# Interface Identification constrained by Local-to-Nonlocal Coupling

Matthias Schuster  Universitaet Trier, D-54286 Trier, Germany; Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Volker
Schulz11footnotemark: 1

Abstract. Models of physical phenomena that use nonlocal operators are better
suited for some applications than their classical counterparts that employ
partial differential operators. However, the numerical solution of these
nonlocal problems can be quite expensive. Therefore, Local-to-Nonlocal
couplings have emerged that combine partial differential operators with
nonlocal operators. In this work, we make use of an energy-based Local-to-
Nonlocal coupling that serves as a constraint for an interface identification
problem.
Keywords. Shape optimization, energy-based Local-to-Nonlocal coupling,
interface identification, Schwarz method.

##  1 Introduction

Nonlocal models have been successfully applied to a broad range of
applications, e.g. anomalous or fractional diffusion[10, 39, 15],
peridynamics[35, 23], finance and jump processes[27, 17, 12, 21], machine
learning[13, 42] and image denoising[11, 19]. Typically, nonlocal operators
are integral operators that are dependent on a function $\gamma$, which is
called _kernel_. As a result, nonlocal operators allow interactions between
different points in space and therefore some physical phenomena are better
described in this nonlocal regime. One drawback is, that the computation of
solutions regarding nonlocal equations can be quite costly. If, e.g. we apply
the finite element method we need to assemble a stiffness matrix that is
typically dense for nonlocal operators. Additionally, the weak formulation, as
we see in Section 2, consists of double integrals, which are computationally
more demanding than single integrals that typically represent weak
formulations of PDEs. Since these stiffness matrices are sparse for PDEs and
some physical processes can also be modeled by PDEs, the coupling of nonlocal
and ’local’ partial differential operators is one possibility to develop a
model that is accurate but at the same time computationally not too expensive.
For more information on these Local-to-Nonlocal couplings we refer to the
review paper [18] and the references therein. As a result, the question
arises, on which part of the domain the nonlocal operator should be deployed,
which leads us to shape optimization and specifically to an interface
identification problem as a first step to model this problem. An overview on
the topic of shape optimization can be found in, e.g. [36, 14, 4, 22], and a
similar interface identification problem constrained by partial differential
equations is investigated in [41]. For an analysis of the interface
identification problem constrained by completely nonlocal equations we refer
to [33].
The paper is organized as follows: First, we recall nonlocal Dirichlet
problems and their well-posedness of the corresponding weak formulation. A
nonlocal Dirichlet problem is part of the energy-based Local-to-Nonlocal(LtN)
coupling that is outlined in Section 3.1, which is followed by a small
subsection on how to solve this LtN coupling with Schwarz methods. After that,
we introduce an interface identification problem that is in our case
constrained by the energy-based LtN coupling of Section 3.3. Chapter 4 is then
dedicated to shape optimization and contains basic notions of shape
optimization as well as the development of the shape derivative for the
reduced objective functional corresponding to the interface identification
problem from Section 3.3. Additionally, we describe a well-known shape
optimization algorithm in Chapter 4.4, which we apply in Section 5 to solve
the interface identification problem constrained by an energy-based LtN
coupling for two numerical examples.

##  2 Nonlocal Terminology and Framework

Let $\Omega\subset\mathbb{R}^{d}$ be an open, connected and bounded domain.
Moreover, we denote by
$\gamma:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty)$ a nonnegative
_(interaction) kernel_ and we let the _nonlocal boundary_ $\mathcal{I}$ of
$\Omega$ be defined in the following way

$\displaystyle\mathcal{I}:=\\{\mathbf{y}\in\mathbb{R}^{d}\setminus\Omega:\int_{\Omega}\gamma(\mathbf{x},\mathbf{y})+\gamma(\mathbf{y},\mathbf{x})~{}d\mathbf{x}>0\\}.$
(1)

As in [16, 40], the kernel $\gamma$ is supposed to fulfill the next two
requirements.

  * (K1)

There exist $\delta\in(0,\infty)$ and
$\phi:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty)$ with

$\gamma(\mathbf{x},\mathbf{y})=\phi(\mathbf{x},\mathbf{y})\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y}).$

  * (K2)

There exist $\gamma_{0}\in(0,\infty)$ and $\epsilon\in(0,\delta)$ such that

$\displaystyle\gamma_{0}\leq\gamma(\mathbf{x},\mathbf{y})\text{ for
}\mathbf{y}\in B_{\epsilon}(\mathbf{x})\text{ and }\mathbf{x}\in\Omega.$

Then, we call an operator $\mathcal{L}$ of the form

$\displaystyle-\mathcal{L}u(\mathbf{x})=\int\limits_{\mathbb{R}^{d}}\left(u(\mathbf{x})\gamma(\mathbf{x},\mathbf{y})-u(\mathbf{y})\gamma(\mathbf{y},\mathbf{x})\right)~{}d\mathbf{y}$
(2)

a _nonlocal convection-diffusion operator_. With an appropriate forcing term
$f$ and suitable boundary data $g$, which will both be specified later, we are
now able to formulate a _nonhomogeneous steady-state Dirichlet problem with
boundary constraints_ as

$\displaystyle\begin{split}-\mathcal{L}u&=f\text{ on }\Omega\\\ u&=g\text{ on
}\mathcal{I}.\end{split}$ (3)

Similar to the case of partial differential equations, the theory of nonlocal
problems focuses on finding a weak solution to (3). In order to derive such a
weak formulation we multiply the first equation of (3) by a test function
$v:\Omega\cup\mathcal{I}\rightarrow\mathbb{R}$ with $v=0$ on $\mathcal{I}$ and
then integrate over $\Omega$, which results in

$\displaystyle\left(-\mathcal{L}u,v\right)_{L^{2}(\Omega)}$
$\displaystyle=\left(f,v\right)_{L^{2}(\Omega)}$
$\displaystyle\Leftrightarrow\int_{\Omega}v(\mathbf{x})\int_{\mathbb{R}^{d}}(u(\mathbf{x})\gamma(\mathbf{x},\mathbf{y})$
$\displaystyle-u(\mathbf{y})\gamma(\mathbf{y},\mathbf{x}))~{}d\mathbf{y}d\mathbf{x}=\int_{\Omega}f(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}.$
(4)

Making use of Fubini’s theorem and applying $v(\mathbf{y})=0$ for
$\mathbf{y}\in\mathcal{I}$ yields an equivalent representation for the
integral on the left-hand side

$\displaystyle\int_{\Omega}v(\mathbf{x})\int_{\Omega\cup\mathcal{I}}(u(\mathbf{x})\gamma(\mathbf{x},\mathbf{y})-u(\mathbf{y})\gamma(\mathbf{y},\mathbf{x}))~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\frac{1}{2}\iint_{(\Omega\cup\mathcal{I})^{2}}(v(\mathbf{x})-v(\mathbf{y}))(u(\mathbf{x})\gamma(\mathbf{x},\mathbf{y})-u(\mathbf{y})\gamma(\mathbf{y},\mathbf{x}))~{}d\mathbf{y}d\mathbf{x}.$

As a next step, we define the bilinear operator $A$ and the linear functional
$F$ as follows

$\displaystyle A(u,v)$
$\displaystyle:=\frac{1}{2}\iint_{(\Omega\cup\mathcal{I})^{2}}(v(\mathbf{x})-v(\mathbf{y}))(u(\mathbf{x})\gamma(\mathbf{x},\mathbf{y})-u(\mathbf{y})\gamma(\mathbf{y},\mathbf{x}))~{}d\mathbf{y}d\mathbf{x}\text{
and}$ $\displaystyle F(v)$
$\displaystyle:=\int_{\Omega}f(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}.$

Then, we can introduce a (semi-)norm

$\displaystyle\begin{split}|u|_{V(\Omega\cup\mathcal{I})}&:=\sqrt{A(u,u)}\text{
and a norm }\\\
||u||_{V(\Omega\cup\mathcal{I})}&:=|u|_{V(\Omega\cup\mathcal{I})}+||u||_{L^{2}(\Omega\cup\mathcal{I})},\end{split}$
(5)

which we employ to define the following nonlocal energy spaces

$\displaystyle V(\Omega\cup\mathcal{I})$ $\displaystyle:=\\{u\in
L^{2}(\Omega\cup\mathcal{I}):||u||_{V(\Omega\cup\mathcal{I})}<\infty\\}\text{
and}$ $\displaystyle V_{c}(\Omega\cup\mathcal{I})$ $\displaystyle:=\\{u\in
V(\Omega\cup\mathcal{I}):u=0\text{ on }\mathcal{I}\\}.$

With these definitions we can now formulate a _variational_ or _weak_
formulation of problem (3):

######  Definition 2.1.

Given $f\in L^{2}(\Omega)$ and $g\in V(\Omega\cup\mathcal{I})$, if $u\in
V(\Omega\cup\mathcal{I})$ solves

$\displaystyle\begin{split}A(u,v)&=F(v)\text{ for all }v\in
V_{c}(\Omega\cup\mathcal{I})\text{ and}\\\ u&-g\in
V_{c}(\Omega\cup\mathcal{I}),\end{split}$ (6)

then $u$ is called weak solution of (3).

Since $A(u,v)=A(u-g,v)+A(g,v)$ and $u-g\in V_{c}(\Omega\cup\mathcal{I})$, we
are able to rephrase problem 6 as a homogeneous Dirichlet problem

$\displaystyle\begin{split}\textit{Given }f&\in L^{2}(\Omega)\textit{ and
}g\in V(\Omega\cup\mathcal{I})\textit{, find }\tilde{u}\in
V_{c}(\Omega\cup\mathcal{I})\textit{ s.t.}\\\
A(\tilde{u},v)&=F(v)-A(g,v)\textit{ for all }v\in
V_{c}(\Omega\cup\mathcal{I}),\\\ \textit{then }u&:=\tilde{u}+g\in
V(\Omega\cup\mathcal{I})\textit{ is a solution of
\eqref{weak_formulation}.}\end{split}$ (7)

######  Example 2.2.

Two popular classes of symmetric kernels, where the well-posedness of (6) is
shown in, e.g. [16, 40], are:

  * •

Integrable kernels:
There exist constants $\gamma_{1},\gamma_{2}\in(0,\infty)$ with

$\displaystyle\gamma_{1}\leq\inf_{\mathbf{x}\in\Omega}\int_{\Omega\cup\mathcal{I}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}\text{
and
}\sup_{\mathbf{x}\in\Omega}\int_{\Omega\cup\mathcal{I}}\left(\gamma(\mathbf{x},\mathbf{y})\right)^{2}~{}d\mathbf{y}\leq\left(\gamma_{2}\right)^{2}.$

Then the solution $u\in V(\Omega\cup\mathcal{I})$ for (6) exists, is unique
and in this case the spaces
$\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(L_{c}^{2}(\Omega\cup\mathcal{I}),||\cdot||_{L^{2}(\Omega\cup\mathcal{I})}\right)$
as well as
$\left(V(\Omega\cup\mathcal{I}),||\cdot||_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(L^{2}(\Omega\cup\mathcal{I}),||\cdot||_{L^{2}(\Omega\cup\mathcal{I})}\right)$
are equivalent.

  * •

Singular symmetric kernels:
There exist constants $\gamma_{1},\gamma_{2}\in(0,\infty)$ and $s\in(0,1)$
with

$\displaystyle\gamma_{1}<\gamma(\mathbf{x},\mathbf{y})||\mathbf{x}-\mathbf{y}||_{2}^{d+2s}<\gamma_{2}\text{
for all }\mathbf{y}\in B_{\delta}(\mathbf{x})\text{ and }\mathbf{x}\in\Omega.$

Then the solution to (6) exists, is unique and the spaces
$\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(H^{s}_{c}(\Omega\cup\mathcal{I}),|\cdot|_{H^{s}(\Omega\cup\mathcal{I})}\right)$
as well as
$\left(V(\Omega\cup\mathcal{I}),||\cdot||_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(H^{s}(\Omega\cup\mathcal{I}),||\cdot||_{H^{s}(\Omega\cup\mathcal{I})}\right)$
are equivalent.

In the coupling formulation, that is presented in the Chapter 3.1, only
integrable kernels with some additional assumptions, which will be specified
later, are considered.

In the next Chapter, we present a Local-to-Nonlocal coupling, where we will
observe a nonlocal Dirichlet problem on one subdomain. Additionally, we
recapture under which assumptions this particular coupling is well-posed(see
Lemma 3.5 and Corollary 3.6).

##  3 Interface Identification constrained by an energy-based Local-to-
Nonlocal Coupling

###  3.1 Energy-based Local-to-Nonlocal Coupling

In this section we introduce a specific coupling of the Laplacian operator on
one domain and a nonlocal operator on the other domain, which was first
formulated in [1, 2].
Here, for any two sets
$\hat{\Omega}_{1},\hat{\Omega}_{2}\subset\mathbb{R}^{d}$ we define the
distance of those two domains as

$\displaystyle\operatorname{dist}(\hat{\Omega}_{1},\hat{\Omega}_{2}):=\inf\limits_{\mathbf{x}\in\hat{\Omega}_{1},~{}\mathbf{y}\in\hat{\Omega}_{2}}||\mathbf{x}-\mathbf{y}||_{2}.$

If $\hat{\Omega}_{1}$ only consists of one element, i.e. there exists one
$\mathbf{x}\in\mathbb{R}^{d}$ such that $\hat{\Omega}_{1}=\\{\mathbf{x}\\}$,
we also use the short notation
$\operatorname{dist}(\mathbf{x},\hat{\Omega}_{2}):=\operatorname{dist}(\\{\mathbf{x}\\},\hat{\Omega}_{2})$.
From now on, the set $\Omega$ is assumed to be an open and bounded Lipschitz
domain and can be described as
$\Omega=\left(\overline{\Omega}_{l}\cup\overline{\Omega}_{nl}\right)^{\circ}$,
where $\Omega_{l}$ and $\Omega_{nl}$ are open, bounded, connected, non-empty
and disjoint sets. Moreover, $\Omega_{l}$ is also supposed to be a Lipschitz
domain and, since trivial couplings should be excluded,
$\operatorname{dist}(\Omega_{l},\Omega_{nl})<\delta$ is assumed to hold. An
example configuration is shown in Picture 3.1. From now on, we will refer to
$\Omega_{l}$ as the ’local’ and to $\Omega_{nl}$ as the ’nonlocal’ domain.

$\Omega_{nl}$
---
$\Omega_{l}$
---
$\mathcal{I}_{nl}$
---
Figure 3.1: Here, the nonlocal domain $\Omega_{nl}$ is depicted in the darker
red and the corresponding nonlocal domain $\mathcal{I}_{nl}$ of the nonlocal
domain $\Omega_{nl}$ is colored in the lighter red. Moreover, the ’local’
domain $\Omega_{l}$ consists in this example of the nonlocal boundary
$\mathcal{I}_{nl}$ as well as the gray area.

In the remaining part of this work we denote by $\partial\Omega$,
$\partial\Omega_{l}$ and $\partial\Omega_{nl}$ the _local boundary_ of
$\Omega$, $\Omega_{l}$ and $\Omega_{nl}$, respectively. Moreover, the nonlocal
boundary $\mathcal{I}_{nl}$ regarding the nonlocal domain $\Omega_{nl}$ is
analogously defined as in (1), i.e.

$\displaystyle\mathcal{I}_{nl}:=\\{\mathbf{y}\in\mathbb{R}^{d}\setminus\Omega_{nl}:\int_{\Omega_{nl}}\gamma(\mathbf{x},\mathbf{y})+\gamma(\mathbf{y},\mathbf{x})~{}d\mathbf{y}d\mathbf{x}>0\\}.$

Then, the coupling problem is formulated in the following way:

$\displaystyle-\mathcal{L}_{l}(u)(\mathbf{x})$ $\displaystyle=f(\mathbf{x})$
$\displaystyle\text{for }\mathbf{x}\in\Omega_{l},$
$\displaystyle\partial_{n}u_{l}$ $\displaystyle=0$ $\displaystyle\text{on
}\partial\Omega_{l}\cap\partial\Omega_{nl},$ (8) $\displaystyle u_{l}$
$\displaystyle=0$ $\displaystyle\text{on
}\partial\Omega_{l}\cap\partial\Omega,$
$\displaystyle-\mathcal{L}_{nl}(u)(\mathbf{x})$ $\displaystyle=f(\mathbf{x})$
$\displaystyle\text{for }\mathbf{x}\in\Omega_{nl},$ $\displaystyle u_{nl}$
$\displaystyle=u_{l}$ $\displaystyle\text{on }\mathcal{I}_{nl}\cap\Omega_{l},$
(9) $\displaystyle u_{nl}$ $\displaystyle=0$ $\displaystyle\text{on
}\mathcal{I}_{nl}\cap\mathcal{I},$

where

$\displaystyle-\mathcal{L}_{l}(u)(\mathbf{x}):=-\mathcal{L}_{l}(u_{l},u_{nl})(\mathbf{x}):=$
$\displaystyle-\Delta
u_{l}(\mathbf{x})+\int_{\Omega_{nl}}\left(u_{l}(\mathbf{x})-u_{nl}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}\quad\text{and}$
$\displaystyle-\mathcal{L}_{nl}(u)(\mathbf{x}):=-\mathcal{L}_{nl}(u_{l},u_{nl})(\mathbf{x}):=$
$\displaystyle
2\int\limits_{\Omega_{nl}}\left(u_{nl}(\mathbf{x})-u_{nl}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}$
$\displaystyle+\int\limits_{\mathcal{I}_{nl}\cap\Omega_{l}}\left(u_{nl}(\mathbf{x})-u_{l}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}+\int\limits_{\mathcal{I}_{nl}\cap\mathcal{I}}u_{nl}(\mathbf{x})\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}.$

Basically, we solve a nonlocal Dirichlet problem on the nonlocal domain
$\Omega_{nl}$, where the ’local’ solution $u_{l}$ serves as boundary data on
$\mathcal{I}_{nl}\cap\Omega_{l}$. For the ’local’ problem (8) the dependence
on the nonlocal solution $u_{nl}$ is more implicit. Here, the operator
$\mathcal{L}_{l}(u)(\mathbf{x})$ is equivalent to the Laplacian $\Delta
u_{l}(\mathbf{x})$, if $\operatorname{dist}(\mathbf{x},\Omega_{nl})\geq\delta$
and otherwise, if $\operatorname{dist}(\mathbf{x},\Omega_{nl})<\delta$, some
nonlocal flux is added to the Laplacian. Thus, in the second case the value of
the operator $\mathcal{L}_{l}$ is influenced by the ’nonlocal’ solution
$u_{nl}$. Since $\gamma=0$ almost everywhere in
$\Omega_{nl}\times\left(\mathbb{R}^{d}\setminus\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right)\right)$
we sometimes integrate over $\Omega_{l}$ and $\mathcal{I}$ instead of
$\mathcal{I}_{nl}\cap\Omega_{l}$ and $\mathcal{I}_{nl}\cap\mathcal{I}$ in the
remaining part of this work.

######  Remark 3.1.

According to [2] the assumption that $\Omega_{nl}$ needs to be connected can
be relaxed to $\hat{\delta}$-connectedness for some
$\hat{\delta}\in[0,\delta)$. A domain $\hat{\Omega}$ is said to be
$\hat{\delta}$-connected, if $\hat{\Omega}$ cannot be written as a union of
two disjoint open sets, whose distance is greater than $\hat{\delta}$, i.e.

$\displaystyle\nexists\hat{\Omega}_{1},\hat{\Omega}_{2}\text{ open}:\
\hat{\Omega}_{1}\cap\hat{\Omega}_{2}=\emptyset,\
\hat{\Omega}_{1}\cup\hat{\Omega}_{2}=\hat{\Omega}\text{ and
}\operatorname{dist}(\hat{\Omega}_{1},\hat{\Omega}_{2})>\hat{\delta}.$

However, since $\hat{\delta}$-connectedness has to be checked after every
deformation of $\Gamma$, we only consider the case, where $\Omega_{nl}$ is
(0-)connected. Thus, we do not apply the general concept of
$\hat{\delta}$-connectedness.

In order to formulate a weak formulation of the LtN coupling we define the LtN
bilinear operator $A^{LtN}$ and the linear functional $F$ as

$\displaystyle F(v)$
$\displaystyle:=\int_{\Omega_{l}}f(\mathbf{x})v_{l}(\mathbf{x})~{}d\mathbf{x}-\int_{\Omega_{nl}}f(\mathbf{x})v_{nl}(\mathbf{x})~{}d\mathbf{x}\quad\text{and}$
$\displaystyle A^{LtN}(u,v)$ $\displaystyle:=\int_{\Omega_{l}}\nabla
u_{l}(\mathbf{x})\nabla
v_{l}(\mathbf{x})~{}d\mathbf{x}+\int_{\Omega_{nl}}\int_{\Omega_{nl}}\left(u_{nl}(\mathbf{x})-u_{nl}(\mathbf{y})\right)\left(v_{nl}(\mathbf{x})-v_{nl}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int_{\Omega_{nl}}\int_{\Omega_{l}}(u_{nl}(\mathbf{x})-u_{l}(\mathbf{y}))(v_{nl}(\mathbf{x})-v_{l}(\mathbf{y}))\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int_{\Omega_{nl}}u_{nl}(\mathbf{x})v_{nl}(\mathbf{x})\int_{\mathcal{I}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}.$

Then we can define an inner product $<\cdot,\cdot\cdot>_{H}$ and the
corresponding norm $||\cdot||_{H}$ as

$\displaystyle<u,v>_{H}:=A^{LtN}(u,v)\quad\text{and}\quad||u||_{H}:=\sqrt{A^{LtN}(u,u)}.$

######  Definition 3.2.

The energy space $(H,||\cdot||_{H})$ is defined as

$\displaystyle H$ $\displaystyle:=\\{u=(u_{l},u_{nl}):u_{l}\in
H_{\partial\Omega\cap\partial\Omega_{l}}^{1}(\Omega_{l}),\ u_{nl}\in
L_{c}^{2}\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right)\\},$

where the spaces $H_{\partial\Omega\cap\partial\Omega_{l}}^{1}(\Omega_{l})$
and $L_{c}^{2}\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right)$ are set as follows

$\displaystyle H_{\partial\Omega\cap\partial\Omega_{l}}^{1}(\Omega_{l})$
$\displaystyle:=\\{v\in H^{1}(\Omega_{l}):v=0\text{ on
}\partial\Omega\cap\partial\Omega_{l}\\}\quad\text{and}$ $\displaystyle
L_{c}^{2}\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right)$ $\displaystyle:=\\{v\in
L^{2}(\Omega_{nl}\cup\mathcal{I}_{nl}):v=0\text{ on }\mathcal{I}_{nl}\\}.$

######  Definition 3.3.

Given $f\in L^{2}(\Omega)$, we call $u=(u_{l},u_{nl})\in H$ a weak solution to
the coupling that consists of equations (8) and (9), if $u$ fulfills

$\displaystyle A^{LtN}(u,v)=F(v)\text{ for all }v=(v_{l},v_{nl})\in H.$ (10)

Thus, equation (10) is the variational formulation of the combined problem.
Next, we would like to point out, why this coupling is called energy-based.
Therefore, we want to minimize the energy functional

$\displaystyle\min_{u\in H}E(u)$
$\displaystyle:=\frac{1}{2}A^{LtN}(u,u)-F(v),$

and derive the first order necessary condition

$\displaystyle dE(u)(v)=A^{LtN}(u,v)-F(v)=0\text{ for all }v=(v_{l},v_{nl})\in
H.$

which is equivalent to the variational formulation (10). Now, if we set the
local component of the test vector $v_{l}=0$, we get the variational
formulation regarding the nonlocal Dirichlet problem (9) of the LtN coupling

$\displaystyle\int_{\Omega_{nl}}\int_{\Omega_{nl}}\left(v_{nl}(\mathbf{x})-v_{nl}(\mathbf{y})\right)\left(u_{nl}(\mathbf{x})-u_{nl}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int_{\Omega_{nl}}v_{nl}(\mathbf{x})\int_{\Omega_{l}}\left(u_{nl}(\mathbf{x})-u_{l}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}+\int_{\Omega_{nl}}v_{nl}(\mathbf{x})u_{nl}(\mathbf{x})\int_{\mathcal{I}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\int_{\Omega_{nl}}f(\mathbf{x})v_{nl}(\mathbf{x})~{}d\mathbf{x}\text{
for all }v_{nl}\in
V_{c}(\Omega_{nl}\cup\mathcal{I}_{nl})=L^{2}_{c}(\Omega_{nl}\cup\mathcal{I}_{nl}).$

Analogously, by setting the nonlocal component of the test vector $v_{nl}=0$,
we derive the weak formulation for the local subproblem (8) of the LtN
coupling

$\displaystyle\int_{\Omega_{l}}\nabla u_{l}(\mathbf{x})\nabla
v_{l}(\mathbf{x})~{}d\mathbf{x}+\int_{\Omega_{l}}v_{l}(\mathbf{x})\int_{\Omega_{nl}}\left(u_{l}(\mathbf{x})-u_{nl}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\int_{\Omega_{l}}f(\mathbf{x})v_{l}(\mathbf{x})~{}d\mathbf{x}\text{
for all }v_{l}\in H_{\partial\Omega_{l}\cap\partial\Omega}^{1}(\Omega_{l}).$

All in all, the coupling is called energy-based, because the weak formulations
for (8), (9) and the combined problem can be derived by investigating first
order necessary conditions for the minimization of a specific energy
functional, where we only consider certain test functions.
For the purpose of having a well-posed LtN coupling, we need further
restrictions. In addition to the assumptions (K1) and (K2) from Chapter 2, the
kernel $\gamma$ is supposed to be

  * (K3)

bounded and

  * (K4)

translation invariant, i.e. there exists a function
$J:\mathbb{R}^{d}\rightarrow\mathbb{R}$ and

$\gamma(\mathbf{x},\mathbf{y})=J(\mathbf{x}-\mathbf{y}).$

######  Remark 3.4.

In contrast to this work, in [2] the kernel $\gamma$ is not assumed to be
truncated. Instead they assume additionally the following conditions:

  * •

The kernel $\gamma$ is integrable.

  * •

(Compactness): The operator $C:L^{2}(\Omega_{nl})\rightarrow
L^{2}(\Omega_{nl})$ with

$C(u)(\mathbf{x})=\int_{\Omega_{nl}}J(\mathbf{x}-\mathbf{y})u(\mathbf{y})~{}d\mathbf{y}$

is compact.

Then, in [2, Lemma 2.6] it is shown that the space
$(H,<\cdot,\cdot\cdot>_{H})$ is actually a Hilbert space.
In our case the truncatedness and boundedness of $\gamma$ imply $\gamma\in
L^{1}((\Omega_{nl}\cup\mathcal{I}_{nl})\times(\Omega_{nl}\cup\mathcal{I}_{nl}))$
and $\gamma\in
L^{2}((\Omega_{nl}\cup\mathcal{I}_{nl})\times(\Omega_{nl}\cup\mathcal{I}_{nl}))$
since the domain $\Omega_{nl}$ and therefore $\Omega_{nl}\cup\mathcal{I}_{nl}$
are also bounded. As a result, the compactness condition holds, because the
operator $C$ is in this case a Hilbert-Schmidt operator and therefore
compact(see [9, Theorem 6.12]).

Consequently, we derive the next Lemma.

######  Lemma 3.5 (After [2, Lemma 2.6]).

Let $\gamma$ fulfill (K1)-(K4). Then, the space $(H,<\cdot,\cdot\cdot>_{H})$
is a Hilbert space and the norm $||\cdot||_{H}$ is equivalent to the norm of
the product space
$\left(H^{1}(\Omega_{l}),||\cdot||_{H^{1}(\Omega_{l})}\right)\times\left(L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl}),||\cdot||_{L^{2}(\Omega_{nl})}\right)$.

###### Proof.

Follows directly from [2, Lemma 2.6] and Remark 3.4. ∎

######  Corollary 3.6.

As a consequence of Lemma 3.5 and the Riesz representation theorem, there
exists a unique solution to (10).

Moreover, we will make use of Lemma 3.5 in order to show that the averaged
adjoint method, which will be introduced in Section 4.2, can be applied. Next,
we present a way to iteratively compute the solution to (10).

###  3.2 Schwarz Methods

In Chapter 5, we solve the energy-based LtN coupling by applying the
multiplicative Schwarz method as described in [2]. Schwarz methods were
developed by H.A. Schwarz around 1870 in order to solve a PDE on an
’irregular’ domain[34]. In the following, we present the Schwarz method in the
context of the LtN coupling problem. The basic idea is to solve problems (8)
and (9) by finding solutions to the local and the nonlocal problem in an
alternating fashion as it is shown in Algorithm 1.

1 Given: $u^{0}=(u_{l}^{0},u_{nl}^{0})$.

2 while  _termination criterion not fulfilled_ do

3 Find $u_{l}^{k}$, s.t.
$\displaystyle-\mathcal{L}_{l}(u_{l}^{k},u_{nl}^{k-1})$ $\displaystyle=f\quad$
$\displaystyle\text{on }\Omega_{l},$ $\displaystyle\partial_{n}u_{l}^{k}$
$\displaystyle=0$ $\displaystyle\text{on
}\partial\Omega_{l}\cap\partial\Omega_{nl},$ $\displaystyle u_{l}^{k}$
$\displaystyle=0$ $\displaystyle\text{on }\partial\Omega_{l}\cap\Omega.$ Find
$u_{nl}^{k}$, s.t.  $\displaystyle-\mathcal{L}_{nl}(u_{l}^{k},u_{nl}^{k})$
$\displaystyle=f$ $\displaystyle\text{on }\Omega_{nl},$ $\displaystyle
u_{nl}^{k}$ $\displaystyle=u_{l}^{k}\quad$ $\displaystyle\text{on
}\mathcal{I}_{nl}\cap\Omega_{l},$ $\displaystyle u_{nl}^{k}$ $\displaystyle=0$
$\displaystyle\text{on }\mathcal{I}_{nl}\cap\mathcal{I}.$ $k\leftarrow k+1$.

4 end while

Algorithm 1 Multiplicative Schwarz Method for LtN coupling

Given the solution $u^{k-1}=(u^{k-1}_{l},u^{k-1}_{nl})$ from the previous
iteration, we first compute the solution $u_{l}^{k}$ of the local problem with
$u^{k-1}_{nl}$ as a second argument of $-\mathcal{L}_{l}$ and after that, we
find $u_{nl}^{k}$ by solving the nonlocal problem with boundary data
$u_{l}^{k}$. So, we use the new solution $u_{l}^{k}$ immediately in the same
iteration.

######  Theorem 3.7 ([2, Lemma 2.9]).

Let $u=(u_{l},u_{nl})\in H$ be a solution to (10). Algorithm 1 yields a
sequence $u^{k}=(u^{k}_{l},u^{k}_{nl})$, such that

$\displaystyle u_{l}^{k}\rightarrow u_{l}\text{ in
}H^{1}_{\partial\Omega\cap\Omega_{l}}(\Omega_{l})\text{ and}$ $\displaystyle
u_{nl}^{k}\rightarrow u_{nl}\text{ in
}L^{2}_{c}(\Omega_{nl}\cup\mathcal{I}_{nl}).$

Additionally, this method is geometrically convergent, i.e. there exists a
constant $\varepsilon\in(0,1)$ with

$\displaystyle\max\\{||u_{l}^{k}-u_{l}||_{H^{1}_{\partial\Omega\cap\partial\Omega_{l}}(\Omega_{l})},||u_{nl}^{k}-u_{nl}||_{L^{2}(\Omega_{nl})}\\}\leq
C\varepsilon^{k},$

where
$C:=\max\\{||u_{l}^{0}-u_{l}||_{H^{1}_{\partial\Omega\cap\partial\Omega_{l}}(\Omega_{l})},||u_{nl}^{0}-u_{nl}||_{L^{2}(\Omega_{nl})}\\}$.

Alternatively, we could also follow the approach of the _additive Schwarz
method_ and employ only the former iterate $u^{k-1}$ inside one iteration and
switch to $u^{k}$ as the current solution after the iteration, which enables
us to solve both subproblems in parallel.
In both cases, we repeat this alternating process until some termination
criterion is met. In this work, since we apply the finite element method to
solve the subproblems, we stop when the Euclidean norm of the residual is
smaller than some given tolerance.

###  3.3 Problem Formulation

Interface identification is a popular problem in PDE-constrained shape
optimization, see, e.g. [41, 38, 14]. Here, as mentioned earlier, we consider
an open, bounded and connected domain $\Omega$ with Lipschitz domain as it is
depicted in Picture 3.1. We then call
$\Gamma:=\partial\Omega_{nl}\cap\partial\Omega_{l}$ _interface_ and assume
that $\Gamma$ is a closed curve that decomposes $\Omega$ into the Lipschitz
domains $\Omega_{nl}$ and $\Omega_{l}$, i.e.
$\Omega=\Omega_{nl}\dot{\cup}\Gamma\dot{\cup}\Omega_{l}$. Therefore, we denote
by $\Omega(\Gamma)$ that $\Omega$ is decomposed by $\Gamma$ as described
above. Moreover, the space $H$ and the norm $||\cdot||_{H}$ are also dependent
on $\Gamma$, which we indicate by $||\cdot||_{H(\Gamma)}$ and $H(\Gamma)$,
respectively.
From now on, let $f_{\Gamma}:=f_{l}\chi_{\Omega_{l}}+f_{nl}\chi_{\Omega_{nl}}$
with $f_{l},f_{nl}\in H^{1}(\Omega)$. Given some data $\bar{u}\in
H^{1}(\Omega)$, that we would like to approximate by a weak solution $u$ of
the LtN coupling (10), we then call

$\displaystyle\begin{split}\min_{u,\Gamma}J(u,\Gamma)&:=j(u,\Gamma)+j_{per}(\Gamma)=\int_{\Omega(\Gamma)}\left(u(\mathbf{x})-\bar{u}(\mathbf{x})\right)^{2}~{}d\mathbf{x}+\nu\int_{\Gamma}1~{}ds\\\
\text{s.t.}\quad&A^{LtN}_{\Gamma}(u,v)=F_{\Gamma}(v)\text{ for all }v\in
H(\Gamma),\end{split}$ (11)

an _interface identification problem_ , which is in our case constrained by a
Local-to-Nonlocal coupling. Here, the subscripts of $F_{\Gamma}$ and
$A^{LtN}_{\Gamma}$ indicate the dependence of these functions on $\Gamma$.
Since every admissible choice of $\Gamma\subset\Omega$ yields a unique
solution, this problem can be seen as only dependent on the choice of the
interface $\Gamma$, i.e. we would like to find a shape $\Gamma$, such that the
corresponding solution approximates our data $\bar{u}$ as good as possible.
The second term in the objective functional $j_{per}$ is called perimeter
regularization with parameter $\nu>0$ and is often applied in optimization to
avoid an ill-posedness of the problem[6].

##  4 Shape Optimization

In this Chapter, we follow the steps in [33], where an interface
identification constrained by purely nonlocal equations is investigated.

###  4.1 Basics of Shape Optimization

In order to introduce the definition of a shape derivative we need a family of
mappings $\\{\mathbf{F}_{\mathbf{t}}\\}_{t\in[0,T]}$ with
$\mathbf{F}_{\mathbf{t}}:\overline{\Omega}\rightarrow\mathbb{R}^{d}$ and
$\mathbf{F}_{0}={\bf Id}$ for a given sufficiently small constant
$T\in(0,\infty)$. We can then deform a domain $\Gamma\subset\Omega$ according
to $\\{\mathbf{F}_{\mathbf{t}}\\}_{t\in[0,T]}$ and get a family of perturbed
shapes $\\{\Gamma^{t}\\}_{t\in[0,T]}$, where
$\Gamma^{t}:=\mathbf{F}_{\mathbf{t}}(\Gamma)$. In this work, as a family of
mappings we will only use the so-called _perturbation of identity_.

######  Definition 4.1 (Perturbation of Identity).

For a vector field $\mathbf{V}\in C_{0}^{k}(\Omega,\mathbb{R})$,
$k\in\mathbb{N}$ we define the _perturbation of identity_ as

$\displaystyle\mathbf{F}_{\mathbf{t}}:\overline{\Omega}\rightarrow\mathbb{R},\quad\mathbf{F}_{\mathbf{t}}(\mathbf{x})=\mathbf{x}+t\mathbf{V}(\mathbf{x})=\left({\bf
Id}+t\mathbf{V}\right)(\mathbf{x}),$ (12)

where ${\bf Id}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is the identity
function.

In Figure 4.1 we present an example how the perturbation of identity maps
$\Gamma$ to a new interface $\mathbf{F}_{\mathbf{t}}(\Gamma)$.

$\mathbf{F}_{\mathbf{t}}$
---
$\mathbf{x}$
---
$\mathbf{F}_{\mathbf{t}}(\mathbf{x})$
---
$\mathbf{F}_{\mathbf{t}}(\Gamma)$
---
$\Gamma$
---
Figure 4.1: The perturbation of identity moves a point $\mathbf{x}$ in the
direction of a vector field $\mathbf{V}$ to the point
$\mathbf{F}_{\mathbf{t}}(\mathbf{x})$. Applied on $\Gamma$, this interface is
shifted and deformed to a new interface $\mathbf{F}_{\mathbf{t}}(\Gamma)$.

If $t$ is sufficiently small, such that
$||t\mathbf{V}||_{W^{1,\infty}(\Omega,\mathbb{R}^{d})}<1$, then the
perturbation of identity
$\mathbf{F}_{\mathbf{t}}:\overline{\Omega}\rightarrow\overline{\Omega}$ is
Lipschitz, invertible and the inverse is also Lipschitz[22]. As a result, if
$\Gamma$ is a Lipschitz domain, the perturbed domain $\Gamma^{t}$ will also
have a Lipschitz boundary. We are now able to present the definition of a
shape derivative.

######  Definition 4.2 (Shape Derivative).

Given a shape functional $J:\mathcal{A}\rightarrow\mathbb{R}$, where
$\mathcal{A}$ is an appropriate shape space, the _Eulerian_ or _directional
derivative_ is defined as

$\displaystyle D_{\Gamma}J(\Gamma)[\mathbf{V}]=\lim_{t\searrow
0}\frac{J(\mathbf{F}_{\mathbf{t}}(\Gamma))-J(\Gamma)}{t}.$ (13)

If for every $\mathbf{V}\in C_{0}^{k}(\Omega,\mathbb{R}^{d})$ the derivative
$D_{\Gamma}J(\Gamma)[\mathbf{V}]$ exists and the function $\mathbf{V}\mapsto
D_{\Gamma}J(\Gamma)[\mathbf{V}]$ is linear and continuous, the derivative
$D_{\Gamma}J(\Gamma)[\mathbf{V}]$ is called _shape derivative_ at $\Gamma$ in
the direction $\mathbf{V}$.

In this work, we use the same framework as in [30] and, starting from an
initial shape $\Gamma^{0}$, we define the corresponding shape space
$\mathcal{A}(\Gamma^{0})$ as follows:

######  Definition 4.3 (Shape Space).

Given a domain $\Gamma^{0}\subset\Omega$ we set the corresponding _shape
space_ $\mathcal{A}(\Gamma^{0})$ as

$\displaystyle\mathcal{A}(\Gamma^{0}):=\\{\mathbf{T}(\Gamma^{0})|~{}\mathbf{T}:\overline{\Omega}\rightarrow\overline{\Omega}\text{
invertible}\\}.$

Since the composition
$\mathbf{T}^{n}\circ\mathbf{T}^{n-1}\circ...\circ\mathbf{T}^{1}$ of invertible
functions $\mathbf{T}^{k}:\overline{\Omega}\rightarrow\overline{\Omega}$ for
$k=1,...,n$ and $n\in\mathbb{N}$ is again an invertible function, the shape
space $\mathcal{A}(\Gamma^{0})$ contains only domains that can be obtained by
a finite number of suitable deformations of $\Gamma^{0}$. Because we only
employ the perturbation of identity, the function $\mathbf{T}^{k}={\bf
Id}+t\mathbf{V}$ is invertible, if
$||t\mathbf{V}||_{W^{1,\infty}(\Omega,\mathbb{R}^{d})}<1$ for $k\in\mathbb{N}$
as mentioned above.

######  Remark 4.4.

Since it could be the case that parts of the nonlocal boundary
$\mathcal{I}_{nl}$ are outside of $\Omega$, i.e.
$\mathcal{I}\cap\mathcal{I}_{nl}\neq\emptyset$, we extend every $\mathbf{V}\in
C_{0}^{k}(\Omega,\mathbb{R}^{d})$ to $\overline{\Omega}\cup\mathcal{I}$ by
zero. So, the boundary $\mathcal{I}$ stays unchanged and only points inside
$\Omega$ can be moved according to

$\mathbf{V}\in
C_{0}^{k}(\Omega\cup\mathcal{I},\mathbb{R}^{d}):=\\{\mathbf{V}:\overline{\Omega}\cup\mathcal{I}\rightarrow\mathbb{R}^{d}|~{}\mathbf{V}=0\text{
on }\mathcal{I}\text{ and }\left.\mathbf{V}\right|_{\overline{\Omega}}\in
C_{0}^{k}(\Omega,\mathbb{R}^{d})\\}.$

In the following, we only need vector fields that are one times continuously
differentiable, i.e. $\mathbf{V}\in
C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})$.

###  4.2 Shape Derivative via the Averaged Adjoint Method

As mentioned in Section 3.3, we assume the existence of a unique solution
$u(\Gamma)$ for every admissible shape $\Gamma$, i.e. $u(\Gamma)$ satisfies
$A^{LtN}_{\Gamma}(u(\Gamma),v)=F_{\Gamma}(v)$ for all $v\in H(\Gamma)$.
Therefore, we investigate the so-called _reduced problem_

$\displaystyle\min\limits_{\Gamma\in\mathcal{A}(\Gamma^{0})}\;$ $\displaystyle
J^{red}(\Gamma):=J(u(\Gamma),\Gamma).$ (14)

Since we would like to apply derivative based minimization algorithms, we need
to compute the shape derivative of the reduced objective functional $J^{red}$.
There are several ways how the shape derivative of the objective functional
can be computed, which are discussed in [38]. We choose the _averaged adjoint
method (AAM)_ of [37, 25, 38], which is a material derivative-free approach to
rigorously access the shape derivative of (14). Therefore, we need the so-
called _Lagrangian functional_

$\displaystyle
L(u,\Gamma,v):=J(u,\Gamma)+A^{LtN}_{\Gamma}(u,v)-F_{\Gamma}(v).$

With the Lagrangian we can express the reduced functional as

$\displaystyle J^{red}(\Gamma)=L(u(\Gamma),\Gamma,v),\quad\text{for an
arbitrary }v\in H(\Gamma),$

where we apply that $u(\Gamma)$ is a weak solution of the LtN coupling (10)
regarding the interface $\Gamma$. We now fix $\Gamma$ and denote by
$\Gamma^{t}:=\mathbf{F}_{\mathbf{t}}(\Gamma)$,
$\Omega_{l}^{t}:=\mathbf{F}_{\mathbf{t}}(\Omega_{l})$ and
$\Omega_{nl}^{t}:=\mathbf{F}_{\mathbf{t}}(\Omega_{nl})$ the deformed interior
boundary and the deformed domains, respectively. Furthermore we indicate by
writing $\Omega(\Gamma^{t})$ that we use the decomposition
$\Omega(\Gamma^{t})=\Omega_{1}^{t}\dot{\cup}\Gamma^{t}\dot{\cup}\Omega_{2}^{t}\left(=\Omega\right)$.
As a consequence, the norm $||\cdot||_{H(\Gamma^{t})}$ and therefore the space
$H(\Gamma^{t})$ differ from $||\cdot||_{H(\Gamma)}$ and $H(\Gamma)$, since
$\Gamma^{t}(=\partial\Omega_{l}^{t}\cap\partial\Omega_{nl}^{t})$ indicates how
$\Omega$ is decomposed into $\Omega_{l}^{t}$ and $\Omega_{nl}^{t}$, which
determines the integration domains that are employed to describe the operator
$A^{LtN}_{\Gamma}$. Then, the reduced objective functional regarding
$\Gamma^{t}$ can be written as

$\displaystyle
J^{red}(\Gamma^{t})=L(u(\Gamma^{t}),\Gamma^{t},v),\quad\text{for any }v\in
H(\Gamma^{t}),$ (15)

where $u(\Gamma^{t})\in H(\Gamma^{t})$. In order to differentiate $L$ with
respect to $t$ to compute the shape derivative, we would have to access the
derivative for $u(\Gamma^{t})\circ\mathbf{F}_{\mathbf{t}}$ and
$v\circ\mathbf{F}_{\mathbf{t}}$, where $u(\Gamma^{t}),v\in H(\Gamma^{t})$ may
not be differentiable. Moreover, the norm $||\cdot||_{H(\Gamma^{t})}$ and
consequently the space $H(\Gamma^{t})$ are also dependent on $t$. Instead, we
can circumvent differentiating $u(\Gamma^{t})\circ\mathbf{F}_{\mathbf{t}}$ and
additionally stay in $\left(H(\Gamma),||\cdot||_{H(\Gamma)}\right)$ by using
$\mathbf{F}_{\mathbf{t}}$, which is a homeomorphism, as a so-called _pull-back
function_. Then, for $u,v\in H(\Gamma^{t})$ there exist functions
$\tilde{u},\tilde{v}\in H(\Gamma)$, such that

$\displaystyle
u=\tilde{u}\circ\mathbf{F}_{\mathbf{t}}^{-1}~{}\text{and}~{}v=\tilde{v}\circ\mathbf{F}_{\mathbf{t}}^{-1}.$

Moreover, for a sufficiently small $T\in(0,\infty)$, we can define

$\displaystyle J:[0,T]\times$ $\displaystyle H(\Gamma)\rightarrow\mathbb{R},$
$\displaystyle J(t,u):=J(u\circ\mathbf{F}_{\mathbf{t}}^{-1},\Gamma^{t}),$
$\displaystyle A^{LtN}:[0,T]\times$ $\displaystyle H(\Gamma)\times
H(\Gamma)\rightarrow\mathbb{R},$ $\displaystyle
A^{LtN}(t,u,v):=A^{LtN}_{\Gamma^{t}}(u\circ\mathbf{F}_{\mathbf{t}}^{-1},v\circ\mathbf{F}_{\mathbf{t}}^{-1}),$
$\displaystyle F:[0,T]\times$ $\displaystyle H(\Gamma)\rightarrow\mathbb{R},$
$\displaystyle F(t,v):=F_{\Gamma^{t}}(v\circ\mathbf{F}_{\mathbf{t}}^{-1}),$
$\displaystyle\begin{split}G:[0,T]\times&H(\Gamma)\times
H(\Gamma)\rightarrow\mathbb{R},\\\
&G(t,u,v):=L(u\circ\mathbf{F}_{\mathbf{t}}^{-1},\Gamma^{t},v\circ\mathbf{F}_{\mathbf{t}}^{-1})=J(t,u)+A^{LtN}(t,u,v)-F(t,v).\end{split}$
(16)

Now, let us briefly mention that the Local-to-Nonlocal bilinear form regarding
$\Gamma^{t}$ can be written as

$\displaystyle\begin{split}&A^{LtN}(t,u^{0},v^{0})=A^{LtN}_{\Gamma^{t}}(u^{0}\circ\mathbf{F}_{\mathbf{t}}^{-1},v^{0}\circ\mathbf{F}_{\mathbf{t}}^{-1})=\int\limits_{\Omega_{l}}\left(\mathcal{B}(t)(\mathbf{x})\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)~{}d\mathbf{x}\\\
&+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\gamma^{t}(\mathbf{x},\mathbf{y})\xi^{t}(\mathbf{x})\xi^{t}(\mathbf{y})~{}d\mathbf{y}d\mathbf{x},\end{split}$
(17)

where

$\displaystyle\xi^{t}$ $\displaystyle:=\det D\mathbf{F}_{\mathbf{t}},\
\gamma^{t}(\mathbf{x},\mathbf{y}):=\gamma(\mathbf{F}_{\mathbf{t}}(\mathbf{x}),\mathbf{F}_{\mathbf{t}}(\mathbf{y}))\text{
and }$ $\displaystyle\mathcal{B}(t)(\mathbf{x})$
$\displaystyle:=\xi^{t}(\mathbf{x})D\mathbf{F}_{\mathbf{t}}^{-1}(\mathbf{F}_{\mathbf{t}}(\mathbf{x}))D\mathbf{F}_{\mathbf{t}}^{-\top}(\mathbf{F}_{\mathbf{t}}(\mathbf{x}))=\xi^{t}(\mathbf{x})\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-\top}.$

Then, (15) can be expressed as

$\displaystyle J^{red}(\Gamma^{t})=G(t,u^{t},v),\quad\text{for any }v\in
H(\Gamma),$

where $u^{t}\in H(\Gamma)$ is defined as the unique solution of the LtN
coupling (10) for $\Gamma^{t}$, i.e. $u^{t}$ solves

$\displaystyle A^{LtN}(t,u^{t},v)-F(t,v)=0,\quad\forall v\in H(\Gamma).$ (18)

Obviously, $A^{LtN}(t,u,v)-F(t,v)$ is linear in $v$ for all
$(t,u)\in[0,T]\times H(\Gamma)$, which is one necessary condition of AAM.
Additionally, the following prerequisites have to hold in order to apply AAM.

  * •

Assumption (H0): For every $(t,v)\in[0,T]\times H(\Gamma)$

    1. 1.

$[0,1]\ni s\mapsto G(t,su^{t}+(1-s)u^{0},v)$ is absolutely continuous and

    2. 2.

$[0,1]\ni s\mapsto d_{u}G(t,su^{t}+(1-s)u^{0},v)[\tilde{u}]\in L^{1}((0,1))$
for all $\tilde{u}\in H(\Gamma)$.

  * •

For every $t\in[0,T]$ there exists a unique solution $v^{t}\in H(\Gamma)$ for
the averaged adjoint equation

$\int_{0}^{1}d_{u}G(t,su^{t}+(1-s)u^{0},v^{t})[\tilde{u}]ds=0\quad\text{for
all }\tilde{u}\in H(\Gamma).$ (19)

  * •

Assumption (H1):
The following equation holds

$\lim_{t\searrow
0}\frac{G(t,u^{0},v^{t})-G(0,u^{0},v^{t})}{t}=\partial_{t}G(0,u^{0},v^{0}).$

Since $A^{LtN}(t,\tilde{u},v^{t})$ is linear in the second argument, the left-
hand side of the averaged adjoint equation (19) can be written as

$\displaystyle\int_{0}^{1}d_{u}G(t,su^{t}+(1-s)u^{0},v^{t})[\tilde{u}]~{}ds=A^{LtN}(t,\tilde{u},v^{t})+\int_{\Omega}\left(\frac{1}{2}(u^{t}+u^{0})-\bar{u}^{t}\right)\tilde{u}\xi^{t}~{}d\mathbf{x},$

where $\bar{u}^{t}(\mathbf{x}):=\bar{u}(\mathbf{F}_{\mathbf{t}}(\mathbf{x}))$.
Therefore, (19) is equivalent to

$\displaystyle
A^{LtN}(t,\tilde{u},v^{t})=-\int_{\Omega}\left(\frac{1}{2}\left(u^{t}+u^{0}\right)-\bar{u}^{t}\right)\tilde{u}\xi^{t}~{}d\mathbf{x}\quad\forall\tilde{u}\in
H(\Gamma).$ (20)

We will make use of (20) in the proof of Lemma 4.9, where we show that, under
some mild additional requirements, Assumptions (H0) and (H1) are fulfilled.
Moreover, for $t=0$ we call the averaged adjoint equation

$\displaystyle A^{LtN}(0,\tilde{u},v^{0})$
$\displaystyle=-\int_{\Omega}(u^{0}-\bar{u})\tilde{u}~{}d\mathbf{x}\quad\forall\tilde{u}\in
H(\Gamma)$ (21)

adjoint equation and the function $v^{0}$, which solves (21), adjoint
solution. So we omit the word ’averaged’ in this case. Moreover, the LtN
problem (18) for $t=0$ is titled state equation and the solution $u^{0}$ is
named state solution. Finally, we can state how to derive the shape derivative
of the reduced functional.

######  Theorem 4.5 ([25, Theorem 3.1]).

Let the Assumptions (H0) and (H1) be fulfilled, $u^{t}$ solve (18) and let
$v^{t}$ be the unique solution to the averaged adjoint equation (19) for
$t\in[0,T]$. Then, for $v\in H(\Gamma)$ we obtain

$\displaystyle
D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]=\left.\frac{d}{dt}\right|_{t=0^{+}}J^{red}(\Gamma^{t})=\left.\frac{d}{dt}\right|_{t=0^{+}}G(t,u^{t},v)=\partial_{t}G(0,u^{0},v^{0}).$
(22)

###### Proof.

See proof of [25, Theorem 3.1]. ∎

###  4.3 Deriving the Shape Derivative of the Reduced Functional

In order to rigorously compute the shape derivative we make use of the
following Lemma.

######  Lemma 4.6 (After [36, Proposition 2.32]).

Let $D\subset\mathbb{R}^{d}$ be a bounded domain with nonzero measure, $f\in
W^{1,1}(D,\mathbb{R})$ and $\mathbf{V}\in C_{0}^{1}(D,\mathbb{R}^{d})$, where
$k\in\mathbb{N}$. Moreover, assume that $T>0$ is sufficiently small, such that
$\mathbf{F}_{\mathbf{t}}:\overline{D}\rightarrow\overline{D}$ is bijective for
all $t\in[0,T]$. Then, $t\mapsto f\circ\mathbf{F}_{\mathbf{t}}$ is
differentiable in $L^{1}(D)$ with

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}\left(f\circ\mathbf{F}_{\mathbf{t}}\right)=\nabla
f^{\top}\mathbf{V}.$

###### Proof.

Since $C^{\infty}(D)$ is dense in $W^{1,1}(D,\mathbb{R})$[3, Theorem 3.17], we
only need to show the result for $f\in C^{\infty}(D)$. Now, given $f\in
C^{\infty}(D)$, applying the mean value theorem yields

$\displaystyle
f(\mathbf{F}_{\mathbf{t}}(\mathbf{x}))-f(\mathbf{x})=\int_{0}^{1}t\nabla
f(\mathbf{x}+st\mathbf{V}(\mathbf{x}))^{\top}\mathbf{V}(\mathbf{x})~{}ds$

for $\mathbf{x}\in D$. Consequently, we derive

$\displaystyle\int_{D}\left|\frac{1}{t}\left(f(\mathbf{F}_{\mathbf{t}}(\mathbf{x}))-f(\mathbf{x})\right)-\nabla
f(\mathbf{x})^{\top}\mathbf{V}(\mathbf{x})\right|~{}d\mathbf{x}\leq\int_{D}\int_{0}^{1}\left|\left(\nabla
f(\mathbf{x}+st\mathbf{V}(\mathbf{x}))-\nabla
f(\mathbf{x})\right)^{\top}\mathbf{V}(\mathbf{x})\right|~{}ds~{}d\mathbf{x}$
$\displaystyle\leq\int_{0}^{1}\int_{D}\left|\left(\nabla
f(\mathbf{x}+st\mathbf{V}(\mathbf{x}))-\nabla
f(\mathbf{x})\right)^{\top}\mathbf{V}(\mathbf{x})\right|~{}d\mathbf{x}~{}ds,$
(23)

where we changed the order of integration in the last step. Now, we just have
to show, that the double integral (23) vanishes, when $t\rightarrow 0$. For
that reason we would like to show, that the inner integral converges to zero.
Since $\mathbf{V}\in C_{0}^{1}(D,\mathbb{R}^{d})$ is continuous and thus
bounded on $\overline{D}$ and $\nabla f(\mathbf{F}_{\mathbf{t}}(\mathbf{x}))$
converges in $L^{2}(D,\mathbb{R}^{d})$ and therefore also in
$L^{1}(D,\mathbb{R}^{d})$ to $\nabla f$ due to Lemma A.2, we conclude by
employing the dominated convergence theorem that

$\displaystyle\lim_{t\rightarrow 0}\int_{D}\left|\left(\nabla
f(\mathbf{x}+st\mathbf{V}(\mathbf{x}))-\nabla
f(\mathbf{x})\right)^{\top}\mathbf{V}(\mathbf{x})\right|~{}d\mathbf{x}=0\quad\text{for
all }s\in[0,1].$

Again, applying the dominated convergence theorem yields

$\displaystyle\lim_{t\rightarrow 0}\int_{0}^{1}\int_{D}\left|\left(\nabla
f(\mathbf{x}+st\mathbf{V}(\mathbf{x}))-\nabla
f(\mathbf{x})\right)^{\top}\mathbf{V}(\mathbf{x})\right|~{}d\mathbf{x}~{}ds=0.$

∎

######  Corollary 4.7.

Let $D\subset\mathbb{R}^{d}\times\mathbb{R}^{d}$ be an open and bounded domain
with nonzero measure. If $\gamma\in W^{1,1}(D,\mathbb{R})$ and
$\tilde{\mathbf{V}}\in C_{0}^{1}(D,\mathbb{R}^{d}\times\mathbb{R}^{d})$, then
$t\mapsto\gamma\circ\widetilde{\mathbf{F}}_{t}$, where
$\widetilde{\mathbf{F}}_{t}:=(\mathbf{x},\mathbf{y})+t\tilde{\mathbf{V}}(\mathbf{x},\mathbf{y})$,
is Fréchet differentiable in $L^{1}(D,\mathbb{R})$ at $t=0$ and its derivative
is given by

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}\gamma\circ\widetilde{\mathbf{F}}_{t}=\nabla\gamma^{T}\tilde{\mathbf{V}}.$

######  Remark 4.8.

The statement of Lemma 4.6 and Corollary 4.7 is still valid, if $\mathbf{V}\in
C_{0}^{1}(\hat{D},\mathbb{R}^{d})$, where the superset $\hat{D}\supset D$ is a
bounded and open domain.

In our case, we set
$\tilde{\mathbf{V}}(\mathbf{x},\mathbf{y}):=\left(\mathbf{V}(\mathbf{x}),\mathbf{V}(\mathbf{y})\right)$
in order to use Corollary 4.7 to derive the shape derivative of the LtN-
operator $A^{LtN}_{\Gamma}$. Now, we are able to prove that the assumptions of
the averaged adjoint method hold for the interface identification problem
constrained by the energy-based LtN coupling under an additional condition.

######  Theorem 4.9.

Let the kernel $\gamma$ be weakly differentiable and assume this weak
derivative to be essentially bounded, i.e. $\gamma\in
W^{1,\infty}(\left(\Omega\cup\mathcal{I}\right)\times\left(\Omega\cup\mathcal{I}\right))$.
Moreover, $\gamma$ is assumed to fulfill (K1)-(K4) and we define $G$ as in
(16). Then, the Assumptions (H0) and (H1) of the averaged adjoint method hold
and there exists a unique solution to the averaged adjoint equation (19) for
every $t\in[0,T]$.

###### Proof.

See Appendix A. ∎

######  Corollary 4.10.

Let $\gamma\in
W^{1,\infty}(\left(\Omega\cup\mathcal{I}\right)\times\left(\Omega\cup\mathcal{I}\right))$
fulfill (K1)-(K4) and let the functions $G$ and $J^{red}$ be defined as in
(16) and (14), respectively. Then, the shape derivative of the reduced cost
functional exists and can be computed by

$\displaystyle
D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]=\partial_{t}G(0,u^{0},v^{0}),$

where $u^{0}$ is the solution to the state equation (18) and $v^{0}$ solves
the adjoint equation (21).

In the remaining part of this subsection we would like to derive the shape
derivative of the reduced functional

$\displaystyle
D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]=D_{\Gamma}J(u^{0},\Gamma)[\mathbf{V}]+D_{\Gamma}A^{LtN}_{\Gamma}(u^{0},v^{0})[\mathbf{V}]-D_{\Gamma}F_{\Gamma}(v^{0})[\mathbf{V}]$

by separately computing the shape derivative of the objective functional $J$,
the bilinear operator $A^{LtN}_{\Gamma}$ and the linear functional
$F_{\Gamma}$.

######  Theorem 4.11.

Let $J$ and $F_{\Gamma}$ be defined as before. Then the shape derivatives of
$J$ and $F_{\Gamma}$ in the direction of a vector field $\mathbf{V}\in
C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})$ can be expressed as

$\displaystyle D_{\Gamma}J(u^{0})[\mathbf{V}]$
$\displaystyle=\int_{\Omega}-\left(u^{0}-\bar{u}\right)\nabla\bar{u}^{\top}\mathbf{V}+\left(u^{0}-\bar{u}\right)^{2}\operatorname{div}\mathbf{V}~{}d\mathbf{x}+\nu\int_{\Gamma}\operatorname{div}\mathbf{V}-\mathbf{n}^{\top}\nabla\mathbf{V}^{\top}\mathbf{n}~{}ds\text{
and}$ $\displaystyle D_{\Gamma}F_{\Gamma}(v^{0})[\mathbf{V}]$
$\displaystyle=\int_{\Omega}\left(\nabla
f_{\Gamma}^{\top}\mathbf{V}\right)v^{0}+f_{\Gamma}v^{0}\operatorname{div}\mathbf{V}~{}d\mathbf{x}.$

###### Proof.

First, from, e.g. [29], we derive $\xi^{0}(x)=\det
D\mathbf{F}_{0}(\mathbf{x})=\det({\bf Id})=1$ and
$\left.\frac{d}{dt}\right|_{t=0^{+}}\xi^{t}=\operatorname{div}\mathbf{V}$.
Thus, the shape derivative of the right-hand side $F_{\Gamma}$ can be computed
by applying Lemma 4.6 and the product rule of Fréchet derivatives as follows

$\displaystyle D_{\Gamma}F_{\Gamma}(v^{0})[\mathbf{V}]$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0^{+}}F_{\Gamma^{t}}(v^{0}\circ\mathbf{F}_{\mathbf{t}}^{-1})=\int_{\Omega}\left.\frac{d}{dt}\right|_{t=0^{+}}(f_{\Gamma}\circ\mathbf{F}_{\mathbf{t}})v^{0}\xi^{t}~{}d\mathbf{x}$
$\displaystyle=\int_{\Omega}\left(\nabla
f_{\Gamma}^{\top}\mathbf{V}\right)v^{0}~{}d\mathbf{x}+\int_{\Omega}f_{\Gamma}v^{0}~{}\operatorname{div}\mathbf{V}~{}d\mathbf{x}.$

Moreover, the shape derivative of the objective functional can be written as

$\displaystyle
D_{\Gamma}J(u^{0},\Gamma)[\mathbf{V}]=D_{\Gamma}j(u^{0},\Gamma)[\mathbf{V}]+D_{\Gamma}j_{per}(\Gamma)[\mathbf{V}]=\left.\frac{d}{dt}\right|_{t=0^{+}}j(u^{0}\circ\mathbf{F}_{\mathbf{t}}^{-1},\Gamma^{t})+\left.\frac{d}{dt}\right|_{t=0^{+}}j_{per}(\Gamma^{t}).$

Here, the shape derivative of the regularization term is given by[41, Theorem
4.13]

$\displaystyle
D_{\Gamma}j_{per}(u^{0},\Gamma)[\mathbf{V}]=\nu\int_{{\Gamma}}\operatorname{div}_{\Gamma}\mathbf{V}~{}ds=\nu\int_{{\Gamma}}\operatorname{div}\mathbf{V}-\mathbf{n}^{\top}\nabla\mathbf{V}^{\top}\mathbf{n}~{}ds,$

where $\mathbf{n}$ denotes the outer normal of
$\partial\Omega_{nl}\cap\partial\Omega_{l}$. Additionally, we obtain for the
shape derivative of the tracking-type functional

$\displaystyle D_{\Gamma}j(u^{0},\Gamma)[\mathbf{V}]$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0^{+}}j(u^{0}\circ\mathbf{F}_{\mathbf{t}}^{-1},\Gamma^{t})=\frac{1}{2}\left.\frac{d}{dt}\right|_{t=0^{+}}\int_{\mathbf{F}_{\mathbf{t}}(\Omega)}(u^{0}\circ\mathbf{F}_{\mathbf{t}}^{-1}-\bar{u})^{2}~{}d\mathbf{x}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\left.\frac{d}{dt}\right|_{t=0^{+}}(u^{0}-\bar{u}\circ\mathbf{F}_{\mathbf{t}})^{2}\xi^{t}~{}d\mathbf{x}=\int_{\Omega}-(u^{0}-\bar{u})\nabla\bar{u}^{\top}\mathbf{V}+(u^{0}-\bar{u})^{2}\operatorname{div}\mathbf{V}~{}d\mathbf{x}.$

∎

######  Theorem 4.12 (Shape Derivative of the Local-to-Nonlocal Operator).

Let $A^{LtN}_{\Gamma}$ be defined as before and let $\gamma\in
W^{1,\infty}(\left(\Omega\cup\mathcal{I}\right)\times\left(\Omega\cup\mathcal{I}\right))$
fulfill (K1)-(K4). Then, for a vector field $\mathbf{V}\in
C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})$ we can compute the shape
derivative via

$\displaystyle D_{\Gamma}$ $\displaystyle
A^{LtN}_{\Gamma}(u^{0},v^{0})[\mathbf{V}]=\int\limits_{\Omega_{l}}-\left(\left(\nabla\mathbf{V}(\mathbf{x})+\nabla\mathbf{V}^{\top}(\mathbf{x})\right)\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)+\left(\nabla
u^{0}(\mathbf{x}),\nabla
v^{0}(\mathbf{x})\right)\operatorname{div}\mathbf{V}(\mathbf{x})~{}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\nabla_{\mathbf{x}}\gamma(\mathbf{x},\mathbf{y})^{\top}\mathbf{V}(\mathbf{x})+\nabla_{\mathbf{y}}\gamma(\mathbf{x},\mathbf{y})^{\top}\mathbf{V}(\mathbf{y})\right)~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\operatorname{div}\mathbf{V}(\mathbf{x})+\operatorname{div}\mathbf{V}(\mathbf{y})\right)~{}d\mathbf{y}d\mathbf{x}.$
(24)

###### Proof.

We compute the shape derivative of the LtN-operator
(LABEL:eq:transformed_LtN_Op) in two steps. Therefore, we start by
differentiating the nonlocal part of $A^{LtN}$, i.e. the double integral, and
then we develop the shape derivative of the remaining integral.
In, e.g. [29] it is shown, that $\xi^{t}$ is continuously differentiable with
$\left.\frac{d}{dt}\right|_{t=0^{+}}\xi^{t}(\mathbf{x})=\operatorname{div}\mathbf{V}(\mathbf{x})$.
Corollary 4.7 combined with the product rule of Fréchet derivatives yields the
Fréchet derivative

$\displaystyle\left.\frac{d}{dt}\right|_{t=0^{+}}\left(\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\gamma^{t}(\mathbf{x},\mathbf{y})\xi^{t}(\mathbf{x})\xi^{t}(\mathbf{x})\right)$
$\displaystyle=\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\nabla_{\mathbf{x}}\gamma(\mathbf{x},\mathbf{y})^{\top}\mathbf{V}(\mathbf{x})+\nabla_{\mathbf{y}}\gamma(\mathbf{x},\mathbf{y})^{\top}\mathbf{V}(\mathbf{y})\right)$
$\displaystyle+\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})\left(\operatorname{div}\mathbf{V}(\mathbf{x})+\operatorname{div}\mathbf{V}(\mathbf{y})\right)\text{
in
}L^{1}(\Omega_{nl}\times\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right),\mathbb{R}).$

Moreover, according to, e.g. [38, Lemma 2.14]

$\displaystyle\left.\frac{d}{dt}\right|_{t=s}\mathcal{B}(t)(\mathbf{x})=$
$\displaystyle\operatorname{trace}\left(D\mathbf{V}(\mathbf{x})\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}\right)\mathcal{B}(t)(\mathbf{x})$
$\displaystyle-\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}D\mathbf{V}(\mathbf{x})\mathcal{B}(t)(\mathbf{x})-\left(\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}D\mathbf{V}(\mathbf{x})\mathcal{B}(t)(\mathbf{x})\right)^{\top},$

and therefore
$\left.\frac{d}{dt}\right|_{t=0^{+}}\mathcal{B}(t)=\operatorname{div}\mathbf{V}-D\mathbf{V}-D\mathbf{V}^{\top}$.
Finally, we derive by again utilizing the product rule of Fréchet derivatives

$\displaystyle\left.\frac{d}{dt}\right|_{t=0^{+}}\left(\mathcal{B}(t)(\mathbf{x})\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)$
$\displaystyle=-\left(\left(\nabla\mathbf{V}(\mathbf{x})+\nabla\mathbf{V}(\mathbf{x})^{\top}\right)\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)+\left(\nabla
u^{0}(\mathbf{x}),\nabla
v^{0}(\mathbf{x})\right)\operatorname{div}\mathbf{V}(\mathbf{x})\text{ in
}L^{1}(\Omega_{l},\mathbb{R}).$

∎

Combining Theorems 4.11 and 4.12 results in a formula for the shape derivative
of the reduced objective functional $D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]$.

######  Corollary 4.13.

Let the reduced objective functional $J$ be defined as in (14) and $\gamma\in
W^{1,\infty}(\left(\Omega\cup\mathcal{I}\right)\times\left(\Omega\cup\mathcal{I}\right))$
is supposed to satisfy (K1)-(K4).
Then, for $\mathbf{V}\in C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})$ we
can express the shape derivative $D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]$ as

$\displaystyle
D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]=D_{\Gamma}J(u^{0},\Gamma)[\mathbf{V}]+D_{\Gamma}A^{LtN}_{\Gamma}(u^{0},v^{0})[\mathbf{V}]-D_{\Gamma}F_{\Gamma}(v^{0})[\mathbf{V}]$
$\displaystyle=\int_{\Omega}-\left(u^{0}-\bar{u}\right)\nabla\bar{u}^{\top}\mathbf{V}+\left(u^{0}-\bar{u}\right)^{2}\operatorname{div}\mathbf{V}~{}d\mathbf{x}+\nu\int_{\Gamma}\operatorname{div}\mathbf{V}-\mathbf{n}^{\top}\nabla\mathbf{V}^{\top}\mathbf{n}~{}ds$
$\displaystyle+\int\limits_{\Omega_{l}}-\left(\left(\nabla\mathbf{V}+\nabla\mathbf{V}^{\top}\right)\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)+\left(\nabla
u^{0}(\mathbf{x}),\nabla
v^{0}(\mathbf{x})\right)\operatorname{div}\mathbf{V}(\mathbf{x})~{}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\nabla_{\mathbf{x}}\gamma(\mathbf{x},\mathbf{y})^{\top}\mathbf{V}(\mathbf{x})+\nabla_{\mathbf{y}}\gamma(\mathbf{x},\mathbf{y})^{\top}\mathbf{V}(\mathbf{y})\right)~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\operatorname{div}\mathbf{V}(\mathbf{x})+\operatorname{div}\mathbf{V}(\mathbf{y})\right)~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle-\int_{\Omega}\left(\nabla
f_{\Gamma}^{\top}\mathbf{V}\right)v^{0}+f_{\Gamma}v^{0}\operatorname{div}\mathbf{V}~{}d\mathbf{x}.$

###  4.4 Optimization Algorithm

In this section we outline the optimization procedure to solve the interface
identification problem constrained by Local-to-Nonlocal coupling as described
in Section 3.3. As shown in [30], shape optimization techniques can be
identified with corresponding methods in the Hilbert space
$\left(\mathcal{H},g\right)$, where
$\mathcal{H}\subset\\{T|T:\Omega\rightarrow\mathbb{R}^{d}\\}$ and $g$ is an
inner product on $\mathcal{H}$. Therefore, shape optimization problems can be
investigated as optimization problems in $\left(\mathcal{H},g\right)$, see
[30].
In the optimization algorithm, that we will deploy, we would like to derive a
Riesz-representation of the shape gradient $\nabla J^{red}$ for a scalar
product $b_{\Gamma}$ that can depend on the current shape $\Gamma$. Then,
given the state $u^{0}$ and adjoint solution $v^{0}$, we compute $\nabla
J^{red}$ by solving

$\displaystyle b_{\Gamma}(\nabla
J^{red},\mathbf{V})=D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]\quad\text{for all
}\mathbf{V}\in C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d}).$ (25)

In our case, we choose the linear elasticity operator as an inner product,
i.e.

$\displaystyle
b_{\Gamma}:C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})\times
C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})\rightarrow\mathbb{R},\quad
g(\mathbf{U},\mathbf{V}):=\int_{\Omega}\sigma(\mathbf{U}):\epsilon(\mathbf{V})~{}d\mathbf{x},\text{
where}$ $\displaystyle\sigma(\mathbf{U}):=\lambda tr(\epsilon(\mathbf{U})){\bf
Id}+2\mu\epsilon(\mathbf{U})\text{ and
}\epsilon(\mathbf{U}):=\frac{1}{2}\left(\nabla\mathbf{U}+\nabla\mathbf{U}^{\top}\right).$

Here, $\sigma(\mathbf{U})$ is called the strain and $\epsilon(\mathbf{U})$
stress tensor of $\mathbf{U}$ with Lame parameters $\lambda$ and $\mu$.
According to [31] a locally varying choice for $\mu$ can have a stabilizing
effect on the mesh. So, we follow [31] and set $\lambda=0$ and
$\mu(\Gamma):=\mu$ as the solution to

$\displaystyle\Delta\mu$ $\displaystyle=0\quad$ $\displaystyle\text{on
}\Omega,$ $\displaystyle\mu$ $\displaystyle=\mu_{min}\quad$
$\displaystyle\text{on }\partial\Omega,$ (26) $\displaystyle\mu$
$\displaystyle=\mu_{max}\quad$ $\displaystyle\text{on }\Gamma.$

As a consequence $\mu(\Gamma)$ is dependent on the current shape $\Gamma$ and
therefore the inner product $b_{\Gamma}$ is also depending on the interface
$\Gamma$. Moreover, the choice of $\mu_{min}$, $\mu_{max}\in\mathbb{R}$ has a
strong effect on the step size and the speed of convergence of the
optimization algorithm.
Further, we apply the finite element method to solve the described interface
identification problem. Accordingly, the domain $\Omega$ is represented by a
mesh and the boundary $\Gamma$ is characterized by edges. The triangle
vertices and, as a consequence, also the edges are then deformed by a vector
$\mathbf{U}_{k}$, which resembles the finite element approximation of the
descent direction in the linear space of deformations. Here, $\mathbf{U}_{k}$
is a linear combination of piecewise linear and continuous basis functions,
i.e. we apply continuous Galerkin. On the other hand, the state and adjoint
solution are chosen to be constructed by a linear combination of piecewise
linear and continuous basis functions on $\Omega_{nl}$ and on $\Omega_{l}$. As
a result, the state and adjoint can be discontinuous on the interface
$\Gamma$.
Furthermore, for vector fields that do not deform the interface $\Gamma$, i.e.
$\operatorname{supp}(\mathbf{V})\cap\Gamma=\emptyset$, it should hold that
$D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]=0$. However, in the finite element
setting it can happen that for such test vector fields $\mathbf{V}$ the
corresponding shape derivative $D_{\Gamma}J^{red}[\mathbf{V}]$ is not zero due
to discretization errors. Following [32], we therefore set in every iteration

$\displaystyle D_{\Gamma}J^{red}(\Gamma)[\mathbf{V}]=0\text{ for
}\mathbf{V}\in C_{0}^{1}(\Omega\cup\mathcal{I},\mathbb{R}^{d})\text{ with
}\operatorname{supp}(\mathbf{V})\cap\Gamma=\emptyset.$

The complete method is shown in Algorithm 2.

1 Given: $\bar{u}$, $\gamma$, $f_{\Gamma}$, $\texttt{maxiter}\in\mathbb{N}$,
$\texttt{tol}\in(0,\infty)$.

2 while  _$k\leq\texttt{maxiter}$ and
$||D_{\Gamma}J^{red}(\Gamma_{k})||>\texttt{tol}$_ do

3 Interpolate data $\bar{u}$ onto the current finite element mesh $\Omega_{k}$

4 $u(\Gamma_{k})\leftarrow$ solve state equation w.r.t. $\Gamma_{k}$.

5 $v(\Gamma_{k})\leftarrow$ Solve adjoint equation w.r.t. $\Gamma_{k}$.

6 $D_{\Gamma}J^{red}(\Gamma_{k})[\mathbf{V}]\leftarrow$ Assemble the shape
derivative for all basis functions $\mathbf{V}$.

7 Set $D_{\Gamma}J^{red}(\Gamma_{k})[\mathbf{V}]=0$ if
$\operatorname{supp}(\mathbf{V})\cap\Gamma_{k}=\emptyset$.

8 $\mu\leftarrow$ Derive the locally varying Lamé parameter.

9 $\nabla J^{red}(\Gamma_{k})\leftarrow$ Solve (25) to get Riesz-
representation of the shape gradient.

10 if  _curvature condition fulfilled_ then

11 $\mathbf{U}_{k}$ = L-BFGS-Update

12 else

13 $\mathbf{U}_{k}=-\nabla J^{red}(\Gamma_{k})$

14 end if

15 while  _$J^{red}(({\bf Id}+\alpha_{k}\mathbf{U}_{k})(\Gamma_{k}))
>J^{red}(\Gamma_{k})+cD_{\Gamma}J^{red}(\Gamma_{k})[\mathbf{U}_{k}]$_ do

16 $\alpha=\tau\alpha$

17 end while

18 $\alpha_{k}\leftarrow\alpha$

19 $\Omega_{k+1}=\left({\bf Id}+\alpha_{k}\mathbf{U}_{k}\right)(\Omega_{k})$

20 $k=k+1$

21 end while

Algorithm 2 Shape optimization algorithm

Additionally, after we have found a shape gradient $\nabla
J^{red}(\Gamma_{k})$ we make use of an limited memory BFGS-update, if the
corresponding curvature condition is satisfied. Otherwise we take the negative
gradient as the descent direction. For more information on how to apply
L-BFGS, we refer to [28]. After we have found a descent direction
$\mathbf{U}_{k}$ we compute the step length by applying a backtracking line
search in Lines 2-2, where we stop after the sufficient decrease condition is
satisfied. The parameter $c$ is chosen to be $c=10^{-4}$ as it is suggested in
[28].

######  Remark 4.14.

Since we use an iterative solver to compute solutions to the state and adjoint
equation, the application of (multi-step) one-shot methods as in [7, 8], where
state, adjoint and the shape are simultaneously calculated, seems to be a
natural way to solve the interface identification problem in a faster time.
However, the assembly of the stiffness matrix needed to compute the state or
adjoint solution is quite costly and compared to that the time saved by
stopping the algorithm for calculating $u(\Gamma_{k})$ and $v(\Gamma_{k})$
after a few iterations is rather small. Consequently, if we need more outer
iteration steps, (multi-step) one-shot methods do not yield a faster way to
solve the interface identification problem constrained by a LtN coupling. On
the other hand, we can only reduce the time by a few seconds, if we slightly
increase the tolerance for solving the state and adjoint equation such that we
still need the same number of outer iterations. Therefore, we keep the
standard approach introduced in this Section and do not use one-shot methods.

##  5 Numerical Experiments

In this section we demonstrate the shape optimization algorithm that was
introduced in the previous Section 4.4 on two examples. Snapshots of the
algorithm are depicted in Figure 5.1 for the first and in Figure 5.2 for the
second example.

|  |  |
---|---|---|---
Start setup | Iteration 3 | Iteration 5 | Iteration 14
Figure 5.1: Example 1 |  |  |
---|---|---|---
Start setup | Iteration 5 | Iteration 5 remeshed | Iteration 10 remeshed
|  |  |
| Iteration 15 | Iteration 20 |
Figure 5.2: Example 2

In both cases, we choose a small perimeter regularization with parameter
$\nu=0.001$ and the forcing term

$\displaystyle
f_{\Gamma}(\mathbf{x})=10\chi_{\Omega_{nl}}(\mathbf{x})-10\chi_{\Omega_{l}}(\mathbf{x}).$

In the first experiment, we use the kernel $\gamma_{1}$ and in the second
example the kernel $\gamma_{2}$, which are defined as follows

$\displaystyle\gamma_{1}(\mathbf{x},\mathbf{y})=\frac{4}{\pi\delta^{4}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})\quad\text{and}\quad\gamma_{2}(\mathbf{x},\mathbf{y})=\frac{4}{\pi\delta^{4}}\left(1-0.5\frac{||\mathbf{x}-\mathbf{y}||^{2}}{\delta^{2}}\right)\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y}).$

Since in the first case the kernel $\gamma_{1}$ is piecewise constant, the
shape derivative of the corresponding LtN-operator (24) reduces to

$\displaystyle D_{\Gamma}$ $\displaystyle
A^{LtN}_{\Gamma}(u^{0},v^{0})[\mathbf{V}]=\int\limits_{\Omega_{l}}-\left(\left(\nabla\mathbf{V}+\nabla\mathbf{V}^{\top}\right)\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)+\left(\nabla
u^{0}(\mathbf{x}),\nabla
v^{0}(\mathbf{x})\right)\operatorname{div}\mathbf{V}(\mathbf{x})~{}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{0}(\mathbf{x})-v^{0}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\operatorname{div}\mathbf{V}(\mathbf{x})+\operatorname{div}\mathbf{V}(\mathbf{y})\right)~{}d\mathbf{y}d\mathbf{x}.$

Now, we solve the interface identification problem described in Section 3.3 by
applying the finite element method, where we choose to employ piecewise linear
and continuous basis functions. Here, the underlying mesh is generated by Gmsh
[20]. The nonlocal components for the stiffness matrix regarding
$A^{LtN}_{\Gamma}$ that is used in the Schwarz method, see Section 3.2, as
well as the shape derivative $D_{\Gamma}A^{LtN}_{\Gamma}$ are computed by a
modified version of the Python package nlfem [24]. The local components and
the shape derivatives of the objective functional $D_{\Gamma}J$ and of the
forcing term $D_{\Gamma}F_{\Gamma}$ are assembled by using FEniCS [5, 26].
Then, the data $\bar{u}$ is computed as the solution of the Local-to-Nonlocal
coupling regarding a circle as the interface $\Gamma$, i.e. the boundary of
the nonlocal domain. This circle can be observed in black on every picture.
After that, we start with a different interface $\Gamma_{0}$ and try to
approximate $\bar{u}$ by following Algorithm 2, where we stop, if
$||D_{\Gamma}J^{red}(\Gamma_{k})||<5\cdot 10^{-5}$ or if $k>25$. As a result,
the nonlocal domain $\Omega_{nl}$, which is colored in blue, should roughly
end up as the disc with the black circle as the boundary.
In the first example the algorithm terminates after 14 iterations. Here, the
edges of the initial shape $\Gamma_{0}$ disappear due to the perimeter
regularization and the nonlocal domain $\Omega_{nl}$ has approximately the
above described target shape. As mentioned in Section 4.4 the convergence
speed and therefore the number of iterations is highly dependent on the choice
of Lamé parameters, which we derive from (26) with $\mu_{min}=0.0$ and
$\mu_{max}=1.0$ for both experiments.
In the second example, our initial shape is small compared to the target shape
and needs to move a bit to the upper right and therefore the mesh quality
decreases. Also, nodes in $\Omega$ are not allowed to be pushed outside of
$\Omega$, which is tested in the code by a function. So, in the beginning of
the algorithm some nodes are moved towards the boundary and as a result the
algorithm stagnates, because they cannot be pushed further. Therefore, we
remesh after the fifth and tenth iteration. In the end, the algorithm stops
after 20 iterations and we are able to recover the target shape quite well,
which can be seen in Picture 5.2.
The development of the objective function values and the norm of the shape
gradients for both examples are presented in Figure 5.3.

Figure 5.3: On the left hand side we see the development of the objective
functional values during the algorithm for each example. Here, the starting
solution of the first example is quite near the optimal solution such that the
objective function value decreases only a bit. In the second example the
objective function value is reduced quite fast in the first twelve iterations.
After that there is also only a small improvement regarding the objective
function value. On the right hand side, we can see the history of the
$L^{2}(\Omega)$ norm of the shape gradients.

##  6 Conclusion

As we have seen in this work, shape optimization techniques can be applied on
interface identification problems that are constrained by an energy-based
Local-to-Nonlocal coupling in order to find out where the nonlocal domain of
this coupling should be located. Here, the shape derivative of the reduced
functional is derived by the averaged adjoint method. In the future it could
be interesting to enlarge this framework to applications in the field of
Peridynamics, where most of these LtN combinations are currently employed.

## Acknowledgements

This work has been supported by the German Research Foundation (DFG) within
the Research Training Group 2126: ’Algorithmic Optimization’.

## Disclosure statement

The authors report there are no competing interests to declare.

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##  Appendix A Proof of Lemma 4.9

For this proof we rely on the next two Lemmata:

######  Lemma A.1.

There exists a positive constant $C\in(0,\infty)$, such that

$\displaystyle A^{LtN}(t,u,u)\geq C||u||_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}^{2}\quad\text{for all }u\in
H(\Gamma)\text{ and }t\in[0,T].$

###### Proof.

Since there exists a positive constant $0<C_{1}<\infty$ with
$C_{1}|\mathbf{x}|^{2}\leq\mathbf{x}^{\top}\mathcal{B}(t)(\mathbf{x})$(see
[14, Chapter 10.2.4]), if $t$ is small enough, and since
$\gamma(\mathbf{x},\mathbf{y})$ is bounded from below by $\gamma_{0}$, if
$\mathbf{y}\in B_{\epsilon}(\mathbf{x})$(see Chapter 2), we set
$C_{2}:=\min\\{C_{1},\gamma_{0}\\}$ and notice that

$\displaystyle A^{LtN}(t,u,u)=$
$\displaystyle\int_{\Omega_{l}}\left(\mathcal{B}(t)\nabla u,\nabla
u\right)~{}d\mathbf{x}+\int_{\Omega_{nl}}\int_{\Omega_{nl}\cup\Omega_{l}}\left(u(\mathbf{x})-u(\mathbf{y})\right)^{2}\gamma^{t}(\mathbf{x},\mathbf{y})\xi^{t}(\mathbf{x})\xi^{t}(\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int_{\Omega_{nl}}u(\mathbf{x})^{2}\int_{\mathcal{I}}\gamma^{t}(\mathbf{x},\mathbf{y})\xi^{t}(\mathbf{x})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle\geq$ $\displaystyle C_{2}\int_{\Omega_{l}}\nabla
u^{2}~{}d\mathbf{x}+\int_{\Omega_{nl}}\int_{\Omega_{nl}\cup\Omega_{l}}\left(u(\mathbf{x})-u(\mathbf{y})\right)^{2}C_{2}\chi_{B_{\varepsilon}(\mathbf{x})}(\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int_{\Omega_{nl}}u(\mathbf{x})^{2}\int_{\mathcal{I}}C_{2}\chi_{B_{\varepsilon}(\mathbf{x})}(\mathbf{y})~{}d\mathbf{y}d\mathbf{x}\geq
C||u||_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}^{2},$

where the last step is a result from Lemma 3.5 applied on
$\tilde{\gamma}(\mathbf{x},\mathbf{y})=C_{2}\chi_{B_{\varepsilon}(\mathbf{x})}(\mathbf{y})$
in the corresponding bilinear form, which yields such a constant
$C\in(0,\infty)$. ∎

######  Lemma A.2 (After [14, Chapter 10.2.4 Lemma 2.1]).

Let $\hat{\Omega}\subset\mathbb{R}^{d}$ be a bounded and open domain with
nonzero measure, $\mathbf{V}\in C_{0}^{1}(\hat{\Omega},\mathbb{R}^{d})$ and
$n\in\mathbb{N}$. Then, for $g\in L^{2}(\hat{\Omega},\mathbb{R}^{n})$, we get

$\displaystyle||g\circ\mathbf{F}_{\mathbf{t}}-g||_{L^{2}(\hat{\Omega},\mathbb{R}^{n})}\rightarrow
0\text{ and
}||g\circ\mathbf{F}_{\mathbf{t}}^{-1}-g||_{L^{2}(\hat{\Omega},\mathbb{R}^{n})}\rightarrow
0\quad\text{for }t\searrow 0.$

###### Proof.

Since $g$ and $\mathbf{V}$ can be extended by zero to functions in
$L^{2}(\mathbb{R}^{d})$ and $C_{0}^{1}(\mathbb{R}^{d},\mathbb{R}^{d})$,
respectively, we refer for the case $n=1$ to the proof of [14, Chapter 10.2.4
Lemma 2.1]. Then, the cases $n\in\mathbb{N}$ with $n\geq 2$ are a direct
consequence. ∎

###### Proof.

Define $f_{\Gamma}^{t}(\mathbf{x}):=f_{\Gamma}(F_{t}(\mathbf{x}))$ and
${\nabla f_{\Gamma}^{t}(\mathbf{x}):=\nabla f_{\Gamma}(F_{t}(\mathbf{x}))}$.
Assumption(H0):
We prove that $G(t,su^{t}+(1-s)u^{0},v)$ is absolutely continuous in $s$ by
showing that
$\tilde{G}:[0,1]\rightarrow\mathbb{R},\tilde{G}(s)=G(t,su^{t}+(1-s)u^{0},v)$
is continuously differentiable. Since $A$ is linear in the second argument, we
can directly conclude

$\displaystyle\partial_{s}\tilde{G}(s)$
$\displaystyle=d_{u}G(t,su^{t}+(1-s)u^{0},v)[u^{t}-u^{0}]$
$\displaystyle=d_{u}A^{LtN}(t,su^{t}+(1-s)u^{0},v)[u^{t}-u^{0}]+d_{u}J(t,su^{t}+(1-s)u^{0})[u^{t}-u^{0}]$
$\displaystyle=A^{LtN}(t,u^{t}-u^{0},v)+\int_{\Omega}\left(su^{t}+(1-s)u^{0}-\bar{u}^{t}\right)\left(u^{t}-u^{0}\right)\xi^{t}~{}d\mathbf{x}$
$\displaystyle=A^{LtN}(t,u^{t}-u^{0},v)+s\int_{\Omega}\left(u^{t}-u^{0}\right)^{2}\xi^{t}~{}d\mathbf{x}+\int_{\Omega}\left(u^{0}-\bar{u}^{t}\right)\left(u^{t}-u^{0}\right)\xi^{t}~{}d\mathbf{x},$

which is clearly continuous in $s$, i.e. $\tilde{G}$ is continuously
differentiable and the first part of Assumption (H0) holds. Furthermore, the
second criterion of (H0) is also fulfilled, since

$\displaystyle\int_{0}^{1}\left|d_{u}G(t,su^{t}+(1-s)u^{0},v)[\tilde{u}]\right|~{}ds=\int_{0}^{1}\left|A^{LtN}(t,\tilde{u},v)+\int_{\Omega}\left(s(u^{t}-u^{0})+(u^{0}-\bar{u}^{t})\right)\tilde{u}\xi^{t}~{}d\mathbf{x}\right|~{}ds$
$\displaystyle\leq\left|A^{LtN}(t,\tilde{u},v)\right|+\frac{1}{2}\left|\int_{\Omega}\left(u^{t}-u^{0}\right)\tilde{u}\xi^{t}~{}d\mathbf{x}\right|+\left|\int_{\Omega}\left(u^{0}-\bar{u}^{t}\right)\tilde{u}\xi^{t}~{}d\mathbf{x}\right|<\infty.$

Notice that the averaged adjoint equation (19) can be rephrased as

$\displaystyle
A^{LtN}(t,\tilde{u},v^{t})=-\int_{\Omega}\left(\frac{1}{2}(u^{t}+u^{0})-\bar{u}^{t}\right)\tilde{u}\xi^{t}~{}d\mathbf{x},$
(27)

where the right-hand side can be interpreted as a linear and continuous
operator with regards to $\tilde{u}$ and therefore, equation (27) is also a
Local-to-Nonlocal coupling problem which has a unique solution $v^{t}\in
H(\Gamma)$.
We now show that $\\{u^{t}\\}_{t\in[0,T]}$ and $\\{v^{t}\\}_{t\in[0,T]}$ are
bounded, which will be used to prove assumption (H1). Therefore Lemma A.1
yields the existence of a constant $C\in(0,\infty)$ with

$\displaystyle||u^{t}||_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}^{2}\leq
CA^{LtN}(t,u^{t},u^{t})=C|\int_{\Omega}u^{t}f_{\Gamma}^{t}\xi^{t}d\mathbf{x}|\leq
C||f_{\Gamma}^{t}\xi^{t}||_{L^{2}(\Omega)}||u^{t}||_{L^{2}(\Omega)}$
$\displaystyle\Rightarrow||u^{t}||_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}\leq C_{1},$ (28)

where we used in the last step, that
$||u^{t}||_{L^{2}(\Omega)}\leq||u^{t}||_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}$ and
$||f_{\Gamma}^{t}\xi^{t}||_{L^{2}(\Omega)}\rightarrow||f_{\Gamma}||_{L^{2}(\Omega)}$(Lemma
A.2), such that consequently
$\\{||f_{\Gamma}^{t}\xi^{t}||_{L^{2}(\Omega)}\\}_{t\in[0,T]}$ is bounded and
(28) holds for some $C_{1}\in(0,\infty)$. Additionally, since also
$\\{||\bar{u}^{t}||_{L^{2}(\Omega)}\\}_{t\in[0,T]}$ is bounded from above by a
positive constant as a consequence of Lemma A.2, we similarly derive the
existence of a constant $C_{2}\in(0,\infty)$ with

$\displaystyle||v^{t}||^{2}_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}\leq
C\left|\left((\frac{1}{2}(u^{t}+u^{0})-\bar{u}^{t})\xi^{t},v^{t}\right)_{L^{2}(\Omega)}\right|\leq\bar{\xi}C||\frac{1}{2}(u^{t}+u^{0})-\bar{u}^{t}||_{L^{2}(\Omega)}||v^{t}||_{L^{2}(\Omega)}$
$\displaystyle\Rightarrow||v^{t}||_{H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})}\leq C_{2}.$

Assumption (H1):
In this part of the proof we make use of the following observation:
Given a domain $\widehat{\Omega}\subset\mathbb{R}^{d}$ for $d\in\mathbb{N}$.
Let a sequence $\\{t_{k}\\}_{k\in\mathbb{N}}\in[0,T]^{\mathbb{N}}$ with
$t_{k}\rightarrow 0$, if $k\rightarrow\infty$. Moreover, assume
$g^{t}$,$h^{t_{k}},h^{0}\in L^{2}(\widehat{\Omega})$ for $t\in[0,T]$,
$k\in\mathbb{N}$ and $h^{t_{k}}\rightharpoonup h^{0}\in
L^{2}(\widehat{\Omega})$ for $k\rightarrow\infty$ as well as $g^{t}\rightarrow
g^{0}\in L^{2}(\widehat{\Omega})$, if $t\searrow 0$. Then we get

$\displaystyle\begin{split}&\left|\int_{\widehat{\Omega}}h^{t_{k}}g^{t}-h^{0}g^{0}~{}d\mathbf{x}\right|\leq\left|\int_{\widehat{\Omega}}h^{t_{k}}\left(g^{t}-g^{0}\right)~{}d\mathbf{x}\right|+\left|\int_{\widehat{\Omega}}\left(h^{t_{k}}-h^{0}\right)g^{0}~{}d\mathbf{x}\right|\\\
&\leq||h^{t_{k}}||_{L^{2}(\widehat{\Omega})}||g^{t}-g^{0}||_{L^{2}(\widehat{\Omega})}+\left|\int_{\widehat{\Omega}}\left(h^{t_{k}}-h^{0}\right)g^{0}~{}d\mathbf{x}\right|\rightarrow
0\text{ for }t\searrow 0\text{ and }k\rightarrow\infty.\end{split}$ (29)

As a first step we will show that $u^{t}\rightharpoonup u^{0}$ and
$v^{t}\rightharpoonup v^{0}$ in $H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})$ and consequently also in
$H(\Gamma)$.
Due to the boundedness of $u^{t}$ and $v^{t}$ in $H^{1}(\Omega_{l})\times
L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})$ for $t\in[0,T]$ there exists for
every sequence $\\{t_{n}\\}_{n\in\mathbb{N}}$ with $t_{n}\rightarrow 0$ for
$n\rightarrow\infty$ two subsequences $\\{t_{n_{k}}\\}_{k\in\mathbb{N}}$,
$\\{t_{n_{l}}\\}_{l\in\mathbb{N}}$ and two functions $q_{1},q_{2}\in
H^{1}(\Omega_{l})\times L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})$ such that
$u^{t_{n_{l}}}\rightharpoonup q_{1}$ and $v^{t_{n_{k}}}\rightharpoonup q_{2}$
in $H^{1}(\Omega_{l})\times L_{c}^{2}(\Omega_{nl}\cup\mathcal{I}_{nl})$.
With Lemma A.2 it holds that $\gamma^{t}\rightarrow\gamma$ in
$L^{2}(\Omega_{nl}\times(\Omega_{nl}\cup\mathcal{I}_{nl}))$ and thus

$\displaystyle\psi^{t}(\mathbf{x},\mathbf{y}):=\left(v(\mathbf{x})-v(\mathbf{y})\right)\gamma^{t}(\mathbf{x},\mathbf{y})\xi^{t}(\mathbf{x})\xi^{t}(\mathbf{y})\rightarrow\left(v(\mathbf{x})-v(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})=\psi^{0}(\mathbf{x},\mathbf{y})$

pointwise for a.e.
$(\mathbf{x},\mathbf{y})\in\Omega_{nl}\times(\Omega_{nl}\cup\mathcal{I}_{nl})$.
Since the kernel $\gamma^{t}$ and $\xi^{t}$ are bounded, we derive by applying
the dominated convergence theorem that $\psi^{t}\rightarrow\psi^{0}$ in
$L^{2}(\Omega_{nl}\times(\Omega_{nl}\cup\mathcal{I}_{nl}))$. Therefore, by
employing (LABEL:weak_convergence_lemma_2) we can conclude

$\displaystyle\int_{\Omega_{nl}}\int_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(u^{t_{n_{l}}}(\mathbf{x})-u^{t_{n_{l}}}(\mathbf{y})\right)\left(v(\mathbf{x})-v(\mathbf{y})\right)\gamma^{t_{n_{l}}}(\mathbf{x},\mathbf{y})\xi^{t_{n_{l}}}(\mathbf{x})\xi^{t_{n_{l}}}(\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\int_{\Omega_{nl}}\int_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(u^{t_{n_{l}}}(\mathbf{x})-u^{t_{n_{l}}}(\mathbf{y})\right)\psi^{t}(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}\rightarrow\int_{\Omega_{nl}}\int_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(q_{1}(\mathbf{x})-q_{1}(\mathbf{y})\right)\psi^{0}(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\int_{\Omega_{nl}}\int_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(q_{1}(\mathbf{x})-q_{1}(\mathbf{y})\right)\left(v(\mathbf{x})-v(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}~{}d\mathbf{x}.$

Moreover, we can directly follow by using (LABEL:weak_convergence_lemma_2) and
the continuity of $\mathcal{B}(t)$ in $t$ that

$\displaystyle\int_{\Omega_{l}}\left(\mathcal{B}(t)\nabla v,\nabla
u^{t_{n_{l}}}\right)~{}d\mathbf{x}\rightarrow\int_{\Omega_{l}}\left(\nabla
v,\nabla q_{1}\right)~{}d\mathbf{x}.$

Since $f_{\Gamma}^{t}\xi^{t}\rightarrow f_{\Gamma}$ in $L^{2}(\Omega)$ due to
Lemma A.2 we get

$\displaystyle
A^{LtN}(0,q_{1},v)=\lim_{l\rightarrow\infty}A^{LtN}(t_{n_{l}},u^{t_{n_{l}}},v)=\lim_{l\rightarrow\infty}\int_{\Omega}f_{\Gamma}^{t_{n_{l}}}\xi^{t_{n_{l}}}v~{}d\mathbf{x}=\int_{\Omega}f_{\Gamma}v~{}d\mathbf{x}.$

Because the solution is unique, we derive $u^{0}=q_{1}$ and
$u^{t}\rightharpoonup u^{0}$. Analogously, one can show

$\displaystyle
A^{LtN}(0,\hat{u},q_{2})=\lim_{k\rightarrow\infty}A^{LtN}(t_{n_{k}},\hat{u},v^{t_{n_{k}}})$
$\displaystyle=-\lim_{k\rightarrow\infty}\int_{\Omega}\left(\frac{1}{2}\left(u^{t_{n_{k}}}+u^{0}\right)-\bar{u}^{t_{n_{k}}}\right)\hat{u}\xi^{t_{n_{k}}}~{}d\mathbf{x}$
$\displaystyle=-\int_{\Omega}\left(u^{0}-\bar{u}\right)\hat{u}~{}d\mathbf{x},$

which yields $q_{2}=v^{0}$ and $v^{t}\rightharpoonup v^{0}$ for $t\searrow 0$.
In order to show Assumption (H1) we make use of the following observation: The
mean value theorem yields the existence of an $s_{t}\in(0,t)$ such that

$\displaystyle\frac{G(t,u^{0},v^{t})-G(0,u^{0},v^{t})}{t}=\partial_{t}G(s_{t},u^{0},v^{t}).$

Therefore, if

$\displaystyle\lim_{s,t\searrow
0}\partial_{t}G(t,u^{0},v^{s})=\partial_{t}G(0,u^{0},v^{0})$ (30)

holds, Assumption (H1) is fulfilled. Since
$\partial_{t}G(t,u^{0}v^{s})=\partial_{t}A^{LtN}(t,u^{0},v^{s})-\partial_{t}F(t,v^{s})+\partial_{t}J(t,u^{0})$
we show the convergence (30) in three steps.
First, we want to mention that $\gamma\in
W^{1,\infty}((\Omega\cup\mathcal{I})\times(\Omega\cup\mathcal{I}))\subset
W^{1,1}((\Omega\cup\mathcal{I})\times(\Omega\cup\mathcal{I}))$,
$f_{\Gamma}\in H^{1}(\Omega)\subset W^{1,1}(\Omega)$, such that we can apply
Lemma 4.6 or Corollary 4.7, and that the functions $\xi^{t}$ and
$\mathcal{B}(t)$ are continuously differentiable for $t\in[0,T]$. As a result,
the following partial derivatives are all derived by by applying the product
rule of Fréchet derivatives in $L^{1}(\Omega)$ and we just have to show the
convergence for $t\searrow 0$. Thus, we can conclude by again using
(LABEL:weak_convergence_lemma_2) that

$\displaystyle\partial_{t}F(t,v^{s})$ $\displaystyle=\int_{\Omega}(\nabla
f_{\Gamma}^{t})^{T}\mathbf{V}v^{s}\xi^{t}+f_{\Gamma}^{t}v^{s}\left.\frac{d}{dr}\right|_{r=t^{+}}\xi^{r}~{}d\mathbf{x}$
$\displaystyle\rightarrow\int_{\Omega}(\nabla
f_{\Gamma})^{T}\mathbf{V}v^{0}+f_{\Gamma}v^{0}\operatorname{div}\mathbf{V}~{}d\mathbf{x}=\partial_{t}F(0,v^{0}),$

since $\nabla f_{\Gamma}^{t}\xi^{t}\rightarrow\nabla f_{\Gamma}$ in
$L^{2}(\Omega,\mathbb{R}^{d})$, $f_{\Gamma}^{t}\rightarrow f_{\Gamma}$ in
$L^{2}(\Omega)$ by utilizing Lemma A.2 and $(\nabla
f_{\Gamma}^{t})^{\top}\mathbf{V}\xi^{t}\rightarrow(\nabla
f_{\Gamma})^{\top}\mathbf{V}$ in $L^{2}(\Omega)$ due to the boundedness of
$\mathbf{V}$. Then, $\lim_{t\searrow
0}\partial_{t}J(t,u^{0})=\partial_{t}J(0,u^{0})$ can be proved in a similar
manner. In the next step, we investigate the partial derivative of the
bilinear form $A^{LtN}$ as

$\displaystyle\partial_{t}A^{LtN}(t,u^{0},v^{s})=\int\limits_{\Omega_{l}}\left(\mathcal{B}^{\prime}(t)(\mathbf{x})\nabla
u^{0}(\mathbf{x}),\nabla v^{s}(\mathbf{x})\right)~{}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{s}(\mathbf{x})-v^{s}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\gamma^{t}(\mathbf{x},\mathbf{y})\left.\frac{d}{dr}\right|_{r=t^{+}}\left(\xi^{r}(\mathbf{x})\xi^{r}(\mathbf{y})\right)~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int\limits_{\Omega_{nl}}\int\limits_{\Omega_{nl}\cup\mathcal{I}_{nl}}\left(v^{s}(\mathbf{x})-v^{s}(\mathbf{y})\right)\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\nabla_{\mathbf{x}}\gamma^{t}(\mathbf{x},\mathbf{y})\mathbf{V}(\mathbf{x})+\nabla_{\mathbf{y}}\gamma^{t}(\mathbf{x},\mathbf{y})\mathbf{V}(\mathbf{y})\right)\xi^{t}(\mathbf{x})\xi^{t}(\mathbf{y})~{}d\mathbf{y}d\mathbf{x},$

where we used that $t\rightarrow\mathcal{B}(t)$ is continuously differentiable
for every $\mathbf{x}\in\Omega_{l}$ with(see [38, Lemma 2.14])

$\displaystyle\mathcal{B}^{\prime}(t)(\mathbf{x})=$
$\displaystyle\operatorname{trace}\left(D\mathbf{V}(\mathbf{x})\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}\right)\mathcal{B}(t)(\mathbf{x})$
$\displaystyle-\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}D\mathbf{V}(\mathbf{x})\mathcal{B}(t)(\mathbf{x})-\left(\left(D\mathbf{F}_{\mathbf{t}}(\mathbf{x})\right)^{-1}D\mathbf{V}(\mathbf{x})\mathcal{B}(t)(\mathbf{x})\right)^{\top},$

Therefore, $\mathcal{B}^{\prime}(t)\nabla
u^{0}\rightarrow\mathcal{B}^{\prime}(0)\nabla u^{0}$ in
$L^{2}(\Omega_{l},\mathbb{R}^{d})$ due to the dominated convergence theorem,
and then (LABEL:weak_convergence_lemma_2) yields

$\displaystyle\lim_{s,t\searrow
0}\int_{\Omega_{l}}\left(\mathcal{B}^{\prime}(t)(\mathbf{x})\nabla
u^{0}(\mathbf{x}),\nabla
v^{s}(\mathbf{x})\right)~{}d\mathbf{x}=\int_{\Omega_{l}}\left(\mathcal{B}^{\prime}(0)(\mathbf{x})\nabla
u^{0}(\mathbf{x}),\nabla v^{0}(\mathbf{x})\right)~{}d\mathbf{x}.$

Moreover, since
$\gamma^{t}(\mathbf{x},\mathbf{y})\rightarrow\gamma(\mathbf{x},\mathbf{y})$ in
$L^{2}(\Omega_{nl}\times\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right))$,
$\nabla\gamma^{t}(\mathbf{x},\mathbf{y})\rightarrow\nabla\gamma(\mathbf{x},\mathbf{y})$
in
$L^{2}(\Omega_{nl}\times\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right),\mathbb{R}^{d}\times\mathbb{R}^{d})$
and $\xi^{t}(\mathbf{x})$ continuous in $t$ for almost every
$\mathbf{x}\in\Omega_{nl}$ we get

$\displaystyle\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\gamma^{t}(\mathbf{x},\mathbf{y})\left.\frac{d}{dr}\right|_{r=t^{+}}\left(\xi^{r}(\mathbf{x})\xi^{r}(\mathbf{y})\right)$
$\displaystyle\rightarrow\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})\left(\operatorname{div}\mathbf{V}(\mathbf{x})+\operatorname{div}\mathbf{V}(\mathbf{y})\right)\text{
in }L^{2}(\Omega_{nl}\times\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right))\text{
for }t\searrow 0\text{ and}$
$\displaystyle\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\nabla_{\mathbf{x}}\gamma^{t}(\mathbf{x},\mathbf{y})\mathbf{V}(\mathbf{x})+\nabla_{\mathbf{y}}\gamma^{t}(\mathbf{x},\mathbf{y})\mathbf{V}(\mathbf{y})\right)\xi^{t}(\mathbf{x})\xi^{t}(\mathbf{y})$
$\displaystyle\rightarrow\left(u^{0}(\mathbf{x})-u^{0}(\mathbf{y})\right)\left(\nabla_{\mathbf{x}}\gamma(\mathbf{x},\mathbf{y})\mathbf{V}(\mathbf{x})+\nabla_{\mathbf{y}}\gamma(\mathbf{x},\mathbf{y})\mathbf{V}(\mathbf{y})\right)\text{
in }L^{2}(\Omega_{nl}\times\left(\Omega_{nl}\cup\mathcal{I}_{nl}\right))\text{
for }t\searrow 0.$

Again using (LABEL:weak_convergence_lemma_2) yields $\lim_{s,t\searrow
0}\partial_{t}A^{LtN}(t,u^{0},v^{s})=A^{LtN}(0,u^{0},v^{0})$. ∎