q^{(0)}_{n})=$ $\displaystyle 0\ \textup{ with
}G=r^{(0)},\textup{ that is, }$ (3.21) $\displaystyle G(q)=$ $\displaystyle
q^{(0)}-q^{(0)}_{0}+r^{(0)}_{\textup{diff}}(q,S(q);q_{0},S(q_{0}))$
$\displaystyle=$ $\displaystyle
q^{(0)}-q^{(0)}_{0}+B_{0}^{-1}\Bigl{(}[B(S(q))-B(S(q_{0}))-B^{\prime}(S(q_{0}))(S(q)-S(q_{0}))]q_{0}$
$\displaystyle\qquad\qquad\qquad\qquad+[B(S(q))-B(S(q_{0}))](q-q_{0})\Bigr{)}.$

To resolve (3.21) with respect to $q_{n}^{(0)}$ and show its convergence as a
consequence of (3.19) and (3.20), we use the Implicit Function Theorem. For
this purpose note that by

$\displaystyle\frac{\partial G}{\partial q^{(0)}}(q)\underline{dq}^{(0)}=$
$\displaystyle\underline{dq}^{(0)}-B_{0}^{-1}\Bigl{\\{}\bigl{[}\bigl{(}B^{\prime}(S(q))-B^{\prime}(S(q_{0}))\bigr{)}\frac{\partial
S}{\partial q^{(0)}}(q)\underline{dq}^{(0)}\bigr{]}q_{0}$ (3.22)
$\displaystyle+[B^{\prime}(S(q))\frac{\partial S}{\partial
q^{(0)}}(q)\underline{dq}^{(0)}](q-q_{0})+[B^{(0)}(S(q))-B^{(0)}(S(q_{0}))]\underline{dq}^{(0)}\Bigr{\\}}$

where as in (3.10)

$\frac{\partial S}{\partial
q^{(0)}}(q)\underline{dq}^{(0)}=-\Bigl{(}D+[B^{\prime}(S(q))\cdot]q\Bigr{)}^{-1}B^{(0)}(S(q))\underline{dq}^{(0)}.$
(3.23)

Thus $G(q_{0})=0$ and $\frac{\partial G}{\partial
q^{(0)}}(q_{0})=\textup{Id}$. Application of the Implicit Function Theorem
yields existence of neighborhoods $U_{0}$ of $q_{0}^{(0)}$ and $U^{\prime}$ of
$q_{0}^{(1,\ldots,I)}:=(q_{0}^{(1)},\ldots,q_{0}^{(I)})$ and a continuously
differentiable function $\phi:U^{\prime}\to U_{0}$ such that

$\displaystyle\forall q^{(1,\ldots,I)}:=(q^{(1)},\ldots,q^{(I)})\in
U^{\prime}\,:\ G(\phi(q^{(1,\ldots,I)}),q^{(1,\ldots,I)})=0$
$\displaystyle\forall q^{(0)}\in U_{0}\,,\ q^{(1,\ldots,I)}\in U^{\prime}\,:\
G(q^{(0)},q^{(1,\ldots,I)})=0\ \Rightarrow\ q^{(0)}=\phi(q^{(1,\ldots,I)}).$

We can therefore define $q^{(0)}_{n}:=\phi(q^{(1,\ldots,I)}_{n})$,
$q^{(0)}_{*}:=\phi(q^{(1,\ldots,I)}_{*})$. Continuity of the function $\phi$
implies the desired convergence
$q^{(0)}_{n}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}q^{(0)}_{*}$.
By
$G(q_{n})=\hat{r}^{(0)}(q_{n})\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}\hat{r}^{(0)}$
and
$G(q_{n})=G(\phi(q^{(1,\ldots,I)}_{n}),q^{(1,\ldots,I)}_{n})\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}G(\phi(q^{(1,\ldots,I)}_{*}),q^{(1,\ldots,I)}_{*})=r_{q}^{(0)}(q^{(0)}_{*},q^{(1,\ldots,I)}_{*})$
we get $r_{q}^{(0)}(q_{*})=\hat{r}_{q}^{(0)}$. Note that the Implicit Function
Theorem also yields local injectivity of $r$. ∎

######  Remark 4.

Note that due to the relaxed definition (2.6) of regularized approximations,
no $\tilde{\mathcal{T}}$ lower semicontinuity of $\mathcal{R}$ is needed. This
enables the choice of a stronger regularization functional $\mathcal{R}$,
which in its turn enables the use of the strong topology in $Q$ as mandated by
the use of the Implicit Function Theorem in the reduced setting (which will
not be needed in the all-at-once setting).

For the frozen Newton method (2.9), from Theorem 2 we conclude

######  Corollary 3.2 (frozen Newton, reduced).

Let $q_{0}\in
U:=\mathcal{B}_{\rho}^{Q_{J}}(q^{\dagger})\subseteq\mathcal{D}(\mathbf{F})$
for some $\rho>0$ sufficiently small and assume that $B:V_{J}\to
L(Q_{J},W^{*}_{J})$ is continuously differentiable, $D\in L(V_{J},W^{*}_{J})$,
and $\widetilde{Q}_{J}=Q_{J}$. Moreover, let
$q_{0}-q^{\dagger}\in\bigl{(}\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))\cap\textup{nsp}(P)\bigr{)}^{\bot}$,
(2.13) hold with $K=\mathbf{F}^{\prime}(q_{0})$, (3.10) and the stopping index
$n_{*}=n_{*}(\delta)$ be chosen according to (2.15).
Then the iterates $(q_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are
well-defined by (2.9), (3.9), remain in
$\mathcal{B}_{\rho}^{Q_{J}}(q^{\dagger})$ and converge in $Q_{J}$,
$\|q_{n_{*}(\delta)}^{\delta}-q^{\dagger}\|_{Q_{J}}\to 0$ as $\delta\to 0$. In
the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we have
$\|q_{n}-q^{\dagger}\|_{Q_{J}}\to 0$ as $n\to\infty$.

###### Proof.

To establish (2.10), it suffices to estimate the difference between the values
of $r^{(0)}_{\textup{diff}}(q,S(q);q_{0},S(q_{0}))$ at $q$ and at
$q^{\dagger}$. Indeed, with $u_{0}=S(q_{0})$, $u=S(q)$,
$u^{\dagger}=S(q^{\dagger})$,

$\displaystyle
r^{(0)}_{\textup{diff}}(q^{\dagger},u^{\dagger};q_{0},u_{0})-r^{(0)}_{\textup{diff}}(q,u;q_{0},u_{0})$
(3.24)
$\displaystyle=B_{0}^{-1}\Bigl{\\{}\bigl{[}B(u)-B(u_{0})-B^{\prime}(u_{0})(u-u_{0})-\bigl{(}B(u^{\dagger})-B(u_{0})-B^{\prime}(u_{0})(u^{\dagger}-u_{0})\bigr{)}\bigr{]}q_{0}$
$\displaystyle\hskip
113.81102pt+[B(u)-B(u_{0})](q-q_{0})-[B(u^{\dagger})-B(u_{0})](q-q_{0})\Bigr{)}\Bigr{\\}}$
$\displaystyle=B_{0}^{-1}\Bigl{\\{}[B(u)-B(u^{\dagger})-B^{\prime}(u_{0})(u-u^{\dagger})]q_{0}$
$\displaystyle\hskip
113.81102pt+[B(u)-B(u^{\dagger})](q-q_{0})+[B(u^{\dagger})-B(u_{0})](q-q^{\dagger})\Bigr{)}\Bigr{\\}},$

where according to (3.14)

$\displaystyle\|u-u^{\dagger}\|_{V_{J}}=\|S(q)-S(q^{\dagger})\|_{V_{J}}$
(3.25)
$\displaystyle\leq\|(D+[B^{\prime}(q^{\dagger})\cdot]q^{\dagger})^{-1}\|_{W^{*}_{J}\to
V_{J}}\|[B(u)-B(u^{\dagger})-B^{\prime}(u)(u-u^{\dagger})]q^{\dagger}+B(u)(q-q^{\dagger})\|_{W^{*}_{J}}.$

Due to continuous differentiability of $B$ we have
$[B(u)-B(u^{\dagger})-B^{\prime}(u)(u-u^{\dagger})]q^{\dagger}\|_{W^{*}_{J}}=o(\|u-u^{\dagger}\|_{V_{J}})$
and thus, by (3.25), in a sufficiently small neighborhood of $q^{\dagger}$,
there exists a constant $M>0$ such that $\|u-u^{\dagger}\|_{V_{J}}\leq
M\|q-q^{\dagger}\|_{Q_{J}}.$ Thus, from (3.24), for $\rho$ sufficiently small
we obtain (2.10). ∎

###  3.2 All-at-once formulation

Here some proofs are simpler and require less assumptions; in particular, the
Implicit Function Theorem is not needed.

We use (1.1) with

$X=Q_{J}\times V_{J},\quad Y=W^{*}_{J}\times Y_{J}$

with $Q_{J}$, $V_{J}$, $W_{J}$, $Y_{J}$ as in Section 3.1 (and a slight
overload of notation on $y$), as well as

$\mathbb{F}(q,u)=\left(\begin{array}[]{l}Du+B(u)q-f\\\ Cu\end{array}\right).$
(3.26)

Thus we have

$\mathbb{F}^{\prime}(q,u)(\underline{dq},\underline{du})=\left(\begin{array}[]{l}D\underline{du}+[B^{\prime}(u)\underline{du}]q+B(u)\underline{dq}\\\
C\underline{du}\end{array}\right),$ (3.27)

where due to

$\displaystyle\|\mathbb{F}^{\prime}(q,u)(\underline{dq},\underline{du})\|_{W^{*}_{J}\times
V_{J}}^{2}\leq$ $\displaystyle
4\|B(u_{0})\|_{L(\widetilde{Q}_{J},W^{*}_{J})}^{2}\|\underline{dq}\|_{\widetilde{Q}_{J}}^{2}$
$\displaystyle+\bigl{(}4\|D\|_{L(V_{J},W^{*}_{J})}^{2}+4\|[B^{\prime}(u_{0})\cdot]q_{0}\|_{L(V_{J},W^{*}_{J})}^{2}+\|C\|_{L(V_{J},Y_{J})}^{2}\bigr{)}\|\underline{du}\|_{V_{J}}^{2}$

we have

$\mathbf{F}^{\prime}(q_{0},u_{0})\in L(\widetilde{Q}_{J}\times
V_{J},W^{*}_{J}\times Y_{J}).$

Therefore (2.1) with $K=\mathbb{F}^{\prime}(q_{0},u_{0})$ and
$(q^{\dagger},u^{\dagger})\in U\subseteq X=Q_{J}\times V_{J}$,
$r(q,u)=(r_{q}(q,u),r_{u}(q,u))$ is equivalent to

$\displaystyle
D(u-u_{0})+B(u)q-B(u_{0})q_{0}=Dr_{u}(q,u)+[B^{\prime}(u)r_{u}(q,u)]q+B(u)r_{q}(q,u)$
$\displaystyle C(u-u_{0})=Cr_{u}(q,u).$

A choice that enables this under condition (3.3) for an arbitrary linear
observation operator $C$ is

$\displaystyle r_{q}^{(0)}(q,u):=$ $\displaystyle
q^{(0)}-q^{(0)}_{0}+r^{(0)}_{\textup{diff}}(q,u;q_{0},u_{0})$ (3.28)
$\displaystyle r_{q}^{(i)}(q,u):=$ $\displaystyle
r^{(i)}(q^{(i)})=q^{(i)}-q^{(i)}_{0},\quad i\in\\{1,\ldots,I\\}$
$\displaystyle r_{u}(q,u):=$ $\displaystyle r_{u}(u)=u-u_{0}$

with $r^{(0)}_{\textup{diff}}(q,u;q_{0},u_{0})$ defined as in (3.17).

To prove convergence of the all-at-once version of the variational method
(2.6) from Theorem 1 we assume

$\left.\begin{aligned} &Q^{(0)}_{J},\ Q^{(i)},\ V_{J},\ Y_{J}\textup{ are
reflexive or duals of separable spaces;}\\\
&\tilde{\mathcal{T}}:=\mathcal{T}:=\prod_{i=0}^{I}\mathcal{T}^{(i)}\times\mathcal{T}_{V},\
\mathcal{T}_{Y}\textup{ the weak(*) topologies;}\\\ &\textup{Sublevel sets of
$\mathcal{R}$ are $\tilde{\mathcal{T}}$ compact}\\\ &\textup{The functional
$\mathcal{P}$ is $\mathcal{T}$ lower semicontinuous;}\\\ &\textup{The operator
$B$ is differentiable on $\\{u\in V\ :\ \exists q\in Q_{J}\,:\ (q,u)\in
U$\\};}\\\ &\textup{For all $i\in\\{1,\ldots,I\\}$, the operators $u\mapsto
B^{(i)}(\cdot)q_{0}$ are $\mathcal{T}_{V}$-to-$W^{*}_{J}$ continuous and}\\\
&\textup{the operators $(q^{(i)},u)\mapsto B^{(i)}(u)q^{(i)}$ are
$\mathcal{T}^{(i)}\times\mathcal{T}_{V}$-to-$W^{*}_{J}$
continuous.}\end{aligned}\right\\}$ (3.29)

The conditions on $\mathcal{R}$ and $\mathcal{P}$ can, e.g., be satisfied by
defining them by means of norms
$\mathcal{R}(\hat{r}_{q},\hat{r}_{u})=\|\hat{r}_{q}^{(0)}\|_{\widetilde{Q}^{(0)}_{J}}^{2}+\sum_{i=1}^{I}\|\hat{r}_{q}^{(i)}\|_{Q^{(i)}}^{2}+\|\hat{r}_{u}\|_{V_{J}}^{2}$,
$\mathcal{P}(x)=\|Px\|_{Z}^{2}$ for some $P\in L(X,Z)$.

######  Corollary 3.3 (variational, all-at-once).

Under conditions (3.2), (3.3), (3.29), $(q_{0},u_{0})\in U$, $D\in
L(V_{J},W^{*}_{J})$, $\widetilde{Q}_{J}=Q_{J}$, the approximations
$x^{\delta}=(q^{\delta},u^{\delta})$ defined by (2.6), (3.26) with (2.7)
converge $\mathcal{T}$ subseqentially to a solution
$(\hat{r}^{\dagger},q^{\dagger},u^{\dagger})$ of (1.3), (3.26) with
$\mathcal{P}(q,u)=0$, If the $\mathcal{R}$ minimizing solution
$(\hat{r}^{\dagger},q^{\dagger},u^{\dagger})$ of (1.3), (3.26) with
$\mathcal{P}(q,u)=0$ is unique, then
$\hat{r}^{\delta}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}r(q^{\dagger},u^{\dagger})$
and
$(q^{\delta},u^{\delta})\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}(q^{\dagger},u^{\dagger})$.

###### Proof.

Again, items (i), (ii), (iii), (v) in Assumption 2.2 directly follow from the
assumptions made in the corollary.

In order to verify item (iv), we assume that

$\hat{r}^{(i)}_{q,n}:=r_{q}^{(i)}(q_{n},u_{n})\stackrel{{\scriptstyle\mathcal{T}^{(i)}}}{{\longrightarrow}}\hat{r}_{q}^{(i)},\
i\in\\{0,\ldots,I\\},\quad\hat{r}_{u,n}:=r_{u}(q_{n},u_{n})\stackrel{{\scriptstyle\mathcal{T}_{V}}}{{\longrightarrow}}\hat{r}_{u}$

and have to show existence of a subsequence
$(q_{n_{k}},u_{n_{k}})_{k\in\mathbb{N}}$ such that
$q^{(i)}_{n_{k}}\stackrel{{\scriptstyle\mathcal{T}^{(i)}}}{{\longrightarrow}}q^{(i)}$,
$u_{n_{k}}\stackrel{{\scriptstyle\mathcal{T}_{V}}}{{\longrightarrow}}u$ and
$r_{q}^{(i)}(q,u)=\hat{r}_{q}^{(i)}$, $r_{u}(q,u)=\hat{r}_{u})$. Again, by the
definition (3.28) of $r$, this is nontrivial only for $q^{(0)}$ but can easily
established due to the identity

$\displaystyle
q^{(0)}_{n}=q^{(0)}_{0}+\hat{r}^{(0)}_{q,n}-r^{(0)}_{\textup{diff}}(q_{n},u_{n};q_{0},u_{0})$

since our assumptions imply
$\prod_{i=1}^{I}\mathcal{T}^{(i)}\times\mathcal{T}_{V}$-to-$\mathcal{T}^{(0)}$
continuity of $r^{(0)}_{\textup{diff}}(\cdot,\cdot;q_{0},u_{0})$. ∎

######  Corollary 3.4 (frozen Newton, all-at-once).

Let $(q_{0},u_{0})\in U:=\mathcal{B}_{\rho}^{Q_{J}\times
V_{J}}(q^{\dagger},u^{\dagger})\subseteq Q_{J}\times V_{J}$ for some $\rho>0$
sufficiently small, assume that $B:V_{J}\to L(Q_{J},W^{*}_{J})$ is
continuously differentiable, $D\in L(V_{J},W^{*}_{J})$, and
$\widetilde{Q}_{J}=Q_{J}$. Moreover, let
$(q_{0},u_{0})-(q^{\dagger},u^{\dagger})\in\bigl{(}\textup{nsp}(\mathbb{F}^{\prime}(q_{0},u_{0}))\cap\textup{nsp}(P)\bigr{)}^{\bot}$,
(2.13) hold with $K=\mathbb{F}^{\prime}(q_{0},u_{0})$ and the stopping index
$n_{*}=n_{*}(\delta)$ be chosen according to (2.15).
Then the iterates
$(q_{n}^{\delta},u_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are well-
defined by (2.9), (3.26), remain in $\mathcal{B}_{\rho}^{Q_{J}\times
V_{J}}(q^{\dagger},u^{\dagger})$ and converge in $Q_{J}\times V_{J}$,
$\|q_{n_{*}(\delta)}^{\delta}-q^{\dagger}\|_{Q_{J}}\to 0$,
$\|u_{n_{*}(\delta)}^{\delta}-u^{\dagger}\|_{V_{J}}\to 0$ as $\delta\to 0$. In
the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we have
$\|q_{n}-q^{\dagger}\|_{Q_{J}}\to 0$, $\|u_{n}-u^{\dagger}\|_{V_{J}}\to 0$ as
$n\to\infty$.

###### Proof.

Here we can just use (3.24) (without $u$, $u^{\dagger}$ depending on $q$) to
verify

$\|r^{(0)}_{\textup{diff}}(q^{\dagger},u^{\dagger};q_{0},u_{0})-r^{(0)}_{\textup{diff}}(q,u;q_{0},u_{0})\|_{Q^{(0)}_{J}}\leq
c\Bigl{(}\|q-q^{\dagger}\|_{Q_{J}}^{2}+\|u-u^{\dagger}\|_{V_{J}}^{2}\Bigr{)}^{1/2},$

by continuous differentiability of $B$ and smallness of $\rho$, which together
with the definition (3.28) of $r$ implies (2.10). ∎

##  4 Some concrete examples

######  Example 5 (identification of a potential revisited).

Examples 1 and 2

$-\Delta_{N}u+q\cdot u=f+\bar{h},\qquad
D_{t}^{M}u-\Delta_{N}u+q\cdot(u+\bar{h})=0$

are covered by the framework of Section 3 with

$\displaystyle D=-\Delta_{N}$ $\displaystyle B(u)q=q\cdot u-f-\bar{h}$ in the
elliptic case $\displaystyle D=D_{t}^{M}-\Delta_{N}$ $\displaystyle
B(u)q=q\cdot(u+\bar{h})$ in the transient case

$Q:=L^{2}(\Omega)$, and $V$, $W^{*}$ as in (A.5), (A.8), respectively; First
of all we consider the single experiment case $J=0$ but later on, in order to
also obtain uniqueness, will describe multiple observations with
$J=\mathbb{N}$ in the elliptic and $J=(0,T)$ in the transient case.

We wish to recall the fact that (2.1) is here much easier to verify than
(1.10), since instead of needing $S^{\prime}(q)$ in (1.14) in the reduced
setting, the mapping $r$ is simply defined by pointwise multiplication

$\displaystyle r(q)=\frac{S(q)}{S(q_{0})}\cdot(q-q_{0})\textup{ in the reduced
setting},$ $\displaystyle r_{q}(q,u)=\frac{u}{u_{0}}\cdot(q-q_{0})\textup{ in
the all-at-once setting.}$

(where in the transient case we have to add $\bar{h}$ in the numerator and
denominator). As mentioned above, boundedness away from zero of the
denominators can be taken care of in the elliptic and transient case, e.g., by
maximum principles and a proper choice of the boundary conditions.

Alternatively, division by $S(q_{0})$ or $u_{0}$ can be avoided by redefining
the norm in $\widetilde{Q}$ as a weighted $L^{2}$ norm
$\|\hat{r}\|_{\tilde{Q}}:=\|\hat{r}\cdot u_{0}\|_{L^{2}(\Omega)}$ while using
the ordinary $L^{2}$ norm on $Q$, i.e., $\|q\|_{Q}:=\|q\|_{L^{2}(\Omega)}$,
see (3.4). However, the requirement of assuming $S(q_{0})$ or $u_{0}$ to be
bounded away from zero returns when proving differentiability of $B$ as an
operator $V_{J}\to L(\widetilde{Q}_{J},W^{*}_{J})$.

A proper choice of $P$ for verifying (a) in (2.13) heavily depends on
uniqueness arguments and therefore on the type of observations, cf. Remark 3.
We will now comment on this with the following settings known from the
literature, as well as the corresponding extension sets $J$

  * •

elliptic case: Neumann-to-Dirichlet (N-t-D) map, $J=\mathbb{N}$;

  * •

transient case: time trace measurements, $J=(0,T)$;

We target our investigations at the two frozen Newton methods, where by
Corollaries 3.2, C.2, 3.4, C.4, and due to $r^{\prime}(q_{0})=\textup{Id}$ we
only have to verify the relatively simple nullspace condition (3.6). With the
penalty operators defined by

$\displaystyle(Pq)(j)=$ $\displaystyle
q(j)-\frac{\sum_{\ell\in\mathbb{N}}\ell^{-2}q(\ell)}{\sum_{\ell\in\mathbb{N}}\ell^{-2}}\mbox{
for }J=\mathbb{N}$ $\displaystyle(Pq)(j)=$ $\displaystyle
q(j)-\frac{\int_{0}^{T}(1+\ell)^{-2}q(\ell)\,d\ell}{\int_{0}^{T}(1+\ell)^{-2}\,d\ell}\mbox{
for }J=(0,T)$

(possibly including $T=\infty$), verification of (3.6) amounts to proving
linearized uniqueness of a potential depending only on $x$, which can be done
by well-known arguments that we here shortly recall for the convenience of the
reader.

##### Elliptic case.

The observations are defined by boundary observations of $u$ according to the
N-t-D map $\Lambda\in L(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$,
defined by

$\Lambda h=\textup{tr}_{\partial\Omega}u\textup{ where $u$ solves $-\Delta
u+q(x)\cdot u=f$ with $\partial_{\nu}u=h$ on $\partial\Omega$}$ (4.1)

Considering the N-t-D map as an operator $\Lambda\in
L(H^{-1/2}(\partial\Omega),L^{2}(\partial\Omega))$ (in view of the fact that
only values and not derivatives can be measured) it is compact and thus can be
represented by means of the sequence of the images of a complete orthonormal
sequence $(\varphi^{j})_{j\in\mathbb{N}}$ in the separable Hilbert space
$H^{-1/2}(\partial\Omega)$,

$g^{j}=\textup{tr}_{\partial\Omega}u^{j}\textup{ where $u^{j}$ solves $-\Delta
u+q(x)\cdot u=f$ with $\partial_{\nu}u^{j}=\varphi^{j}$ on
$\partial\Omega$}\,,\quad j\in\mathbb{N}.$

The sequence $(\varphi^{j})_{j\in\mathbb{N}}$ can be obtained from a complete
(in $H^{-1/2}(\partial\Omega)$) sequence of given excitations
$(h^{j})_{j\in\mathbb{N}}$ by means of Gram-Schmidt orthonormalization
(skipping linearly dependent elements in the sequence
$(h^{j})_{j\in\mathbb{N}}$).

Using the weak form of the PDE, we get the model equations

$\langle A^{j}(q,u^{j}),w\rangle_{W^{*},W}:=\int_{\Omega}\Bigl{(}\nabla
u^{j}\cdot\nabla
w+qu^{j}w\Bigr{)}\,dx-\int_{\partial\Omega}\varphi^{j}w\,d\Gamma(x)\,,w\in
W:=H^{1}(\Omega)$ (4.2)

for $u^{j}\in V:=H^{1}(\Omega)$, and the observation equations

$C^{j}(u^{j})=\textup{tr}_{\partial\Omega}u^{j}.$ (4.3)

Thus we can use the formalism from Section 3 with $I=0$, $q^{(0)}=q$,
$J=\mathbb{N}$, $D^{j}=-\Delta_{N}$, $\langle
B^{0,j}(u^{j})q,w\rangle_{W^{*},W}=\int_{\Omega}q\,u^{j}\,w\,dx$.

Assuming $q\in\textup{nsp}(P)$ means that the artificial dependency of $q$ on
$j$ is formally undone and to verify (3.6), we can proceed by proving that an
only space dependent $q$ lying in $\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))$
must vanish. (Due to the identity (3.7) this also covers the all-at-once
case.) For the details we refer to Section D.1 in Appendix D.

##### Transient case.

The proof of (3.6) we choose to present here relies on taking the Laplace
transform (to this end, we assume to have observations on the whole positive
time line $T=\infty$) and applies to a general transient model with

$Du=\sum_{m=1}^{M}\mathcal{M}_{m}*\partial_{t}^{m}u+\sum_{n=0}^{N}\mathcal{N}_{n}*\mathcal{A}^{\beta_{n}}\partial_{t}^{n}u$

where $\mathcal{A}$ is a selfadjoint (with respect to a possible weighted
$L^{2}$ inner product $L^{2}_{w}$ on $\Omega)$), nonnegative definite operator
with domain $V\subseteq L^{\infty}(\Omega)$, eigensystem
$(\lambda_{i},\varphi_{i}^{k})_{i\in\mathbb{N},k\in K^{i}}$ and spectral
powers $\mathcal{A}^{\beta_{n}}$ defined by
$\mathcal{A}^{\beta}v=\sum_{i\in\mathbb{N}}\lambda_{i}^{\beta}\sum_{k\in
K^{i}}\langle v,\varphi_{i}^{k}\rangle_{L^{2}_{w}(\Omega)}\varphi_{i}^{k}$,
the star $*$ denotes convolution $(v*w)(t):=\int_{0}^{t}v(t-s)w(s)\,ds$, and
the kernel functions $\mathcal{M}_{m}$, $\mathcal{N}_{n}$ are assumed to have
Laplace transforms $\widehat{\mathcal{M}}_{m}$, $\widehat{\mathcal{N}}_{n}$
lying in $L^{\infty}(\mathbb{C})$. Besides the parabolic and hyperbolic cases
with $N=0$, $\mathcal{N}_{0}=\delta$ (the delta distribution centered a zero
with $\widehat{\mathcal{N}}_{0}\equiv 1$), $\beta_{0}=1$, and either $M=1$,
$\mathcal{M}_{1}=\delta$ or $M=2$, $\mathcal{M}_{1}=0$,
$\mathcal{M}_{2}=\delta$ respectively, this also covers many viscoelastic
models, see, e.g., [4, 6, 33, 46, 53] and fractional damping models in
acoustics see, e.g., [25, 37, 35, 58].
A uniqueness proof only needing a finite time interval but more specific to
the parabolic or hyperbolic setting could be carried out along the lines of
proofs in, e.g., [28] and the citing literature.
We set $J=(0,\infty)$, $V_{J}\subseteq\\{v\in
C([0,\infty);H):\frac{\partial^{\ell}v}{\partial t^{\ell}}(t=0)=0\\}$ for some
Hilbert space $H\supseteq V$ (enforcing homogeneous initial conditions;
inhomogeneous ones cound be incorporated into the right hand side $f$ and the
offet $\bar{h}$). Moreover we assume that $u_{0}+\bar{h}$ is a space-time
separable function $(u_{0}+\bar{h})(x,t)=\phi(x)\psi(t)$; in the reduced case
where $u_{0}=S(q_{0})$ this can be achieved by an appropriate choice of the
excitation $f$, $\bar{h}$, see, e.g., [38]. The observation operator is
supposed to be time invariant, bounded and linear $(Cu)(t)=C_{0}u(t)$ with
$C_{0}\in L(H,Y)$. In this setting, the intersection of the nullspaces of
$\mathbb{F}^{\prime}(q_{0},u_{0})$ and $P$ can be shown to be trivial, see
Section D.2 in Appendix D, which implies (3.6).

######  Remark 5.

Instead of time trace observations, final observations in all of $\Omega$ for
a fixed time instance would actually be simpler, allowing to set $J=\\{0\\}$
and employ an easy proof of linearized uniqueness; also, the inverse problem
of recovering $q(x)$ is known to be less ill-posed than with time trace data,
cf. e.g., [11]. Since our main motivation was to allow for iterative or
variational reconstruction from boundary measurements, we do not go into
further detail about this here.

######  Example 6 (combined diffusion and absorption identification).

Identify $a(x)$ and $c(x)$ (that is, $q=(a,c)$) in

$-\nabla\cdot(a\nabla u)+cu=0\textup{ in }\Omega$ (4.4)

in a domain satisfying (A.1) from boundary observations of $u$. This problem
arises, e.g., in steady-state diffuse optical tomography, see, e.g., [2, 17,
22]; for an analysis of a finite element approxiation of this problem we refer
to [49].

To give an idea on uniqueness for this problem, we quote some identifiability
results from the literature:

  * •

uniqueness of $a$ and $c$ from N-t-D maps $\Lambda_{\lambda}\in
L(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$ for $-\nabla\cdot(a\nabla
u)+(c-\lambda)u=0$, for all $\lambda\geq\lambda_{*}$, and some fixed
$\lambda_{*}\in\mathbb{R}$, provided $a\in W^{2,\infty}(\Omega)$, $c\in
L^{\infty}(\Omega)$ and $\nabla a$ is known on $\partial\Omega$); see [12,
proof of Corollary 1.7]555he same result would also provide uniqueness from
spectral data, but this cannot written by means of a linear operator $C$;

  * •

uniqueness of effective absorption
$\tilde{c}:=\frac{\Delta\sqrt{a}}{\sqrt{a}}+\frac{c}{a}$ as well as the jumps
of $a$ and $\partial_{\nu}a$ on the discontinuity set from a single N-t-D map
$\Lambda=\Lambda_{0}\in L(L^{2}(\partial\Omega),L^{2}(\partial\Omega))$ for
piecewise analytic $a$ and $c$; see Theorem 2.2 in [23]; a counterexample on
uniqueness for general $a$, $c$ can be found in [1];

  * •

uniqueness of $a$ or of $c$ alone from a single N-t-D map
$\Lambda=\Lambda_{0}\in L(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$,

    * –

for $c\in L^{p}(\Omega)$, $p>\frac{d}{2}$ if $d\geq 3$, $p>1$ if $d=2$; see
[28] for $d=2$, [45] for $d\geq 3$;

    * –

for $a\in C^{2}(\Omega)$ if $d\geq 3$ [57], for $a\in L^{\infty}(\Omega)$ if
$d=2$ [3].

To recover $a$ and $c$ simultaneously, we therefore consider the N-t-D maps
$\Lambda_{\lambda}\in L(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$ for
all $\lambda\geq 0$. As in in the elliptic case of Example 5, we think of the
N-t-D map as represented by means of the sequence of the images of a basis
$(\varphi^{n})_{n\in\mathbb{N}}$ of $H^{-1/2}(\partial\Omega)$,

$\displaystyle g^{\lambda,n}=\textup{tr}_{\partial\Omega}u^{\lambda,n}$ (4.5)
$\displaystyle\textup{where $u^{\lambda,n}$ solves $-\nabla\cdot(a\nabla
u)+(c-\lambda)u=0$ with $\partial_{\nu}u^{n}=\varphi^{n}$ on
$\partial\Omega$}\,,\quad n\in\mathbb{N},\quad\lambda\geq 0.$

Using the weak form of (4.4), we get the model equations

$\displaystyle\langle
A^{\lambda,n}(c,a,u^{\lambda,n}),w\rangle_{W^{*},W}:=\int_{\Omega}\Bigl{(}a\nabla
u^{\lambda,n}\cdot\nabla
w+(c-\lambda)u^{\lambda,n}w\Bigr{)}\,dx-\int_{\partial\Omega}\varphi^{n}w\,d\Gamma(x)\,,$
$\displaystyle w\in W:=H^{1}(\Omega)$

for $u^{\lambda,n}\in V:=H^{1}(\Omega)$, and the observation equations

$C^{\lambda,n}(u^{\lambda,n})=\textup{tr}_{\partial\Omega}u^{\lambda,n}.$

Thus we can use the formulation from Section 3 with $I=1$, $q^{(0)}=c$,
$q^{(1)}=a$, $J=[0,\infty)\times\mathbb{N}$, $D^{\lambda,n}=-\lambda\cdot$,
$\langle
B^{0,\lambda,n}(u^{\lambda,n})c,w\rangle_{W^{*},W}=\int_{\Omega}c\,u^{\lambda,n}\,w\,dx$,
$\langle
B^{1,\lambda,n}(u^{\lambda,n})a,w\rangle_{W^{*},W}=\int_{\Omega}a\nabla
u^{\lambda,n}\cdot\nabla w\,dx$.

It is readily checked that the range invariance condition (2.1) holds with
$K=\mathbf{F}^{\prime}(c_{0},a_{0})$ (or
$K=\mathbb{F}^{\prime}(c_{0},a_{0},u_{0})$) and

$r(c,a)=(c-c_{0}-\frac{1}{u_{0}^{\lambda,n}}\Bigl{(}-\nabla\cdot((a-a_{0})\nabla(u^{\lambda,n}-u_{0}^{\lambda,n}))+(c-c_{0})\cdot(u^{\lambda,n}-u_{0}^{\lambda,n})\Bigr{)},a-a_{0})$

(or $r(c,a,u^{\lambda,n})=(r(c,a),u^{\lambda,n}-u^{\lambda,n}_{0})$) where in
the reduced case we have to use $u^{\lambda,n}:=S^{\lambda,n}(c,a)$ as in
(4.5), cf. (3.16), (3.28). Extension only affects the absorption coefficient
$c(x)$, which becomes $c(x,\lambda,n)$, while the diffusion coefficient $a$
remains only $x$ dependent.
We use the spaces $Q_{J}=L^{2}(J;L^{2}(\Omega))\times(L^{\infty}(\Omega)\cap
W^{1,2P/(P-2)}(\Omega)$ with $P\in(2,\infty)$,
$1-\frac{d}{2}\geq-\frac{d}{P}$, $V=H^{2}(\Omega)$, $W^{*}=L^{2}(\Omega)$,
$Y=L^{2}(\partial\Omega)$.
With the penalty operator defined by

$\displaystyle(Pc)(\lambda,n)=c(\lambda,n)-\frac{\sum_{\ell\in\mathbb{N}}\ell^{-2}\int_{(0,\infty)}(1+\kappa)^{-2}c(\kappa,\ell)\,d\kappa}{\sum_{\ell\in\mathbb{N}}\ell^{-2}\int_{(0,\infty)}(1+\kappa)^{-2}\,d\kappa},$

verification of (3.6) amounts to proving linearized uniqueness of a diffusion
coefficient and a potential, both depending only on $x$, which can be done
similarly to the elliptic case of recovering a potential, see Section D.3 in
Appendix D.

######  Example 7 (Reconstruction of a boundary coefficient).

As a final example in which we actually have uniqueness without needing to
extend the coefficient we consider the problem of recovering the Robin
coefficient $q$ in the elliptic boundary value problem

$\displaystyle-\Delta u$ $\displaystyle=\ell\mbox{ in }\Omega$ (4.6)
$\displaystyle\partial_{\nu}u+q\cdot\Phi(u)$ $\displaystyle=h\mbox{ on
}\Gamma_{R}\subseteq\partial\Omega$ $\displaystyle\partial_{\nu}u$
$\displaystyle=h\mbox{ on
}\Gamma_{N}\subseteq\partial\Omega\setminus\Gamma_{R}$ $\displaystyle u$
$\displaystyle=0\mbox{ on
}\Gamma_{D}:=\partial\Omega\setminus(\Gamma_{R}\cup\Gamma_{N})$

from observations $y=Cu$ of $u$, for example Dirichlet data on some part of
the boundary $\partial\Omega$ of $\Omega$ or on some other surface $\Sigma$
contained in the closure of the PDE domain $\Omega$. Since we will not appeal
to elliptic regularity, it suffices to have a Lipschitz domain
$\Omega\subseteq\mathbb{R}^{d}$, $d\in\\{1,2,3\\}$ here.

This has applications for example in corrosion detection, and heat conduction,
see, e.g., [13, 26], where the latter application also motivates the use of a
potentially nonlinear term $\Phi(u)$ in the boundary condition. The mentioned
papers and the references therein also provide uniqueness of $q$ from a single
boundary observation.

The weak form of (4.6) reads as

$\displaystyle u\in V:=H_{D}^{1}(\Omega)\textup{ and }\forall v\in
W:=H_{D}^{1}(\Omega)$ (4.7) $\displaystyle\langle
A(q,u),v\rangle_{W^{*},W}:=\int_{\Omega}\nabla u\cdot\nabla
v\,dx+\int_{\Gamma_{R}}q\cdot\Phi(u)\,v\,d\Gamma(x)-\int_{\Omega}\ell
v\,dx-\int_{\Gamma_{R}\cup\Gamma_{N}}h\,v\,d\Gamma(x)=0$

where $H_{D}^{1}(\Omega)=\\{v\in
H^{1}(\Omega)\,:\,\textup{tr}_{\Gamma_{D}}v=0\\}$. Here $\Phi:L^{4}(\Omega)\to
L^{4}(\Omega)$ is supposed to be a known nonlinear operator, for example a
superposition (or Nemitzky) operator induced by some function
$\phi:\mathbb{R}\to\mathbb{R}$ with at most linear growth at zero and
infinity.

This problem is therefore covered by the framework of Section 3 with

$\displaystyle\langle Du,v\rangle_{W^{*},W}=\int_{\Omega}\nabla u\cdot\nabla
v\,dx,\quad\langle
B(u)q,v\rangle_{W^{*},W}=\int_{\Gamma_{R}}q\,\Phi(u)\,v\,d\Gamma(x),\quad$
$\displaystyle\langle f,v\rangle_{W^{*},W}=\int_{\Omega}\ell
v\,dx+\int_{\Gamma_{R}\cup\Gamma_{N}}h\,v\,d\Gamma(x).$

We set $K=\mathbf{F}^{\prime}(q_{0})$ (or
$K=\mathbb{F}^{\prime}(q_{0},u_{0})$) and according to (3.16) (or (3.28)) set

$r(q)=(q-q_{0}+\frac{1}{\Phi(u_{0})}\Bigl{(}(\Phi(u)-\Phi(u_{0}))q-\Phi^{\prime}(u_{0})(u-u_{0})q_{0}\Bigr{)}$

(or $r(q,u)=(r(q),u-u_{0}))$), where in the reduced case $u$ has to solve
(4.6) and likewise for $u_{0}$ with $q_{0}$ in place of $q$.

In the reduced setting we have to take care of well-posedness of the forward
problem by restricting the domain to (similarly to (A.6))

$\displaystyle\mathcal{D}(\mathbf{F}):=\\{q\in L^{2}(\partial\Omega):$
$\displaystyle\exists\bar{q}\in L^{\infty}(\Omega),\bar{q}\geq 0\textup{ a.e.
and
}C_{\textup{PF}}L\Bigl{(}\|\textup{tr}_{\Gamma_{R}}\|_{H_{D}^{1},H^{1/2}}C_{H^{1/2},L^{4}}^{\Gamma_{R}}\Bigr{)}^{2}\|q-\bar{q}\|_{L^{2}(\partial\Omega)}<1\\}.$

with $C_{\textup{PF}}$, $L$ as in (4.8), (4.9) below. Here for simplicity we
have assumed that the $d-1$ dimensional measure
$\textup{meas}^{d-1}(\Gamma_{D})>0$ so that a Poincaré-Friedrichs equation

$\forall v\in H_{D}^{1}(\Omega)\,:\ \|v\|_{H^{1}(\Omega)}\leq
C_{\textup{PF}}\|v\|_{H^{1}_{D}(\Omega)}$ (4.8)

holds. Moreover we impose the following monotonicity and continuity conditions
on $\Phi$

$\displaystyle\forall v,w\in L^{4}(\Gamma_{R})\,\stackrel{{\scriptstyle
a.e.}}{{\forall}}x\in\Omega:\ (\Phi(v)-\Phi(w))\cdot(v-w)\geq 0\textup{ and
}|\Phi(v)-\Phi(w)|\leq L|v-w|$ (4.9) $\displaystyle\Phi:L^{4}(\Gamma_{R})\to
L^{4}(\Gamma_{R})^{*}\textup{ hemicontinuous.}$

(Note that $\Phi(u)$ is supposed to act as a pointwise multiplication but
$\Phi$ does not necessarily need to be a Nemitzky operator, thus might as well
be nonlocal.) From these properties we can conclude (strict, due to the $D$
term) monotonicity, coercivity and hemicontinuity of the operator
$D+B(\cdot)\bar{q}$ for any nonegative $\bar{q}\in L^{\infty}(\Omega)$. By the
Browder-Minty Theorem on the separable Banach space $H_{D}^{1}(\Omega)$ this
implies existence and uniqueness (and by the elliptic $D$ part also
continuity) of $S(\bar{q}):=(D+B(\cdot)\bar{q})^{-1}(\tilde{f})$ for any
$\tilde{f}\in H_{D}^{1}(\Omega)^{*}$. Defining the fixed point map
$\mathcal{M}:u\mapsto u_{+}:=(D+B(\cdot)\bar{q})^{-1}(f-B(u)(q-\bar{q}))$ on
$H_{D}^{1}(\Omega)$ and proving its contractivity on a sufficiently small
neighborhood of $\bar{q}$ yields well-definedness of
$S(q)=(D+B(\cdot)q)^{-1}(f)$. Indeed, by testing the PDE for the difference

$D(u_{1+}-u_{2+})+(B(u_{1+})-B(u_{2+}))\bar{q}=(B(u_{1})-B(u_{2}))(\bar{q}-q)$

with $u_{1+}-u_{2+}$ and using (4.8), (4.9) we obtain

$\displaystyle\frac{1}{C_{\textup{PF}}}\|u_{1+}-u_{2+}\|_{H^{1}(\Omega)}^{2}\leq\langle
D(u_{1+}-u_{2+})+(B(u_{1+})-B(u_{2+}))\bar{q},u_{1+}-u_{2+}\rangle_{W^{*},W}$
$\displaystyle=\langle(B(u_{1})-B(u_{2}))(\bar{q}-q),u_{1+}-u_{2+}\rangle_{W^{*},W}=\int_{\Gamma_{R}}(\bar{q}-q)(\Phi(u_{1})-\Phi(u_{2}))(u_{1+}-u_{2+})\,d\Gamma$
$\displaystyle\leq
L\Bigl{(}\|\textup{tr}_{\Gamma_{R}}\|_{H_{D}^{1},H^{1/2}}C_{H^{1/2},L^{4}}^{\Gamma_{R}}\Bigr{)}^{2}\|q-\bar{q}\|_{L^{2}(\partial\Omega)}\|u_{1}-u_{2}\|_{H^{1}(\Omega)}\|u_{1+}-u_{2+}\|_{H^{1}(\Omega)}.$

If additionally to (4.9), $\Phi:L^{4}(\Gamma_{R})\to L^{4}(\Gamma_{R})$ is
continuously differentiable with $\Phi^{\prime}(S(q_{0}))$ bijective and
$\Phi^{\prime}(S(q_{0}))^{-1}\in L(L^{4}(\Gamma_{R}),L^{4}(\Gamma_{R}))$, then
in the function space setting

$Q=L^{2}(\Gamma_{R}),\quad V=W=H_{D}^{1}(\Omega)$

with $Y$ such that $C\in L(V,Y)$ the conditions of Corollaries 3.1, 3.2, C.2
are satisfied with, a proper choice of $\mathcal{R}$ (e.g.,
$\mathcal{R}(q)=H^{r}(\Gamma_{R})$ for some $r>0$) and $\mathcal{P}$. Indeed,
we can set $\mathcal{P}=0$ in case of boundary observations
$C=\textup{tr}_{\Gamma}$ for some open subset $\Gamma\subseteq\Gamma_{N}$, and
verify triviality of $\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))$, where
$\mathbf{F}^{\prime}(q_{0})$ is the mapping that takes $\underline{dq}$ to
$Cv$ with $v$ solving

$\displaystyle-\Delta v$ $\displaystyle=0\mbox{ in }\Omega$
$\displaystyle\partial_{\nu}v+q_{0}\cdot\Phi^{\prime}(u_{0})v$
$\displaystyle=-\underline{dq}\cdot\Phi(u_{0})\mbox{ on }\Gamma_{R}$
$\displaystyle\partial_{\nu}v$ $\displaystyle=0\mbox{ on }\Gamma_{N}$
$\displaystyle v$ $\displaystyle=0\mbox{ on }\Gamma_{D}.$

Thus if $Cv=\textup{tr}_{\Gamma}v=0$ then $v$ is harmonic in $\Omega$ with
vanishing Cauchy data on $\Gamma$ and thus, due to Holmgren’s Uniqueness
Theorem, $v=0$ in $\Omega$. From this we conclude that $F^{\prime}(q_{0})$ has
trivial nullspace and therefore conditions (2.13), (C.8), (C.14) as well as
$q_{0}-q^{\dagger}\in\bigl{(}\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))\cap\textup{nsp}(P)\bigr{)}^{\bot}$,
and
$r(q^{\dagger})\in\bigl{(}\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))\cap\textup{nsp}(Pr^{\prime}(q_{0})^{-1})\bigr{)}^{\bot}$
are satisfied with the trivial choice $P=0$.

Likewise we can apply Corollaries 3.3, 3.4, C.4 in the all-at-once setting
even without having to assume (4.8) and (4.9).

Further examples satisfying a range invariance condition can be found, e.g.,
in [10, Section 2.2], [9, Section 4], and [39, 40].

## Conclusion and Outlook

While the use of range invariance conditions for proving convergence of
regularization methods is known in the literature, their use has so far
largely been limited to relatively simple model problems – as is the case for
other restrictions on the nonlinearity of the forward operator (such as the
tangential cone condition) used in the analysis of solution methods for ill-
posed problems. In this paper, we suggest to overcome this limitation by
possibly augmenting the dimensions of variability of the searched for
parameter, thus allowing to establish validity of this condition for a much
larger class of inverse problems, which indeed contains practically relevant
examples. The loss of uniqueness, which also impacts convergence of these
methods (since the initial guess needs to be orthogonal to the nullspace of
the linearized forward operator) is overcome by adding a penalty that in the
limit enforces the original lower dimensional variability.

The analysis is carried out in a general framework (Section 2) that covers
both reduced and all-at-once formulations of inverse problems as demonstrated
in Section 3 and contains some practically relevant inverse problems of
reconstruction from boundary observations as detailed in Section 4.

While – for clarity of exposition – we stayed with norm based misfit
functionals in most of this paper, extension to more general functional should
be considered in view of their practical usefulness for, e.g., modeling
different types of data noise.

Also convergence rates under source conditions will be subject of future
research.

## Acknowledgment

This work was partially supported by the Austrian Science Fund fwf under the
grants P30054 and DOC78.

##  Appendix A Details on examples from introduction

######  Example 1 (identification of a potential in an elliptic PDE).

Consider identification of the spatially varying potential $q\in
L^{2}(\Omega)$ in the elliptic boundary value problem (1.11) with $f\in
L^{2}(\Omega)$, $h\in H^{1/2}(\partial\Omega)$, $\partial_{\nu}$ the normal
derivative, from observations $y=Cu$ of $u$, for example measurements at the
boundary of the domain $\Omega$, on which we here and in most of the further
examples assume

$\Omega\subseteq\mathbb{R}^{d},\ d\in\\{1,2,3\\},\quad\partial\Omega\in
C^{1,1}\mbox{ or $\Omega$ polygonal and convex}$ (A.1)

With $q$ sufficiently close to a function that is positive and bounded away
from zero (see (A.6) below) we can avoid the problem of a nontrivial nullspace
of the negative Neumann Laplacian, defined by its weak form
$-\Delta_{N}:H^{2}(\Omega)\to L^{2}(\Omega)$

$\langle-\Delta_{N}u,v\rangle_{H^{1}(\Omega)^{*},H^{1}(\Omega)}:=\int_{\Omega}\nabla
u\cdot\nabla v\,dx\quad\forall u,v\in H^{1}(\Omega).$ (A.2)

With this and $\bar{h}$ defined by

$\langle\bar{h},v\rangle_{H^{1}(\Omega)^{*},H^{1}(\Omega)}:=\int_{\partial\Omega}hv\,d\Gamma(x)\quad\forall
v\in H^{1}(\Omega).$ (A.3)

we can rewrite (1.11) as an operator equation in $L^{2}(\Omega)$

$-\Delta_{N}u+q\cdot u=f+\bar{h},$ (A.4)

where $q\cdot:H^{2}(\Omega)\to L^{2}(\Omega)$ denotes the multiplication
operator, which is bounded for $q\in L^{2}(\Omega)$, due to continuity of the
embedding $H^{2}(\Omega)\to L^{\infty}(\Omega)$ (with constant denoted by
$C_{H^{2},L^{\infty}}^{\Omega}$ below).

We can write the inverse problem in a reduced form (1.2) as $\mathbf{F}(q)=y$,
or in an all-at-once form (1.1) as $\mathbb{F}(q,u)=(0,y)^{T}$ with the
respective reduced and all-at-once forward operators being defined by (1.12).
With

$Q:=L^{2}(\Omega),\quad V=H^{2}(\Omega),\quad W^{*}=L^{2}(\Omega),$ (A.5)

and $Y$ such that $C\in L(V,Y)$, the reduced forward operator $\mathbf{F}:Q\to
Y$ is well-defined on

$\displaystyle\mathcal{D}(\mathbf{F}):=\\{q\in L^{2}(\Omega):$
$\displaystyle\exists\bar{q}\in L^{\infty}(\Omega),\gamma\in\mathbb{R}^{+}\,:\
\bar{q}\geq\gamma\textup{ a.e. and }$ (A.6)
$\displaystyle\|(-\Delta_{N}+\bar{q}\cdot)^{-1}\|_{L^{2}\to
H^{2}}C_{H^{2},L^{\infty}}^{\Omega}\|q-\bar{q}\|_{L^{2}(\Omega)}<1\\}$

due to the fact that by elliptic regularity, the operator
$(-\Delta_{N}+\bar{q}\cdot):H^{2}(\Omega)\to L^{2}(\Omega)$ is boundedly
invertible.
In the all-at-once setting we have more freedom to choose the spaces. Fixing
again $Q=L^{2}(\Omega)$, we may use any $V,W^{*},Y$ such that
$-\Delta_{N}+q\cdot\in L(V,W^{*})$ and $C\in L(V,Y)$ holds. For simplicity, we
again take $V=H^{2}(\Omega)$ and correspondingly choose $W^{*}=L^{2}(\Omega)$.

In this function space setting, the linearizations of $\mathbf{F}$ and
$\mathbb{F}$ given by (1.13) can easily be verified to be Fréchet derivatives.
For any fixed $q_{0}\in L^{2}(\Omega)$, with $\mathbf{R}(q)$ according to
(1.14), the range invariance (1.15) is satisfied. To show that the $R$
operators are close to the identity, using

$\displaystyle(-\Delta_{N}+q_{0}\cdot)(-\Delta_{N}+q\cdot)^{-1}-\textup{id}=(q-q_{0})\cdot(-\Delta_{N}+q\cdot)^{-1}$
$\displaystyle S(q)-S(q_{0})=-(-\Delta_{N}+q_{0}\cdot)^{-1}[(q-q_{0})\cdot
S(q)]$

and the estimates

$\displaystyle\|\mathbf{R}(q)\underline{dq}-\underline{dq}\|_{Q}$
$\displaystyle=\|\tfrac{1}{S(q_{0})}\Bigl{(}\Bigl{\\{}(-\Delta_{N}+q_{0}\cdot)(-\Delta_{N}+q\cdot)^{-1}-\textup{id}\Bigr{\\}}[\underline{dq}\cdot
S(q)]+\underline{dq}\cdot(S(q)-S(q_{0}))\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|\tfrac{1}{S(q_{0})}\|_{L^{\infty}(\Omega)}\|(q_{0}-q)\cdot\bigl{(}(-\Delta_{N}+q\cdot)^{-1}[\underline{dq}\cdot
S(q)]\bigr{)}$ $\displaystyle\hskip
113.81102pt-\underline{dq}\cdot\bigl{(}(-\Delta_{N}+q_{0}\cdot)^{-1}[(q-q_{0})\cdot
S(q)\bigr{)}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|\tfrac{1}{S(q_{0})}\|_{L^{\infty}(\Omega)}\|q_{0}-q\|_{L^{2}(\Omega)}\|\underline{dq}\|_{L^{2}(\Omega)}\|S(q)\|_{L^{\infty}(\Omega)}C_{H^{2},L^{\infty}}^{\Omega}$
$\displaystyle\hskip 56.9055pt\Bigl{(}\|(-\Delta_{N}+q\cdot)^{-1}\|_{L^{2}\to
H^{2}}+\|(-\Delta_{N}+q_{0}\cdot)^{-1}\|_{L^{2}\to H^{2}}\Bigr{)}$
$\displaystyle\|\mathbb{R}(q,u)(\underline{dq},\underline{du})-(\underline{dq},\underline{du})\|_{Q\times
V}$ (A.7) $\displaystyle=\|\tfrac{1}{u_{0}}\cdot\bigl{(}\underline{dq}\cdot
u+(q-q_{0})\cdot\underline{du}\bigr{)}-\underline{dq}\|_{Q}$
$\displaystyle=\|\tfrac{1}{u_{0}}\cdot\bigl{(}\underline{dq}\cdot(u-u_{0}))+(q-q_{0})\cdot\underline{du}\bigr{)}\|_{Q}$
$\displaystyle\leq\|\tfrac{1}{u_{0}}\|_{L^{\infty}(\Omega)}C_{H^{2},L^{\infty}}^{\Omega}\Bigl{(}\|\underline{dq}\|_{L^{2}(\Omega)}\|u-u_{0}\|_{H^{2}(\Omega)}+\|q-q_{0}\|_{L^{2}(\Omega)}\|\underline{du}\|_{H^{2}(\Omega)}\Bigr{)},$

we obtain (1.16).

######  Example 2 (identification of a potential in a time-dependent PDE).

Consider identification of $q=q(x)$ in the initial boundary value problem
(1.17) with $f\in L^{2}(0,T;L^{2}(\Omega))$, $h\in
L^{2}(0,T;H^{1/2}(\Omega))$, and $\Omega$ satisfying (A.1). Using an extension
$\bar{h}$ of the initial and boundary data such that
$\bar{h}_{t}-\Delta\bar{h}=f$ in $\Omega\times(0,T)$,
$\partial_{\nu}\bar{h}=h$ on $\partial\Omega\times(0,T)$,
$\bar{h}(x,0)=u_{0}(x)$, $x\in\Omega$ and $\hat{u}:=u-\bar{h}$, we can write
the forward problem as (1.18).

With the operator

$T(\widetilde{q}):\,V\to W^{*},\quad v\mapsto\Bigl{(}t\mapsto
D_{t}^{M}v(t)-\Delta_{N}v(t)+\widetilde{q}(t)\cdot v(t)\Bigr{)}$

for $\widetilde{q}\in L^{2}(0,T;\mathcal{D}(F))$ we can define the reduced and
all-at-once forward operators by (1.19) on
$\tilde{\mathcal{D}}(\mathbf{F}):=L^{2}(0,T;\mathcal{D}(\mathbf{F}))\subseteq
X=Q=L^{2}(0,T;L^{2}(\Omega))$ with $\mathcal{D}(\mathbf{F})$ defined as in
(A.6)) and
$\tilde{\mathcal{D}}(\mathbb{F}):=\tilde{\mathcal{D}}(\mathbf{F})\times V$,
respectively.

To prove that for any for $\widetilde{q}\in L^{2}(0,T;\mathcal{D}(F))$, the
operator $T(\widetilde{q}):\,V\to W^{*}$ is indeed an isomorphism, the
function spaces $V$ and $W^{*}$ have to be properly chosen. For example, in
the case of a strongly damped wave equation

$D_{t}^{M}u-\Delta_{N}u+qu=u_{tt}-\Delta_{N}u-b\Delta_{N}u_{t}+qu$

with $b>0$ this holds true with the choice

$\displaystyle V=L^{\infty}(0,T;H^{2}(\Omega))\cap
W^{1,\infty}(0,T;H^{1}(\Omega))\cap H_{\diamondsuit}^{2}(0,T;L^{2}(\Omega)),$
(A.8) $\displaystyle W^{*}=L^{2}(0,T;L^{2}(\Omega)),$ $\displaystyle
H_{\diamondsuit}^{2}(0,T;L^{2}(\Omega))=\\{v\in
H^{2}(0,T;L^{2}(\Omega))\,:\,v(0)=0,v_{t}(0)=0\\}.$

cf. e.g. [34]. Since $V\subseteq L^{\infty}(0,T;L^{\infty}(\Omega)$, we can
invoke the elementary estimate

$\|q\cdot
u\|_{L^{2}(0,T;L^{2}(\Omega))}\leq\|q\|_{L^{2}(0,T;L^{2}(\Omega))}\|u\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}.$

for establishing smallness of the difference between the $R$ operators and the
identity similarly to (A.7). We mention in passing that this does not work in
the seemingly simpler parabolic case $M=1$, $D_{t}^{M}=\partial_{t}$, where a
typical function space setting would be

$\displaystyle V=L^{2}(0,T;H^{2}(\Omega))\cap
H_{\diamondsuit}^{1}(0,T;L^{2}(\Omega)),$ (A.9) $\displaystyle
W^{*}=L^{2}(0,T;L^{2}(\Omega)),$ $\displaystyle
H_{\diamondsuit}^{1}(0,T;L^{2}(\Omega))=\\{v\in
H^{1}(0,T;L^{2}(\Omega))\,:\,v(0)=0\\}.$

Here the use of a smooth cutoff function $\Phi$ and replacing the term $q\cdot
u$ by $q\cdot\Phi(u)$ still enables an analysis, which come with additional
technicalities, though.

######  Example 3 (identification of a diffusion coefficient in an elliptic
PDE).

Consider identification of the coefficient $a(x)$ in the elliptic boundary
value problem (1.21) with $f\in H^{1}(\Omega)^{*}$, $h\in
H^{-1/2}(\partial\Omega)$, and $\Omega$ satisfying (A.1) (as a matter of fact,
a Lipschitz domain would suffice here, since we are not employing elliptic
regularity in this example).

The forward operators in the reduced (1.2) and in the all-at-once formulation
(1.1) are given by (1.22) with $\Delta_{a,N}$ defined by (1.23), where
$\bar{h}$ is defined by (A.3). In the reduced case, in order to achieve unique
solvability of the Neumann problem, we assume $f\in L^{1}(\Omega)$, $h\in
L^{1}(\partial\Omega)$,
$\int_{\Omega}f(x)\,dx+\int_{\partial\Omega}h(x)\,d\Gamma(x)=0$, replace
$H^{1}(\Omega)$ by $H_{\circ}^{1}(\Omega):=\\{v\in
H^{1}(\Omega)\,:\,\int_{\Omega}v(x)\,dx=0\\}$ and define the domain of the
forward operator as

$\mathcal{D}(\mathbf{F}):=\\{a\in
L^{\infty}(\Omega)\,:\,\exists\gamma\in\mathbb{R}^{+}\,:\ a\geq\gamma\textup{
a.e.}\\}.$ (A.10)

With $Q:=L^{\infty}(\Omega)$, $V=W=H_{\circ}^{1}(\Omega)$ and $Y$ such that
$C\in L(V,Y)$, the reduced forward operator $\mathbf{F}:Q\to Y$ is well-
defined on $\mathcal{D}(\mathbf{F})$. For simplicity, we again use the same
spaces (but skip the restriction to $\mathcal{D}(\mathbf{F})$) for defining
the all-at-once forward operator $\mathbb{F}$.

##  Appendix B Proofs of some auxiliary results

######  Lemma B.1 ([42], extended to a Banach space setting).

For $X,Y$ Banach spaces, $A,B\in L(X,Y)$ the following equivalence holds

$\textup{rng}(A)=\textup{rng}(B)\ \Longleftrightarrow\ \exists R\in L(X,X):\
R^{-1}\in L(X,X)\mbox{ and }A=BR$

###### Proof.

The implication from right to left is trivial. To show the other direction, we
assume that $\textup{rng}(A)=\textup{rng}(B)$ and first of all define $R:X\to
X/_{\textup{nsp}(A)}$ by assigning to any $x\in X$ the unique element $z:=Rx$
from the quotient space $X/_{\textup{nsp}(A)}$ (which due to the fact that
$\textup{nsp}(A)$ is closed for $A\in L(X,Y)$, is a Banach space as well) such
that $Az=Bx$. To prove that the resulting linear operator $R$ is bounded, we
invoke the Closed Graph Theorem. The graph of $R$ is given by

$\text{Gr}(R)=\\{(x,Rx)\in X\times X/_{\textup{nsp}(A)}\,:\,x\in
X\\}=\\{(x,z)\in X\times X/_{\textup{nsp}(A)}\,:\,Az=Bx\\}.$

For an arbitrary sequence
$(x_{n},z_{n})_{n\in\mathbb{N}}\subseteq\text{Gr}(R)$ with $x_{n}\to x$,
$z_{n}\to z$ in $X$ we have, by continuity of $A$, $B$, that

$\|Az-Bx\|\leq\|Az-Az_{n}\|+\|Bx_{n}-Bx\|\to 0\text{ as }n\to\infty$

and thus $(x,z)\in\text{Gr}(R)$. This implies that $R$ is bounded. Boundedness
of its inverse follows by exchanging the roles of $A$ and $B$. ∎

######  Lemma B.2.

Consider (1.1) and (1.2) under the assumption that $\tfrac{\partial
A}{\partial u}(q,u)$ is an isomorphism for $(q,u)\in U$. Then (1.10) hold for
(1.1) and all $C\in L$ if and only if it holds (1.10) holds for (1.2).

###### Proof.

This can be seen from the chain of equivalences

$\displaystyle\forall C\in L(V,Y)\
:\quad\mathbb{F}^{\prime}(q,u)=\mathbb{F}^{\prime}(q_{0},u_{0})\mathbb{R}(q,u)$
$\displaystyle\Leftrightarrow\ (\underline{dq},\underline{du})\in Q\times V\
:$ $\displaystyle\hskip 42.67912pt\begin{cases}\tfrac{\partial A}{\partial
q}(q,u)\underline{dq}+\tfrac{\partial A}{\partial u}(q,u)\underline{du}\\\
=\tfrac{\partial A}{\partial
q}(q_{0},u_{0})(\mathbb{R}_{qq}(q,u)\underline{dq}+\mathbb{R}_{qu}(q,u)\underline{du})+\tfrac{\partial
A}{\partial
u}(q_{0},u_{0})(\mathbb{R}_{uq}(q,u)\underline{dq}+\mathbb{R}_{uu}(q,u)\underline{du})\\\
C\underline{du}=C(\mathbb{R}_{uq}(q,u)\underline{dq}+\mathbb{R}_{uu}(q,u)\underline{du})\end{cases}$
$\displaystyle\Leftrightarrow\quad\begin{cases}\tfrac{\partial A}{\partial
q}(q,u)=\tfrac{\partial A}{\partial q}(q_{0},u_{0})\mathbb{R}_{qq}(q,u)\\\
\tfrac{\partial A}{\partial u}(q,u)=\tfrac{\partial A}{\partial
q}(q_{0},u_{0})\mathbb{R}_{qu}(q,u)+\tfrac{\partial A}{\partial
u}(q_{0},u_{0})\end{cases}$
$\displaystyle\Leftrightarrow\quad-S^{\prime}(q)=\tfrac{\partial A}{\partial
u}(q,u)^{-1}\tfrac{\partial A}{\partial q}(q,u)=\Bigl{(}\tfrac{\partial
A}{\partial q}(q_{0},u_{0})\mathbb{R}_{qu}(q,u)+\tfrac{\partial A}{\partial
u}(q_{0},u_{0})\Bigr{)}^{-1}\tfrac{\partial A}{\partial
q}(q_{0},u_{0})\mathbb{R}_{qq}(q,u)$ $\displaystyle\hskip
56.9055pt=-\tfrac{\partial A}{\partial u}(q_{0},u_{0})^{-1}\tfrac{\partial
A}{\partial
q}(q_{0},u_{0})\underbrace{\sum_{j=0}^{\infty}\bigl{(}\mathbb{R}_{qu}(q,u)\tfrac{\partial
A}{\partial u}(q_{0},u_{0})^{-1}\tfrac{\partial A}{\partial
q}(q_{0},u_{0})\bigr{)}^{j}\mathbb{R}_{qq}(q,u)}_{=:\mathbf{R}(q)}$
$\displaystyle\hskip 56.9055pt=-S^{\prime}(q^{0})\mathbf{R}(q)$
$\displaystyle\Leftrightarrow\ \forall C\in L(V,Y)\
:\quad\mathbf{F}^{\prime}(q)=\mathbf{F}^{\prime}(q_{0})\mathbf{R}(q)$

for arbitrary $(q,u)=(q,S(q))\in U$, $(q_{0},u_{0})=(q_{0},S(q_{0}))$. Here we
have used the fact that allowing for arbitrary observations enforces the
second component of the operator $\mathbb{R}$ to be the identity on $u$ and
considered the two cases $\underline{du}=0$ and $\underline{dq}=0$ separately.
∎

###### Proof.

(Lemma 2.1) If $\textup{nsp}(K)^{\bot}\subseteq\textup{nsp}(P)$, then we can
estimate

$\displaystyle\|(K^{\star}K+P^{\star}P+\alpha
I)^{-1}K^{\star}Kv\|=\|(K^{\star}K+P^{\star}P+\alpha
I)^{-1}K^{\star}K\textup{Proj}_{\textup{nsp}(K)^{\bot}}v\|$
$\displaystyle=\|(K^{\star}K+P^{\star}P+\alpha
I)^{-1}(K^{\star}K+P^{\star}P)\textup{Proj}_{\textup{nsp}(K)^{\bot}}v\|\leq\|\textup{Proj}_{\textup{nsp}(K)^{\bot}}v\|\leq\|v\|.$

Alternatively, if the symmetric operators $(K^{\star}K)^{1/2}$ and
$P^{\star}P$ commute, then

$\|(K^{\star}K+P^{\star}P+\alpha
I)^{-1}K^{\star}Kv\|=\|(K^{\star}K)^{1/2}(K^{\star}K+P^{\star}P+\alpha
I)^{-1}(K^{\star}K)^{1/2}v\|$

and for $z:=(K^{\star}K+P^{\star}P+\alpha I)^{-1}(K^{\star}K)v$ we have on one
hand $(K^{\star}K+P^{\star}P+\alpha I)z=(K^{\star}K)v$ which is the first
order necessary (and by convexity also sufficient) condition for $z$
minimizing
$\|(K^{\star}K)^{1/2}(\tilde{z}-v)\|^{2}+\|(P^{\star}P)^{1/2}\tilde{z}\|^{2}+\alpha\|\tilde{z}\|^{2}$
with respect to $\tilde{z}$. Comparing the value of this objective function at
$\tilde{z}=z$ with the one at $\tilde{z}=0$, we obtain

$\|(K^{\star}K)^{1/2}(z-v)\|^{2}+\|(P^{\star}P)^{1/2}z\|^{2}+\alpha\|z\|^{2}\leq\|(K^{\star}K)^{1/2}v\|^{2}$

and hence $\|(K^{\star}K)^{1/2}z\|\leq\|(K^{\star}K)^{1/2}v\|$, that is, by
definition of $z$,

$\|(K^{\star}K)^{1/2}(K^{\star}K+P^{\star}P+\alpha
I)^{-1}(K^{\star}K)^{1/2}(K^{\star}K)^{1/2}v\|\leq 2\|(K^{\star}K)^{1/2}v\|$

for any $v\in X$. Extending this estimate by density from the range of
$(K^{\star}K)^{1/2}$ to
$\textup{nsp}((K^{\star}K)^{1/2})^{\bot}=\textup{nsp}(K)^{\bot}$, we arrive at

$\displaystyle\|(K^{\star}K)^{1/2}(K^{\star}K+P^{\star}P+\alpha
I)^{-1}(K^{\star}K)^{1/2}w\|$
$\displaystyle=\|(K^{\star}K)^{1/2}(K^{\star}K+P^{\star}P+\alpha
I)^{-1}(K^{\star}K)^{1/2}\textup{Proj}_{\textup{nsp}((K^{\star}K)^{1/2})^{\bot}}w\|$
$\displaystyle\leq
2\|\textup{Proj}_{\textup{nsp}((K^{\star}K)^{1/2})^{\bot}}w\|\leq
2\|w\|\quad\forall w\in X,$

which yields $\|(K^{\star}K)^{1/2}(K^{\star}K+P^{\star}P+\alpha
I)^{-1}(K^{\star}K)^{1/2}\|\leq 2.$

In both cases of (2.13) we get

$\displaystyle\|K(K^{\star}K+P^{\star}P+\alpha
I)^{-1}v\|^{2}=\langle(K^{\star}K+P^{\star}P+\alpha
I)^{-1}v,K^{\star}K(K^{\star}K+P^{\star}P+\alpha
I)^{-1}v\rangle\leq\frac{C}{\alpha}\|v\|^{2}.$

∎

##  Appendix C Two further Newton type methods

We return to the variational approach (2.4), (2.6), and define an alternative
regularized Newton type method for (2.3) by linearizing under the norm in
(2.4).

Assuming that $r$ is Gâteaux differentiable, $U$ is weakly closed (e.g., $U$
is closed and convex) and $\mathcal{P}$ is weakly lower semicontinuous, the
iterates $(\hat{r}_{n+1}^{\delta},x_{n+1}^{\delta})\in\widetilde{X}\times U$
are well-defined by

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

with some $p,b\in[1,\infty)$, $\alpha_{n},\beta_{n}>0$. As above 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 $n\to\infty$.

Comparison with $(r(x^{\dagger}),x^{\dagger})\in\widetilde{X}\times U$ and
using (1.4), (2.1) and $\mathcal{P}(x^{\dagger})=0$ then yields

$\displaystyle\|K(\hat{r}_{n+1}^{\delta}-r(x^{\dagger}))+y-y^{\delta}\|_{Y}^{p}+\alpha_{n}\Bigl{(}\mathcal{R}(\hat{r}_{n+1}^{\delta})-\mathcal{R}(r(x^{\dagger}))\Bigr{)}$
(C.2)
$\displaystyle+\beta_{n}\Bigl{(}\|r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x^{\dagger})+\textup{res}_{\textup{Taylor}}+r(x^{\dagger})-\hat{r}_{n+1}^{\delta}\|_{\widetilde{X}}^{b}-\|\textup{res}_{\textup{Taylor}}\|_{\widetilde{X}}^{b}\Bigr{)}+\mathcal{P}(x_{n+1}^{\delta})$
$\displaystyle\leq\delta^{p}+\eta_{n},$

where we have used

$\displaystyle\textup{res}_{\textup{Taylor}}:=$ $\displaystyle
r(x_{n}^{\delta})-r(x^{\dagger})+r^{\prime}(x_{n}^{\delta})(x^{\dagger}-x_{n}^{\delta}),$
$\displaystyle\textup{res}_{\textup{Newton}}:=$ $\displaystyle
r(x_{n}^{\delta})-r(x^{\dagger})+r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x_{n}^{\delta})=r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x^{\dagger})+\textup{res}_{\textup{Taylor}}.$

However, the connection between $\hat{r}_{n+1}^{\delta}$ and $r(x^{\dagger})$
by (C.2) is likely too weak to imply smallness of the Taylor remainder
$\textup{res}_{\textup{Taylor}}$. Still, exact penalization, that is, the
choice $b=1$, $\beta_{n}$ sufficiently large, enforces the corresponding
constraint
$r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x_{n}^{\delta})-\hat{r}_{n+1}^{\delta}=0$
and thus, inserting the resulting expression for $\hat{r}_{n+1}^{\delta}$ into
the data misfit term and the regularization term, the definition of the Newton
iterate reduces to

$\displaystyle x_{n+1}^{\delta}\in\mbox{argmin}_{x\in\widetilde{X}\times
U}J_{n,1}^{\delta}(x)$ (C.3) $\displaystyle\mbox{where
}J_{n,1}^{\delta}(x):=\|K(r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x-x_{n}^{\delta}))+F(x_{0})-y^{\delta}\|_{Y}^{p}$
$\displaystyle\hskip
113.81102pt+\alpha_{n}\mathcal{R}(r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x-x_{n}^{\delta}))+\mathcal{P}(x),$

where the data misfit term can be rewritten as
$K(r(x_{n}^{\delta})+r^{\prime}(x_{n}^{\delta})(x-x_{n}^{\delta}))+F(x_{0})-y^{\delta}=Kr^{\prime}(x_{n}^{\delta})(x-x_{n}^{\delta})+F(x_{n}^{\delta})-y^{\delta}=F(x_{n}^{\delta})+F^{\prime}(x_{n}^{\delta})(x-x_{n}^{\delta})-y^{\delta}$.
Thus, as compared to (2.8), we work with an enhanced misfit term and have more
freedom to choose the space $\tilde{X}$. As compared to the Levenberg-
Marquardt Method (or the iteratively regularized Gauss-Newton method), see,
e.g. [19, 20, 30, 50], we work with a more structure adapted regularization
term.

Comparing the value of $J_{n,1}^{\delta}$ at the approximate minimizer
$x_{n+1}^{\delta}$ with the one at $x^{\dagger}$ we obtain

$\displaystyle\|K(r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x^{\dagger})+\textup{res}_{\textup{Taylor}})+y-y^{\delta}\|_{Y}^{p}-\|K\textup{res}_{\textup{Taylor}}+y-y^{\delta}\|_{Y}^{p}$
$\displaystyle+\alpha_{n}\Bigl{(}\mathcal{R}(r(x^{\dagger})+r^{\prime}(x_{n}^{\delta})(x_{n+1}^{\delta}-x^{\dagger})+\textup{res}_{\textup{Taylor}})-\mathcal{R}(r(x^{\dagger})+\textup{res}_{\textup{Taylor}})\Bigr{)}+\mathcal{P}(x_{n+1}^{\delta})\leq\delta^{p}.$

Deriving a contractive recursion on $e_{n}=x_{n}^{\delta}-x^{\dagger}$ from
this would require Taylor remainder estimates of the form

$\|\textup{res}_{\textup{Taylor}}\|_{X}\leq
c\|r^{\prime}(x_{n-1}^{\delta})(x_{n}^{\delta}-x^{\dagger})\|_{\widetilde{X}},\quad\|K\textup{res}_{\textup{Taylor}}\|_{Y}\leq
c\|Kr^{\prime}(x_{n-1}^{\delta})(x_{n}^{\delta}-x^{\dagger})\|_{Y}$

for some $c\in(0,1)$, where the latter is an estimate in image space and thus
requires an assumption similar to the tangential cone condition (1.5). Indeed,
to the best of the author’s knowledge, convergence results of Newton type
methods in a general Banach space setting have so far only been achieved under
conditions of the type (1.5), but not yet under a range invariance condition
on $F^{\prime}$. We will thus, like in Section 2.2.1, also here restrict
ourselves to a Hilbert space setting with $p=2$,
$\mathcal{R}(\hat{r})=\|\hat{r}\|_{\widetilde{X}}^{2}$,
$\mathcal{P}(x)=\|Px\|_{Z}^{2}$, in order to be able to apply spectral theory
for bounded selfadjoint operators. The solution to (C.3) can then be
explicitely expressed via the first order optimality condition

$\displaystyle x_{n+1}^{\delta}=x_{n}^{\delta}+$
$\displaystyle((KR_{n})^{\star}KR_{n}+P^{\star}P+\alpha_{n}R_{n}^{\star}R_{n})^{-1}$
(C.4)
$\displaystyle\Bigl{(}(KR_{n})^{\star}(y^{\delta}-Kr(x_{n}^{\delta})-F(x_{0}))-P^{\star}Px_{n}^{\delta}-\alpha_{n}R_{n}^{\star}r(x_{n}^{\delta})\Bigr{)}$

with $R_{n}:=r^{\prime}(x_{n}^{\delta})$. Due to the fact that
$Px^{\dagger}=0$, the error is thus given by

$\displaystyle x_{n+1}^{\delta}-x^{\dagger}$
$\displaystyle=((KR_{n})^{\star}KR_{n}+P^{\star}P+\alpha_{n}R_{n}^{\star}R_{n})^{-1}\Bigl{(}(KR_{n})^{\star}(y^{\delta}-y)-((KR_{n})^{\star}K+\alpha_{n}R_{n}^{\star})\textup{res}_{\textup{Taylor}}-\alpha_{n}R_{n}^{\star}r(x^{\dagger})\Bigr{)}$
$\displaystyle=(R_{n}^{\star}(K^{\star}K+P_{n}^{\star}P_{n}+\alpha_{n}\textup{id})R_{n})^{-1}\Bigl{(}R_{n}^{\star}K^{\star}(y^{\delta}-y)-R_{n}^{\star}(K^{\star}K+\alpha_{n}\textup{id})\textup{res}_{\textup{Taylor}}-\alpha_{n}R_{n}^{\star}r(x^{\dagger})\Bigr{)}$
$\displaystyle=R_{n}^{-1}(K^{\star}K+P_{n}^{\star}P_{n}+\alpha_{n}\textup{id})^{-1}\Bigl{(}K^{\star}(y^{\delta}-y)-(K^{\star}K+\alpha_{n}\textup{id})\textup{res}_{\textup{Taylor}}-\alpha_{n}r(x^{\dagger})\Bigr{)}$

with $R_{\dagger}=r^{\prime}(x^{\dagger})$, $P_{n}:=PR_{n}^{-1}$. From Lemma
2.1, assuming

$\displaystyle r^{\prime}(x_{0})^{-1}\in L(\widetilde{X},\widehat{X})\textup{
and }$ (C.5) $\displaystyle\exists L_{r}>0\,\forall x\in U\,:\
\|r^{\prime}(x^{\dagger})-r^{\prime}(x)\|_{L(\widehat{X},\widetilde{X})}\leq
L_{r}\|x^{\dagger}-x\|_{\widehat{X}}<1,$

with another auxiliary space $\widehat{X}$, we obtain

$\|x_{n+1}^{\delta}-x^{\dagger}\|_{\widehat{X}}\leq\|R_{n}^{-1}\|\Bigl{(}\frac{\delta}{\sqrt{\alpha_{n}}}+\|\textup{res}_{\textup{Taylor}}\|_{\widetilde{X}}+a_{n}(P_{n})\Bigr{)}$
(C.6)

with

$a_{n}(P_{n})=\alpha_{n}\|(K^{\star}K+P_{n}^{\star}P_{n}+\alpha_{n}\textup{id})^{-1}r(x^{\dagger})\|_{\widetilde{X}},$

provided (cf. Lemma 2.1)

$\textup{(a) }\textup{nsp}(K)^{\bot}\subseteq\textup{nsp}(P_{n})\textup{ or \
(b)
}P_{n}^{\star}P_{n}(K^{\star}K)^{1/2}=(K^{\star}K)^{1/2}P_{n}^{\star}P_{n}.$
(C.7)

From this we can conclude the following convergence result.

######  Theorem 3.

Let $x_{0}\in U:=\mathcal{B}_{\rho}^{\widehat{X}}(x^{\dagger})$ for some
$\rho>0$ sufficiently small. Assume that $F$ satisfies (2.1) with $r$ Gâteaux
differentiable and satisfying (C.5),
$r(x^{\dagger})\in\bigl{(}\mathcal{N}(K)\cap\mathcal{N}(PR(x^{\dagger})^{-1}\bigr{)}^{\bot}$,
$K\in L(\widetilde{X},Y)$ and

$\forall x\in U\,:\ \textup{(a)
}\mathcal{N}(K)^{\bot}=\mathcal{N}(P_{x})\textup{ or \ (b)
}(P_{x}^{\star}P_{x})^{1/2}(K^{\star}K)^{1/2}=(K^{\star}K)^{1/2}(P_{x}^{\star}P_{x})^{1/2}$
(C.8)

with $P_{x}:=Pr^{\prime}(x)^{-1}$. Let the stopping index
$n_{*}=n_{*}(\delta)$ be chosen according to (2.15) with $c$ replaced by
$\bar{c}:=2L_{r}\rho<1$, $L_{r}$ as in (C.5).

Then the iterates $(x_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are
well-defined by (C.1), remain in
$\mathcal{B}_{\rho}^{\widehat{X}}(x^{\dagger})$ and converge in $\widehat{X}$,
$\|x_{n_{*}(\delta)}^{\delta}-x^{\dagger}\|_{\widehat{X}}\to 0$ as $\delta\to
0$. In the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we have
$\|x_{n}-x^{\dagger}\|_{\widehat{X}}\to 0$ as $n\to\infty$.

###### Proof.

We estimate $a_{n}(P_{n})$ by

$\displaystyle a_{n}(P_{n})\leq$
$\displaystyle\|\alpha_{n}(K^{\star}K+P_{n}^{\star}P_{n}+\alpha_{n}\textup{id})^{-1}(P_{\dagger}^{\star}P_{\dagger}-P_{n}^{\star}P_{n})(K^{\star}K+P_{\dagger}^{\star}P_{\dagger}+\alpha_{n}\textup{id})^{-1}r(x^{\dagger})\|_{\widetilde{X}}+a_{n}(P_{\dagger})$

with the shorthand notation $P_{n}=Pr^{\prime}(x_{n}^{\delta})^{-1}$,
$P_{\dagger}=Pr^{\prime}(x^{\dagger})^{-1}$ and

$\displaystyle
a_{n}(P_{\dagger})=\|\alpha_{n}(K^{\star}K+P_{\dagger}^{\star}P_{\dagger}+\alpha_{n}\textup{id})^{-1}r(x^{\dagger})\|_{\widetilde{X}}\to
0\mbox{ as }n\to\infty$ $\displaystyle
P_{\dagger}^{\star}P_{\dagger}-P_{n}^{\star}P_{n}=P_{n}^{\star}P_{n}(r^{\prime}(x_{n})r^{\prime}(x^{\dagger})^{-1}-\textup{id})+(\textup{id}-r^{\prime}(x_{n}^{\delta})^{-1\,\star}r^{\prime}(x^{\dagger})^{\star})P_{\dagger}^{\star}P_{\dagger}.$

Due to Lemma 2.1 (with reversed roles of $K$ and $P$) we have

$\displaystyle\|(K^{\star}K+P_{n}^{\star}P_{n}+\alpha_{n}\textup{id})^{-1}P_{n}^{\star}P_{n}\|\leq
C,\quad$
$\displaystyle\alpha_{n}\|(K^{\star}K+P_{n}^{\star}P_{n}+\alpha_{n}\textup{id})^{-1}\|\leq
1,$
$\displaystyle\|P_{\dagger}^{\star}P_{\dagger}(K^{\star}K+P_{\dagger}^{\star}P_{\dagger}+\alpha_{n}\textup{id})^{-1}\|\leq
C,\quad$
$\displaystyle\alpha_{n}\|(K^{\star}K+P_{\dagger}^{\star}P_{\dagger}+\alpha_{n}\textup{id})^{-1}\|\leq
1.$

Moreover,

$\displaystyle\|r^{\prime}(x_{n}^{\delta})r^{\prime}(x^{\dagger})^{-1}-\textup{id}\|=\|(r^{\prime}(x_{n}^{\delta})-r^{\prime}(x^{\dagger}))r^{\prime}(x^{\dagger})^{-1}\|\leq
c\|r^{\prime}(x^{\dagger})^{-1}\|\,\|x_{n}^{\delta}-x^{\dagger}\|_{X}$
$\displaystyle\|\textup{id}-r^{\prime}(x_{n}^{\delta})^{-1\,\star}r^{\prime}(x^{\dagger})^{\star}\|=\|(r^{\prime}(x_{n}^{\delta})-r^{\prime}(x^{\dagger}))r^{\prime}(x_{n}^{\delta})^{-1}\|\leq
c\|r^{\prime}(x_{n}^{\delta})^{-1}\|\,\|x_{n}^{\delta}-x^{\dagger}\|_{X},$

and due (C.5), the Taylor remainder can be estimated by means of

$\|r(x)-r(x^{\dagger})+r^{\prime}(x)(x^{\dagger}-x)\|_{\widetilde{X}}\leq\frac{L_{r}}{2}\|x^{\dagger}-x\|_{\widehat{X}}^{2},$
(C.9)

which yields an estimate of the form

$\|x_{n+1}^{\delta}-x^{\dagger}\|_{\widehat{X}}\leq
C\Bigl{(}\frac{\delta}{\sqrt{\alpha_{n}}}+\|x_{n}^{\delta}-x^{\dagger}\|_{\widehat{X}}^{2}+a_{n}(P_{\dagger})\Bigr{)}.$

Thus, by choosing $\rho$ sufficiently small, we can conclude from
$x_{n}\in\mathcal{B}_{\rho}(x^{\dagger})$ and (C.6) that for some $C>0$,
$\bar{c}\in(0,1)$ independent of $n$,

$\|x_{n+1}^{\delta}-x^{\dagger}\|_{\widehat{X}}\leq
C\frac{\delta}{\sqrt{\alpha_{n}}}+\bar{c}\|x_{n}^{\delta}-x^{\dagger}\|_{\widehat{X}}+Ca_{n}(P_{\dagger}).$
(C.10)

Thus $x_{n+1}\in\mathcal{B}_{\rho}(x^{\dagger})$ and as in the proof of
Theorem 2, the assertions follow. ∎

To arrive at a frozen Newton method in place of (C.4), we replace
$r^{\prime}(x_{n})$ by $r^{\prime}(x_{0})$ in the above

$\displaystyle x_{n+1}^{\delta}=x_{n}^{\delta}+$
$\displaystyle((KR_{0})^{\star}KR_{0}+P^{\star}P+\alpha_{n}R_{0}^{\star}R_{0})^{-1}$
(C.11)
$\displaystyle\Bigl{(}(KR_{0})^{\star}(y^{\delta}-Kr(x_{n}^{\delta})-F(x_{0}))-P^{\star}Px_{n}^{\delta}-\alpha_{n}R_{0}^{\star}r(x_{n}^{\delta})\Bigr{)}$

with $R_{0}=r^{\prime}(x_{0})$, for which we obtain the estimate

$\|x_{n+1}^{\delta}-x^{\dagger}\|_{\widehat{X}}\leq\|r^{\prime}(x_{0})^{-1}\|_{L(\widetilde{X},\widehat{X})}\Bigl{(}\frac{\delta}{\sqrt{\alpha_{n}}}+c_{0}\|x_{n}^{\delta}-x^{\dagger}\|_{\widehat{X}}+a_{n}(PR_{0}^{-1})\Bigr{)}$
(C.12)

with

$a_{n}(PR_{0}^{-1})=\alpha_{n}\|(K^{\star}K+(PR_{0}^{-1})^{\star}PR_{0}^{-1}+\alpha_{n}\textup{id})^{-1}r(x^{\dagger})\|_{\widetilde{X}}\to
0\mbox{ as }n\to\infty$

provided
$r(x^{\dagger})\in(\textup{nsp}(K)\cap\textup{nsp}(PR_{0}^{-1}))^{\bot}\subseteq\textup{nsp}(K^{\star}K+(PR_{0}^{-1})^{\star}PR_{0}^{-1})^{\bot}$
and bounded invertibility as well as a Taylor remainder estimate hold

$\displaystyle r^{\prime}(x_{0})^{-1}\in L(\widetilde{X},\widehat{X})\textup{
and }$ (C.13) $\displaystyle\exists c_{0}\in(0,1)\,\forall x\in U\,:\
\|r(x)-r(x^{\dagger})+r^{\prime}(x_{0})(x^{\dagger}-x)\|_{\widetilde{X}}\leq
c_{0}\|x^{\dagger}-x\|_{\widehat{X}}.$

Similarly to Theorem 3 we obtain

######  Theorem 4.

Let $x_{0}\in U:=\mathcal{B}_{\rho}^{\widehat{X}}(x^{\dagger})$ for some
$\rho>0$ sufficiently small. Assume that $F$ satisfies (2.1) with (C.13),
$r(x^{\dagger})\in\bigl{(}\textup{nsp}(K)\cap\textup{nsp}(PR(x_{0})^{-1})\bigr{)}^{\bot}$,
$K\in L(\widetilde{X},Y)$, and

$\textup{(a) }\textup{nsp}(K)^{\bot}\subseteq\textup{nsp}(P_{0})\textup{ or \
(b)
}P_{0}^{\star}P_{0}(K^{\star}K)^{1/2}=(K^{\star}K)^{1/2}P_{0}^{\star}P_{0}$
(C.14)

with $P_{0}:=Pr^{\prime}(x_{0})^{-1}$. Let the stopping index
$n_{*}=n_{*}(\delta)$ be chosen according to (2.15) with $c$ replaced by
$c_{0}$ and $c_{0}$ as in (C.13).

Then the iterates $(x_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are
well-defined by (C.11), remain in
$\mathcal{B}_{\rho}^{\widehat{X}}(x^{\dagger})$ and converge in $X$,
$\|x_{n_{*}(\delta)}^{\delta}-x^{\dagger}\|_{\widehat{X}}\to 0$ as $\delta\to
0$. In the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we have
$\|x_{n}-x^{\dagger}\|_{\widehat{X}}\to 0$ as $n\to\infty$.

Note that in case of the canonical choice $K=F^{\prime}(x_{0})$ we have
$r^{\prime}(x_{0})=\textup{id}$ and therefore (C.13) enforces
$\widehat{X}=\widetilde{X}$.

###  C.1 Convergence in the setting of Section 3

In the reduced setting of Section 3.1, Theorems 3, 4 yield the following two
corollaries.

######  Corollary C.1 (Newton, reduced).

Let $q_{0}\in
U:=\mathcal{B}_{\rho}^{\widetilde{Q}_{J}}(q^{\dagger})\subseteq\mathcal{D}(\mathbf{F})$
for some $\rho>0$ sufficiently small, assume that $B:V_{J}\to
L(\widetilde{Q}_{J},W^{*}_{J})$ is Lipschitz continuously differentiable, and
$D\in L(V_{J},W^{*}_{J})$. Moreover, let
$r(q^{\dagger})\in\bigl{(}\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))\cap\textup{nsp}(Pr^{\prime}(q^{\dagger})^{-1})\bigr{)}^{\bot}$,
(C.8) hold with $K=\mathbf{F}^{\prime}(q_{0})$, and the stopping index
$n_{*}=n_{*}(\delta)$ be chosen according to (2.15).
Then the iterates $(q_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are
well-defined by (C.4), (3.9), remain in
$\mathcal{B}_{\rho}^{\widetilde{Q}_{J}}(q^{\dagger})$ and converge in
$\widetilde{Q}_{J}$,
$\|q_{n_{*}(\delta)}^{\delta}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$ as
$\delta\to 0$. In the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we
have $\|q_{n}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$ as $n\to\infty$.

###### Proof.

To establish (C.5), note that $K=\mathbf{F}(q_{0})$, thus
$r^{\prime}(q_{0})=\textup{id}$ and we have set
$\widehat{X}=\widetilde{X}=\widetilde{Q}_{J}$. Moreover, with
$\underline{dr}:=\bigl{(}{r^{(0)}_{\textup{diff}}}^{\prime}(q^{\dagger},S(q^{\dagger});q_{0},S(q_{0}))-{r^{(0)}_{\textup{diff}}}^{\prime}(q,S(q);q_{0},S(q_{0}))\bigr{)}\underline{dq}$
we have

$\|(r^{\prime}(q^{\dagger})-r^{\prime}(q))\underline{dq}\|_{\widetilde{Q}_{J}}=\|\underline{dr}\|_{\widetilde{Q}^{(0)}_{J}}\sim\|B_{0}\underline{dr}\|_{W^{*}_{J}}\,,$

due to the assumed isomorphism property of $B_{0}$. With $u=S(q)$,
$u^{\dagger}=S(q^{\dagger})$, $v=S^{\prime}(q)\underline{dq}$,
$v^{\dagger}=S^{\prime}(q^{\dagger})\underline{dq}$ we can rewrite

$\displaystyle B_{0}\underline{dr}=$
$\displaystyle[B^{\prime}(u^{\dagger})v^{\dagger}]q^{\dagger}-[B^{\prime}(u_{0})v^{\dagger}]q_{0}+B(u^{\dagger})\underline{dq}-([B^{\prime}(u)v]q-[B^{\prime}(u_{0})v]q_{0}+B(u)\underline{dq})$
$\displaystyle=$
$\displaystyle[(B^{\prime}(u^{\dagger})-B^{\prime}(u))v^{\dagger}]q^{\dagger}+[B^{\prime}(u)(v^{\dagger}-v)]q^{\dagger}+[B^{\prime}(u)v](q^{\dagger}-q)$
$\displaystyle-[B^{\prime}(u_{0})(v^{\dagger}-v)]q_{0}+[B(u^{\dagger})-B(u)]\underline{dq}$

where as in (3.14), (3.10)

$\displaystyle
u^{\dagger}-u=-\Bigl{(}D+[\overline{B^{\prime}}\cdot]q\Bigr{)}^{-1}B(u^{\dagger})(q^{\dagger}-q),$
$\displaystyle
v^{\dagger}=-\Bigl{(}D+[B^{\prime}(u^{\dagger})\cdot]q^{\dagger}\Bigr{)}^{-1}B(u^{\dagger})\underline{dq},\quad
v=-\Bigl{(}D+[B^{\prime}(u)\cdot]q\Bigr{)}^{-1}B(u)\underline{dq}$

with
$[\overline{B^{\prime}}\underline{du}]q=\int_{0}^{1}[B^{\prime}(u+\theta(u^{\dagger}-u))\underline{du}]q\,d\theta$.
Thus using our assumption of $B$ being Lipschitz continuously differentiable
as an operator $V_{J}\to L(\widetilde{Q}_{J},W^{*}_{J})$, we can estimate
$\|B_{0}\underline{dr}\|_{W^{*}_{J}}\leq
L_{r}\|q^{\dagger}-q\|_{\widetilde{Q}_{J}}$ for some constant $L_{r}$. ∎

######  Corollary C.2 (alternative frozen Newton, reduced).

Let $q_{0}\in
U:=\mathcal{B}_{\rho}^{\widetilde{Q}_{J}}(q^{\dagger})\subseteq\mathcal{D}(\mathbf{F})$
for some $\rho>0$ sufficiently small, assume that $B:V_{J}\to
L(\widetilde{Q}_{J},W^{*}_{J})$ is continuously differentiable, and $D\in
L(V_{J},W^{*}_{J})$. Moreover, let
$r(q^{\dagger})\in\bigl{(}\textup{nsp}(\mathbf{F}^{\prime}(q_{0}))\cap\textup{nsp}(P)\bigr{)}^{\bot}$,
(C.14) hold with $K=\mathbf{F}^{\prime}(q_{0})$, $P_{0}:=P$ and the stopping
index $n_{*}=n_{*}(\delta)$ be chosen according to (2.15).
Then the iterates $(q_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are
well-defined by (C.11), (3.9), remain in
$\mathcal{B}_{\rho}^{\widetilde{Q}_{J}}(q^{\dagger})$ and converge in
$\widetilde{Q}_{J}$,
$\|q_{n_{*}(\delta)}^{\delta}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$ as
$\delta\to 0$. In the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we
have $\|q_{n}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$ as $n\to\infty$.

###### Proof.

Also here, $r^{\prime}(x_{0})=\textup{id}$ and we only need to estimate the
Taylor remainder (C.13) for the nonlinear part $r^{(0)}_{\textup{diff}}$ of
$r$, which due to the fact that

$\frac{d}{dq}r^{(0)}_{\textup{diff}}(q_{0},S(q_{0});q_{0},u_{0})\underline{dq}=\frac{\partial
r^{(0)}_{\textup{diff}}}{\partial
q}(q_{0},S(q_{0});q_{0},u_{0})\underline{dq}+\frac{\partial
r^{(0)}_{\textup{diff}}}{\partial
u}(q_{0},S(q_{0});q_{0},u_{0})S^{\prime}(q_{0})\underline{dq}$

where

$\displaystyle\frac{\partial r^{(0)}_{\textup{diff}}}{\partial
q}(q_{0},S(q_{0});q_{0},u_{0})\underline{dq}=\Bigl{[}B_{0}^{-1}\Bigl{(}[B(u)-B(u_{0})](q-q_{0})\Bigr{)}\Bigr{]}_{q=q_{0},u=u_{0}}=0$
(C.15) $\displaystyle\frac{\partial r^{(0)}_{\textup{diff}}}{\partial
q}(q_{0},S(q_{0});q_{0},u_{0})\underline{dq}$
$\displaystyle=\Bigl{[}B_{0}^{-1}\Bigl{(}[B^{\prime}(u)\underline{du}-B^{\prime}(u_{0})\underline{du}]q_{0}+[B^{\prime}(u)\underline{du}](q-q_{0})\Bigr{)}\Bigr{]}_{q=q_{0},u=u_{0}}=0$

is just equal to (3.24). Using (3.25), the assumed isomorphism property of
$B_{0}$ and the assumed continuous differentiability of $B$ as an operator
$V_{J}\to L(\widetilde{Q}_{J},W^{*}_{J})$ in (3.24) yields the desired
estimate. ∎

In the all-at-once setting of Section 3.2 we obtain the following.

######  Corollary C.3 (Newton, all-at-once).

Let $(q_{0},u_{0})\in U:=\mathcal{B}_{\rho}^{\widetilde{Q}_{J}\times
V_{J}}(q^{\dagger},u^{\dagger})\subseteq\widetilde{Q}_{J}\times V_{J}$ for
some $\rho>0$ sufficiently small, assume that $B:V_{J}\to
L(\widetilde{Q}_{J},W^{*}_{J})$ is Lipschitz continuously differentiable, and
$D\in L(V_{J},W^{*}_{J})$. Moreover, let
$r(q^{\dagger},u^{\dagger})\in\bigl{(}\textup{nsp}(\mathbb{F}^{\prime}(q_{0},u_{0}))\cap\textup{nsp}(Pr^{\prime}(q^{\dagger},u^{\dagger})^{-1})\bigr{)}^{\bot}$,
(C.8) hold with $K=\mathbb{F}^{\prime}(q_{0},u_{0})$, (3.27) and the stopping
index $n_{*}=n_{*}(\delta)$ be chosen according to (2.15).
Then the iterates
$(q_{n}^{\delta},u_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are well-
defined by (C.4), (3.26), remain in
$\mathcal{B}_{\rho}^{\widetilde{Q}_{J}\times V_{J}}(q^{\dagger},u^{\dagger})$
and converge in $\widetilde{Q}_{J}\times V_{J}$,
$\|q_{n_{*}(\delta)}^{\delta}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$,
$\|u_{n_{*}(\delta)}^{\delta}-u^{\dagger}\|_{V_{J}}\to 0$ as $\delta\to 0$. In
the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we have
$\|q_{n}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$,
$\|u_{n}-u^{\dagger}\|_{V_{J}}\to 0$ as $n\to\infty$.

###### Proof.

We obtain (C.5) by estimating

$\displaystyle\|(r^{\prime}(q^{\dagger},u^{\dagger})-r^{\prime}(q,u))(\underline{dq},\underline{du})\|_{\widetilde{Q}_{J}\times
V_{J}}$
$\displaystyle=\|\bigl{(}{r^{(0)}_{\textup{diff}}}^{\prime}(q^{\dagger},S(q^{\dagger});q_{0},S(q_{0}))-{r^{(0)}_{\textup{diff}}}^{\prime}(q,S(q);q_{0},S(q_{0}))\bigr{)}(\underline{dq},\underline{du})\|_{\widetilde{Q}^{(0)}_{J}}$
$\displaystyle\leq\|B_{0}^{-1}\|_{W^{*},\widetilde{Q}^{(0)}_{J}}\|[(B^{\prime}(u^{\dagger})-B^{\prime}(u))\underline{du}]q^{\dagger}+[B^{\prime}(u)\underline{du}](q^{\dagger}-q)+[B(u^{\dagger})-B(u)]\underline{dq}\|_{W^{*}_{J}}.$

∎

######  Corollary C.4 (alternative frozen Newton, all-at-once).

Let $(q_{0},u_{0})\in U:=$
$\mathcal{B}_{\rho}^{\widetilde{Q}_{J}\times
V_{J}}(q^{\dagger},u^{\dagger})\subseteq\widetilde{Q}_{J}\times V_{J}$ for
some $\rho>0$ sufficiently small, assume that $B:V_{J}\to
L(\widetilde{Q}_{J},W^{*}_{J})$ is continuously differentiable, and $D\in
L(V_{J},W^{*}_{J})$. Moreover, let
$r(q^{\dagger},u^{\dagger})\in\bigl{(}\textup{nsp}(\mathbb{F}^{\prime}(q_{0},u_{0}))\cap\textup{nsp}(Pr^{\prime}(q_{0},u_{0})^{-1})\bigr{)}^{\bot}$,
(C.14) hold with $K=\mathbb{F}^{\prime}(q_{0},u_{0})$,
$P_{0}:=Pr^{\prime}(q_{0},u_{0})^{-1}$ and the stopping index
$n_{*}=n_{*}(\delta)$ be chosen according to (2.15).
Then the iterates
$(q_{n}^{\delta},u_{n}^{\delta})_{n\in\\{1,\ldots,n_{*}(\delta)\\}}$ are well-
defined by (C.11), (3.26), remain in
$\mathcal{B}_{\rho}^{\widetilde{Q}_{J}\times V_{J}}(q^{\dagger},u^{\dagger})$
and converge in $\widetilde{Q}_{J}\times V_{J}$,
$\|q_{n_{*}(\delta)}^{\delta}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$,
$\|u_{n_{*}(\delta)}^{\delta}-u^{\dagger}\|_{V_{J}}\to 0$ as $\delta\to 0$. In
the noise free case $\delta=0$, $n_{*}(\delta)=\infty$ we have
$\|q_{n}-q^{\dagger}\|_{\widetilde{Q}_{J}}\to 0$,
$\|u_{n}-u^{\dagger}\|_{V_{J}}\to 0$ as $n\to\infty$.

###### Proof.

Again the crucial estimate to establish (C.13) is the one on the Taylor
remainder of $\textup{res}_{0\textup{Taylor}}(r^{(0)}_{\textup{diff}})$ and
due to (C.15) equals (3.24). ∎

##  Appendix D Linearized uniqueness proofs

###  D.1 Nullspace condition (3.6) for the elliptic case in Example 5

We use the fact that $\mathbf{F}^{\prime}(q_{0})\underline{dq}=0$ with
$\underline{du}^{j}=(S^{j})^{\prime}(q_{0})\underline{dq}$,
$u_{0}^{j}=S^{j}(q_{0})$ reads as

$\int_{\Omega}\Bigl{(}\nabla\underline{du}^{j}\cdot\nabla
w+q_{0}\underline{du}^{j}w+\underline{dq}u_{0}^{j}w\Bigr{)}\,dx=0\,,\forall
w\in
W:=H^{1}(\Omega),\quad\textup{tr}_{\partial\Omega}\underline{du}^{j}=0\qquad
j\in\mathbb{N}.$

From this, using integration by parts, we conclude that

$\forall j\in\mathbb{N}\,\forall w\in M_{q_{0}}:=\\{v\in
H^{2}(\Omega)\,:\,-\Delta v+q_{0}\cdot
v=0\\}\,:\int_{\Omega}\underline{dq}u^{j}w\,dx=0.$

(Note that no boundary conditions are prescribed on $v\in M_{q_{0}}$.) Since
the boundary values $\varphi^{j}$ of $u_{0}^{j}$ run over a basis of
$H^{-1/2}(\partial\Omega)$, this is equivalent to

$\forall w_{1},\,w_{2}\,\in
M_{q_{0}}:\int_{\Omega}\underline{dq}\,w_{1}\,w_{2}\,dx=0,$