# Stability and reconstruction of a special type of anisotropic conductivity
in magneto-acoustic tomography with magnetic induction

Niall Donlon Romina Gaburro
Shari Moskow Isaac Woods Department of Mathematics and Statistics, University
of Limerick, Ireland. Email<EMAIL_ADDRESS>of Mathematics and
Statistics, Health Research Institute (HRI), CONFIRM, University of Limerick,
Ireland. Email<EMAIL_ADDRESS>of Mathematics, Drexel
University, Philadelphia, PA, USA. Email<EMAIL_ADDRESS>of
Mathematics, Drexel University, Philadelphia, PA, USA. Email<EMAIL_ADDRESS>
###### Abstract

We consider the issues of stability and reconstruction of the electrical
anisotropic conductivity of biological tissues in a domain
$\Omega\subset\mathbb{R}^{3}$ by means of the hybrid inverse problem of
magneto-acoustic tomography with magnetic induction (MAT-MI). The class of
anisotropic conductivities considered here is of type
$\sigma(\cdot)=A(\cdot,\gamma(\cdot))$ in $\Omega$, where
$[\lambda^{-1},\lambda]\ni t\mapsto A(\cdot,t)$ is a one-parameter family of
matrix-valued functions which are a-priori known to be $C^{1,\beta}$, allowing
us to stably reconstruct $\gamma$ in $\Omega$ in terms of an internal
functional $F(\sigma)$. Our results also extend previous results in MAT-MI
where $\sigma(\cdot)=\gamma(\cdot)D(\cdot)$, with $D$ an a-priori known
matrix-valued function on $\Omega$ to a more general anisotropic structure
which depends non-linearly on the scalar function $\gamma$ to be
reconstructed.

##  1 Introduction

In many physical situations, it is important to determine certain physical
properties of the interior of a body that cannot be measured directly. Inverse
problems are employed to infer such information from external observations. If
one is interested in imaging biological tissue inside the human body, its
electrical conductivity distribution can provide valuable information about
its state. A recently developed non-invasive imaging modality based on the
determination of the electrical conductivity distribution of biological tissue
is Magnetoacoustic Tomography with Magnetic Induction (MAT-MI).

If a biological conductive body, modelled by a domain
$\Omega\subset\mathbb{R}^{3}$ with smooth boundary $\partial\Omega$, is placed
in a static magnetic field $B_{0}=(0,0,1)$, it starts to emit ultrasound waves
that can be measured around the body, i.e. at a series of locations on
$\partial\Omega$. To be more precise, to cause an eddy current within the
conductive tissue, a pulsed time-dependent magnetic field $B(t)$ is applied to
excite the tissue in $\Omega$, which, in turn, emits the ultrasound waves. For
the sake of simplicity and to follow the research line already initiated in
[8], [36], we assume here that $B(t)$ is time-independent and that
$B(t)=B_{0}$. MAT-MI is an example of hybrid inverse problems, which typically
combine a high contrast modality with a high resolution one and they typically
involve two steps. In the first step, a well-posed problem involving high
resolution and low contrast modality is solved from the knowledge of boundary
measurements. In the second step, a quantitative reconstruction of the
parameters of interest (describing a physical property of the medium in
question) is solved by knowledge of a so-called internal functional which has
been reconstructed during the first step. Hybrid methods are mainly concerned
with the solution of the second step (assuming that the first step has been
successfully performed). For a review on hybrid imaging modalities we refer to
[11].

The first step in MAT-MI is to retrieve the acoustic source from the
measurements around the object in the scalar wave equation. Then, in the
second step, MAT-MI reconstructs the distribution of electrical conductivity
from acoustic source information (see [8], [36]). This second step is the
focus of this paper, where we study, in the MAT-MI experiment, the issues of
stability and reconstruction for a special type of anisotropic conductivity
$\sigma$ in terms of internal measurements of the acoustic sources (which are
assumed to be known after the first step has been performed) modelled by the
internal functional

(1.1) $F(\sigma)=\nabla\cdot(\sigma E_{\sigma}\times
B_{0}),\qquad\textnormal{in}\quad\Omega,$

where $E_{\sigma}$ solves the Maxwell’s equations

(1.2) $\Bigg{\\{}\begin{array}[]{llll}\nabla\times
E_{\sigma}&=B_{0},&\text{in}\quad\Omega,\\\ \nabla\cdot\sigma
E_{\sigma}&=0,&\text{on}\quad\Omega,\\\ \sigma
E_{\sigma}\cdot\nu&=0,&\text{on}\quad\partial\Omega\end{array}$

and $\sigma\in C^{1,\beta}(\overline{\Omega})$ is symmetric on $\Omega$ and
satisfies a uniform ellipticity condition (the precise formulation of the
problem is given in section 2).

For clinical and research purposes, the electrical conductivity of biological
tissues can provide valuable information as the conductivity varies
significantly within the human body. Other and more established medical
imaging modalities, like Computerized Tomography (CT), Magnetic Resonance
Imaging (MRI) and ultrasound imaging, are typically capable of creating images
of the human body with very high resolution. These modalities often fail to
exhibit a sufficient contrast between different type of tissues. Imaging
modalities like Optical Tomography (OT) and Electrical Impedance Tomography
(EIT) do, on the other hand, display such high contrast at the expense of a
poor resolution ([9], [13], [41]). EIT, similarly to MAT-MI, also provides
information about materials and biological tissues in terms of their
conductivity. The resulting images from measurements of electrostatic voltages
and current flux taken on the surface of the body under investigation, are
often blurred, due to the EIT poor resolution. MAT-MI, on the other hand, has
the potential to overcome this issue by providing images of the conductivity
of a body in terms of internal measurements that have been obtained by means
of a high resolution imaging modality performed in the first step of MAT-MI.
Hence MAT-MI has the potential to provide high contrast images of a body in
terms of its internal conductivity that also benefits from a reasonably good
resolution.

Biological tissues are known to have a have anisotropic conductivity [30]. In
EIT, since its first mathematical formulation by A. Calderón in his 1980
seminal paper [16], a lot of progress has been made, but the problem of
uniquely determining the anisotropic conductivity of a body by means of EIT
measurements is still considered an open problem. Partial results of
uniqueness and stability for this inverse problem have been obtained in [1],
[2], [3], [4], [5], [6], [10], [12], [18], [19], [20] [21], [27], [32], [33],
[34], [35], [39].

In the MAT-MI experiment, results of stability and reconstruction of the
conductivity $\sigma$ in terms of the internal functional $F(\sigma)$ given in
(1.1), have been obtained in [36] and [8], where the isotropic case
$\sigma=\gamma I$ (here $I$ denotes the $3\times 3$ identity matrix and
$\gamma$ is a positive scalar function on $\Omega$) and the special
anisotropic case $\sigma=\gamma D$, with $D$ a known $3\times 3$ symmetric
matrix and $\gamma$ is a positive scalar function on $\Omega$, were
considered, respectively. In the present paper, we extend these results to the
case where the anisotropic conductivity $\sigma$ is of type
$\sigma=A(\cdot,\gamma(\cdot))$, and $A(\cdot,t)$ is known for
$t\in[\lambda^{-1},\lambda]$.

More precisely, we start by considering the simpler case where $\sigma$ is a
symmetric, uniformly positive definite matrix which is a-priori known to have
the structure $\sigma(x)=A(\gamma(x))$, $x\in\overline{\Omega}$, where the
one-parameter family of matrix-valued functions

$t\mapsto A(t),\qquad t\in[\lambda^{-1},\lambda]$

is assumed to be known and to belong to a certain class $\mathcal{A}$ defined
below (Definition 2.2) and $\gamma=\gamma(x)$, $x\in\overline{\Omega}$ is an
unknown scalar function to be determined. The above structure for $\sigma$ is
also generalized to the case $\sigma=A(x,\gamma(x))$, $x\in\overline{\Omega}$,
where the one-parameter family of matrix-valued functions

$t\mapsto A(x,\>t),\qquad\textnormal{for\>any}\quad
x\in\overline{\Omega},\qquad t\in[\lambda^{-1},\lambda]$

is assumed to be known and to belong to a certain class $\mathcal{A^{\prime}}$
defined below (Definition 2.3) and $\gamma=\gamma(x)$, $x\in\overline{\Omega}$
is again an unknown scalar function to be determined. The latter, in
particular, extends the results in [36] and [8], where the problem of
stability and reconstruction in MAT-MI has been addressed in the isotropic
case and in the anisotropic case $\sigma(x)=\gamma(x)D(x)$, where $D$, in this
case, is a known matrix-valued function and $\gamma=\gamma(x)$,
$x\in\overline{\Omega}$ is the unknown scalar function to be determined. For
both the cases when $A\in\mathcal{A}$ and $A\in\mathcal{A^{\prime}}$
considered in this manuscript, we prove that we can stably reconstruct the
scalar function $\gamma$ in terms of $F$.

In this paper we also present several numerical experiments in which we
reconstruct the scalar function $\gamma=\gamma(x)$, $x\in\Omega$ for both
types of anisotropic conductivities $\sigma(x)=A(\gamma(x))$ and
$\sigma(x)=A(x,\gamma(x))$, $x\in\Omega$. We start by considering a number of
examples in which $A\in\mathcal{A}$ only (Examples 1 - 4). The more general
case of $A\in\mathcal{A^{\prime}}$ is exploited in Examples 5 and 6. The
reconstruction is based on an algorithm introduced in [36], [8], which relies
on projecting its iterates into a convex subset of $C^{1,\beta}(\Omega)$ in
order to restore well-posedness in the inverse problem. We note that in the
particular examples considered here, the projection step seems unnecessary for
the convergence of the iterates. Moreover, in two of our examples, the true
conductivity $\gamma_{*}$ we wish to reconstruct is not $C^{1}$, hence is less
regular than the classes of conductivities we considered in our theoretical
framework. This not only allows us to test the performance of the
reconstruction algorithm on non-smooth data, but it also provides insights
about possibly lowering the regularity of the conductivity considered in our
theoretical framework (the stability estimates in 2 and the convergence
analysis we carry out in 4.2). It will be part of future work to extend our
theoretical framework to a class of conductivities with lower regularity
assumptions and a more general anisotropic structure. The anisotropic
structures considered in the current paper are one-parameter families of
symmetric and uniformly positive definite matrices that depend on the unknown
scalar function $\gamma$ in a nonlinear way, extending the results in [8] to
the case of a more realistic dependence of the anisotropic structure on the
unknown scalar function $\gamma$. We wish to stress that this paper aims at
providing a first step in the treatment in MAT-MI of anisotropic structures
that depend nonlinearly on (possibly) a finite number of unknown scalar
functions to be stably reconstructed in the MAT-MI experiment. It will be the
subject of future work to consider the fully anisotropic case and to which
extent the MAT-MI experiment can be employed to determine the anisotropic
structure itself.

The paper is organized as follows. In Section 2 we introduce notation, state
our main assumptions, and define precisely the two classes $\mathcal{A}$ and
$\mathcal{A}^{\prime}$ for $A$, corresponding to both the simpler and
spatially dependent anisotropic case, respectively. The main stability results
are also contained in this section. Section 3 contains the proofs of our main
stability results, and includes estimates for the electromagnetic boundary
value problem. In Section 4 we describe the reconstruction algorithm (Section
4.1) and prove convergence results (Section 4.2), again corresponding to both
classes of $A$. Section 5 contains several numerical experiments demonstrating
accurate reconstructions, including two cases of non-smooth true scalar
functions $\gamma_{*}$ and one fully three dimensional reconstruction. Some
final concluding remarks are included in Section 6.

##  2 Main assumptions and stability estimates

Throughout this paper the medium to be imaged is a bounded domain $\Omega$ in
$\mathbb{R}^{3}$ with $C^{1,\beta}$ boundary $\partial\Omega$ (see definition
2.1) and diameter $\textnormal{diam}(\Omega)=M$, for some constant $M>0$. For
a point $x\in\mathbb{R}^{3}$, we will denote $x=(x^{\prime},x_{3})$, where
$x^{\prime}\in\mathbb{R}^{2}$ and $x_{3}\in\mathbb{R}$. We will also denote by
$B_{r}$, $B^{\prime}_{r}$ the open balls $B_{r}(0)$ and $B^{\prime}_{r}(0)$ in
$\mathbb{R}^{3}$ and $\mathbb{R}^{2}$, respectively and by $Q_{r}$ the
cylinder $Q_{r}(0)$ in $\mathbb{R}^{3}$ defined by

$Q_{r}(0)=B^{\prime}_{r}\times(x_{3}-r,x_{3}+r).$

From here onwards we fix a number $\beta$ satisfying $0<\beta\leq 1$.

######  Definition 2.1.

Given a bounded domain $\Omega\subset\mathbb{R}^{3}$, we say that
$\partial\Omega$ is of class $C^{1,\beta}$ with constants $r_{0},L>0$, if for
every $P\in\partial\Omega$, there exists a rigid transformation of coordinates
under which $P=0$ and

$\Omega\cap Q_{r_{0}}=\left\\{(x^{\prime},x_{n})\in Q_{r_{0}}\quad|\quad
x_{3}>\varphi(x^{\prime})\right\\},$

where $\varphi$ is a $C^{1,\beta}$ function on $B^{\prime}_{r_{0}}$ satisfying

$\varphi(0)=|\nabla\varphi(0)|=0$

and

$||\varphi||_{C^{1,\beta}(B^{\prime}_{r_{0}})}\leq Lr_{0},$

where we use the normalization convention

$||\varphi||_{C^{1,\beta}(B^{\prime}_{r_{0}})}=||\varphi||_{L^{\infty}(B^{\prime}_{r_{0}})}+r_{0}||\nabla\varphi||_{L^{\infty}(B^{\prime}_{r_{0}})}+r_{0}^{1+\beta}\sup_{x^{\prime},y^{\prime}\in
B^{\prime}_{r_{0}},\>x^{\prime}\neq
y^{\prime}}\frac{|\nabla\varphi(x^{\prime})-\nabla\varphi(y^{\prime})|}{|x^{\prime}-y^{\prime}|^{\beta}}.$

Once for all it is understood that $\Omega\subset\mathbb{R}^{3}$ is a bounded
domain with diameter $\textnormal{diam}(\Omega)=M$ and $C^{1,\beta}$ boundary
$\partial\Omega$. We define below the two classes of admissible anisotropic
structures considered in this paper.

######  Definition 2.2.

Given $\Lambda$, $\mathcal{E}_{1}>0$ and denoting by $Sym$ the class of
$3\times 3$ real-valued symmetric matrices, we say that
$A(\cdot)\in\mathcal{A}$ if the following conditions hold:

(2.1) $A\in C^{1,\beta}([\lambda^{-1},\lambda],Sym),$ (2.2)
$\begin{split}&\underset{t\in[\lambda^{-1},\lambda]}{\sup}\Big{|}\frac{dA}{dt}(t)\Big{|}+\underset{t_{1}\neq
t_{2}}{\underset{t_{1},\>t_{2}\in[\lambda^{-1},\lambda]}{\sup}}\frac{|\frac{dA}{dt_{i}}(t_{1})-\frac{dA}{dt_{i}}(t_{2})|}{|t_{1}-t_{2}|^{\beta}}\leq\mathcal{E}_{1}\end{split}$

and

(2.3) $\Lambda^{-1}|\xi|^{2}\leq
A(t)\xi\cdot\xi\leq\Lambda|\xi|^{2},\quad\textnormal{for\>every}\quad
t\in[\lambda^{-1},\lambda],\quad\xi\in\mathbb{R}^{3}.$

######  Proposition 2.1.

If $A\in\mathcal{A}$ and $\gamma$ satisfies

(2.4) $\displaystyle\lambda^{-1}\leq\gamma(x)$ $\displaystyle\leq$
$\displaystyle\lambda,\quad\textnormal{for\>almost\>every}\>x\in\Omega,$ (2.5)
$\displaystyle||\gamma||_{C^{1,\beta}(\bar{\Omega})}$ $\displaystyle\leq$
$\displaystyle K,$

for some constants $\lambda,K>0$, then we have

(2.6) $A(\gamma(\cdot))\in C^{1,\beta}(\bar{\Omega},Sym_{n})$

and furthermore,

(2.7)
$||A(\gamma(\cdot))||_{C^{1,\beta}(\bar{\Omega})}\leq\Lambda+K\mathcal{E}_{1}M(1+2M^{\beta}):=\mathcal{F}_{1},$

where $M$, $\mathcal{E}_{1}$ and $\Lambda$ have been introduced above.

###### Proof.

The proof is a straightforward consequence of equality

(2.8)
$\begin{split}||A(\gamma(\cdot))||_{C^{1,\beta}(\bar{\Omega})}&=||A(\gamma(\cdot))||_{L^{\infty}(\bar{\Omega})}\\\
&+M\underset{1\leq i\leq n}{\sup}\Big{|}\Big{|}\frac{\partial A}{\partial
x_{i}}(\gamma(\cdot))\Big{|}\Big{|}_{L^{\infty}(\bar{\Omega})}\\\
&+M^{1+\beta}\underset{1\leq i\leq n}{\sup}\quad\underset{x\neq
y}{\underset{x,y\in\bar{\Omega}}{\sup}}\frac{|\frac{\partial A}{\partial
x_{i}}(\gamma(x))-\frac{\partial A}{\partial
x_{i}}(\gamma(y))|}{|x-y|^{\beta}}\end{split}$

and our assumptions. ∎

######  Definition 2.3.

Given $\Lambda$, $\mathcal{E}_{2}>0$, we say that
$A(\cdot,\cdot)\in\mathcal{A^{\prime}}$ if the following conditions hold:

(2.9) $A\in C^{1,\beta}(\bar{\Omega}\times[\lambda^{-1},\lambda],Sym),$ (2.10)
$\begin{split}&\underset{t\in[\lambda^{-1},\lambda]}{\sup}\Bigg{\\{}\quad\underset{1\leq
i\leq n}{\sup}\Big{|}\Big{|}\frac{\partial A}{\partial
x_{i}}(\cdot,t)\Big{|}\Big{|}_{L^{\infty}(\bar{\Omega})}+\Big{|}\Big{|}\frac{\partial
A}{\partial t}(\cdot,t)\Big{|}\Big{|}_{L^{\infty}(\bar{\Omega})}\\\
&+\underset{1\leq i\leq n}{\sup}\quad\underset{x\neq
y}{\underset{x,y\in\bar{\Omega}}{\sup}}\frac{|\frac{\partial A}{\partial
x_{i}}(x,t)-\frac{\partial A}{\partial
x_{i}}(y,t)|}{|x-y|^{\beta}}+\underset{x\neq
y}{\underset{x,y\in\bar{\Omega}}{\sup}}\frac{|\frac{\partial A}{\partial
t}(x,t)-\frac{\partial A}{\partial t}(y,t)|}{|x-y|^{\beta}}\Bigg{\\}}\\\
&+\underset{t_{1}\neq
t_{2}}{\underset{t_{1},\>t_{2}\in[\lambda^{-1},\lambda]}{\sup}}\Bigg{\\{}\frac{||\frac{\partial
A}{\partial x_{i}}(\cdot,t_{1})-\frac{\partial A}{\partial
x_{i}}(\cdot,t_{2})||_{L^{\infty}(\bar{\Omega})}}{|t_{1}-t_{2}|^{\beta}}+\frac{||\frac{\partial
A}{\partial t}(\cdot,t_{1})-\frac{\partial A}{\partial
t}(\cdot,t_{2})||_{L^{\infty}(\bar{\Omega})}}{|t_{1}-t_{2}|^{\beta}}\Bigg{\\}}\leq\mathcal{E}_{2}\end{split}$

and

(2.11) $\Lambda^{-1}|\xi|^{2}\leq
A(x,t)\xi\cdot\xi\leq\Lambda|\xi|^{2},\quad\textnormal{for\>a.e.}\>x\in\bar{\Omega},\quad\forall
t\in[\lambda^{-1},\lambda],\quad\forall\xi\in\mathbb{R}^{3}.$

We observe that (2.3), (2.11) are conditions of uniform ellipticity.

######  Proposition 2.2.

If $A\in\mathcal{A^{\prime}}$ and $\gamma$ satisfies (2.4), (2.5), then we
have

(2.12)
$||A(\cdot,\gamma(\cdot))||_{C^{1,\beta}(\bar{\Omega})}\leq\Lambda+M\mathcal{E}_{2}(1+K)(1+2M^{\beta}):=\mathcal{F}_{2},$

where where $M$, $\mathcal{E}_{2}$ and $\Lambda$ have been introduced above.

###### Proof.

This is a straightforward consequence of equalities

(2.13) $\frac{\partial A}{\partial x_{i}}(x,\gamma(x))=\frac{\partial
A}{\partial x_{i}}(x,t)\big{|}_{(x,t)=(x,\gamma(x))}+\frac{\partial
A}{\partial t}(x,t)\big{|}_{(x,t)=(x,\gamma(x))}\frac{\partial\gamma}{\partial
x_{i}}(x),$

together with our assumptions. ∎

Below are our two stability estimates of $\gamma$ in terms of the internal
functional $F$ for the two classes of admissible anisotropic structures
$\mathcal{A}$ and $\mathcal{A^{\prime}}$, respectively.

######  Theorem 2.3.

Let $A\in\mathcal{A}$ and assume $\gamma_{i}$ satisfies (2.4), (2.5), for
$i=1,2$, $\gamma_{1}=\gamma_{2}$ on $\partial\Omega$ and

(2.14) $\displaystyle||\nabla(\gamma_{1}-\gamma_{2})||_{L^{2}(\Omega)}$
$\displaystyle\leq\mathcal{C}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)},$

for some constant $\mathcal{C}>0$. Denoting

(2.15) $F_{i}:=\nabla\cdot(A(\gamma_{i})E_{A(\gamma_{i})}\times
B_{0}),\qquad\textnormal{for}\quad i=1,2,$

if $\mathcal{C}\Big{(}C_{1}(\mathcal{E}_{1}+1)+\Lambda
C_{3}\Big{)}<\frac{1}{2}$, where
$C_{1}=(1+\Lambda\mathcal{F}_{1})(\frac{1}{2}M(\mu(\Omega))^{\frac{1}{2}})$
and $C_{3}=\Lambda\mathcal{E}_{1}C_{1}$, then

(2.16) $||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)}\leq
C||F_{1}-F_{2}||_{L^{2}(\Omega)},$

where $C>0$ is a constant that depends on $\mathcal{C}$, $K$,
$\mathcal{E}_{1}$, $\Lambda$, $M$, $r_{0}$ and $L$ only.

More generally, we obtain the following main stability result.

######  Theorem 2.4.

Let $A\in\mathcal{A}^{\prime}$ and $\gamma_{i}$ be as in Theorem 2.3, for
$i=1,2$. Denoting

(2.17)
$\tilde{F}_{i}:=\nabla\cdot(A(\cdot,\>\gamma_{i})E_{A(\cdot,\>\gamma_{i})}\times
B_{0}),\qquad\textnormal{for}\quad i=1,2,$

if $\mathcal{C}\Big{(}C_{2}(\mathcal{E}_{2}+1)+\Lambda
C_{4}\Big{)}<\frac{1}{2}$, where
$C_{2}=(1+\Lambda\mathcal{F}_{2})(\frac{1}{2}M(\mu(\Omega))^{\frac{1}{2}})$
and $C_{4}=\Lambda\mathcal{E}_{2}C_{2}$, then

(2.18)
$||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)}\leq\tilde{C}||\tilde{F}_{1}-\tilde{F}_{2}||_{L^{2}(\Omega)},$

where $\tilde{C}>0$ is a constant that depends on $\mathcal{C}$, $K$,
$\mathcal{E}_{2}$, $\Lambda$, $M$, $r_{0}$ and $L$ only.

##  3 Proof of the stability estimates

To prove our main stability results, we proceed by reducing the Maxwell system
(1.2) into a Neumann boundary value problem, for which we recall standard
estimates that will be needed for the arguments in the proofs of our stability
results.

###  3.1 Technical results

In what follows we recall standard estimates for the solution of Neumann
boundary value problems of type

(3.1) $\Bigg{\\{}\begin{array}[]{llll}\nabla\cdot(\sigma\nabla
u)&=-\nabla\cdot E,&\text{in}\quad\Omega,\\\ (\sigma\nabla
u+E)\cdot\nu&=0,&\text{on}\quad\partial\Omega.\end{array}$

Such estimates are standard in the theory of elliptic partial differential
equations and we recall them, together with their proof, for the sake of
completeness.

######  Proposition 3.1.

Let $\sigma\in C^{1,\beta}(\overline{\Omega})$ be a matrix-valued function
satisfying the uniform ellipticity condition

(3.2)
$\Lambda^{-1}|\xi|^{2}\leq\sigma(x)\xi\cdot\xi\leq\Lambda|\xi|^{2},\quad\textnormal{for\>a.e.}\>x\in\bar{\Omega},\quad\forall\xi\in\mathbb{R}^{3}$

and let $E\in$ $L^{2}(\Omega)$ be a vector-valued function. The Neumann
problem (3.1) has a unique solution $u\in H^{1}(\Omega)$ (up to an additive
constant) that satisfies

(3.3) $||\nabla u||_{L^{2}(\Omega)}\leq\Lambda||E||_{L^{2}(\Omega)}.$

###### Proof.

As a straighforward consequence of

(3.4) $\int_{\Omega}A\nabla{u}\cdot\nabla
u\>dx=-\int_{\Omega}{E}\cdot\nabla{u}\>dx,$

we obtain

(3.5)
$\displaystyle\Lambda^{-1}||\nabla{u}||^{2}_{L^{2}(\Omega)}\leq\Big{|}-\int_{\Omega}{E}\cdot\nabla{u}dx\Big{|}\leq||{E}||_{L^{2}(\Omega)}||\nabla{u}||_{L^{2}(\Omega)},$

which concludes the proof. ∎

######  Proposition 3.2.

Let $\gamma_{i}$ satisfy (2.4), (2.5), for $i=1,2$.

  1. 1.

If $A\in\mathcal{A}$ and $\sigma_{i}(x)=A(\gamma_{i}(x))$,
$x\in\overline{\Omega}$, for $i=1,2$, then we have

(3.6) $\displaystyle||E_{A(\gamma_{i})}||_{C^{1,\beta}(\Omega)}\leq
C_{1},\qquad\textnormal{for}\>i=1,2,$ (3.7)
$\displaystyle||E_{A(\gamma_{1})}-E_{A(\gamma_{2})}||_{L^{2}(\Omega)}\leq
C_{3}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)},$ (3.8)
$\displaystyle||E_{A(\gamma_{1})}-E_{A(\gamma_{2})}||_{H^{1}(\Omega)}\leq\tilde{C_{3}}||\gamma_{1}-\gamma_{2}||_{H^{1}(\Omega)},$

where $E_{A(\gamma_{i})}$ is the unique solution to (1.2), with
$\sigma_{i}=A(\gamma_{i})$, for $i=1,2$ and $C_{1}$, $C_{3}$, $\tilde{C_{3}}$
are positive constants depending only on $\Lambda$, $r_{0}$ and $L$.

  2. 2.

Similarly, if and $A\in\mathcal{A}^{\prime}$ and
$\sigma_{i}(x)=A(x,\gamma_{i}(x))$, $x\in\overline{\Omega}$, for $i=1,2$, then
we have

(3.9) $\displaystyle||E_{A(\cdot,\gamma_{i})}||_{C^{1,\beta}(\Omega)}\leq
C_{2},\qquad\textnormal{for}\>i=1,2,$ (3.10)
$\displaystyle||E_{A(\cdot,\gamma_{1})}-E_{A(\cdot,\gamma_{2})}||_{L^{2}(\Omega)}\leq
C_{4}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)},$ (3.11)
$\displaystyle||E_{A(\cdot,\gamma_{1})}-E_{A(\cdot,\gamma_{2})}||_{H^{1}(\Omega)}\leq\tilde{C_{4}}||\gamma_{1}-\gamma_{2}||_{H^{1}(\Omega)},$

where $E_{A(\cdot,\gamma_{i})}$ is the unique solution to (1.2), with
$\sigma_{i}=A(\cdot,\gamma_{i})$, for $i=1,2$, with $C_{2}$, $C_{4}$,
$\tilde{C_{4}}$ being positive constants depending only on $\Lambda$, $r_{0}$
and $L$.

###### Proof.

As the proof follows the same line of [8, Proof of Proposition 2], we only
highlight the main steps of case $(a)$ and only point out where case $(b)$
differs from it, as such modifications are minor. For case $(a)$ we start by
noticing that for $\tilde{E}$ = $\frac{1}{2}(-y,x,0)$ we have that
$\nabla\times(E_{A(\gamma_{i})}-\tilde{E})=0$, for $i=1,2$ and
$E_{A(\gamma_{i})}=\tilde{E}+\nabla u$, where $u_{i}$ solves the Neumann
problem

(3.12) $\Bigg{\\{}\begin{array}[]{llll}\nabla\cdot(A(\gamma_{i})\nabla
u_{i})&=-\nabla\cdot(A(\gamma_{i})\tilde{E})&\textnormal{in}\quad\Omega;\\\
(A(\gamma_{i})\nabla
u_{i}+A(\gamma_{i})\tilde{E})\cdot\nu&=0&\textnormal{on}\quad\partial\Omega,\end{array}$

for $i=1,2$. Observing that

(3.13)
$||\tilde{E}||_{L^{2}(\Omega)}\leq\frac{1}{2}M\big{(}\mu(\Omega)\big{)}^{\frac{1}{2}},$

where $\mu(\Omega)$ denotes the Lebesgue measure of $\Omega$, estimate (3.6)
is a straightforward consequence of (2.5) and proposition 2.1 (see also [8]).

To prove (3.7), (3.8), we start by simplifying our notation. We denote
$E_{A(\gamma_{i})}$, simply with $E_{i}$, for $i$ = 1, 2. Noting that
$\nabla\times E_{1}=\nabla\times E_{2}$, we write

(3.14) $\nabla v=E_{1}-E_{2},$

where $v$ is the unique solution to

(3.15) $\Bigg{\\{}\begin{array}[]{llll}\nabla\cdot(A(\gamma_{1})\nabla
v)&=-\nabla\cdot((A(\gamma_{1})-A(\gamma_{2}))E_{2})&\textnormal{in}\quad\Omega,\\\
(A(\gamma_{1})\nabla
v+(A(\gamma_{1})-A(\gamma_{2}))E_{2})\cdot\nu&=0&\textnormal{on}\quad\partial\Omega\end{array}$

and

(3.16) $\int_{\partial\Omega}v=0.$

Thus by Cauchy Schwartz we obtain

(3.17)
$\int_{\Omega}A(\gamma_{1})\nabla{v}\cdot\nabla{v}dx\leq\int_{\Omega}((A(\gamma_{1})-A(\gamma_{2})){E_{2}})\cdot\nabla{v}\>dx,$

which, combined with (3.6) and the uniform ellipticity condition (2.3), leads
to

(3.18) $||\nabla{v}||_{L^{2}(\Omega)}\leq\Lambda
C_{1}||A(\gamma_{1})-A(\gamma_{2})||_{L^{2}(\Omega)}.$

From the Lagrange Theorem, for every $x\in\Omega$, there exists $t(x)$,
$0<t(x)<1$, such that

(3.19)
$A(\gamma_{1}(x))-A(\gamma_{2}(x))=(\gamma_{1}-\gamma_{2})(x)\>\frac{dA(t)}{dt}\Big{|}_{t=c(x)},$

where $c(x)=\gamma_{2}(x)+t(x)\big{(}\gamma_{1}(x)-\gamma_{2}(x)\big{)}$.
Hence we obtain

(3.20) $||\nabla v||_{L^{2}(\Omega)}\leq\Lambda
C_{1}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)}\Big{|}\Big{|}\frac{dA(t)}{dt}\Big{|}_{t=c(\cdot)}\Big{|}\Big{|}_{L^{2}(\Omega)}$

and

(3.21) $||E_{1}-E_{2}||_{L^{2}(\Omega)}\leq
C_{3}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)},$

where $C_{3}=\Lambda\mathcal{E}_{1}C_{1}$. From the standard theory of
elliptic equations

(3.22) $||v||_{H^{2}(\Omega)}\leq
C||\nabla\cdot((A(\gamma_{1})-A(\gamma_{2}))E_{2})||_{L^{2}(\Omega)},$

where $C$ is a positive constant depending only on $\lambda$, $\Lambda$,
$r_{0}$ and $L$, which implies (3.8).

For case $(b)$, where $A\in\mathcal{A^{\prime}}$ and $\gamma_{i}\in
C^{1,\beta}(\overline{\Omega})$ is a scalar function satisfying (2.4), for
$i=1,2$, we have, by (3.13) and proposition 3.1 that

(3.23) $||E_{A(\cdot,\gamma_{i})}||_{L^{2}(\Omega)}\leq
C_{2},\qquad\textnormal{for}\>i=1,2,$

where
$C_{2}=(1+\Lambda\mathcal{F}_{2})(\frac{1}{2}M(\mu(\Omega))^{\frac{1}{2}})$.
With a similar argument to case $(a)$, from the Lagrange Theorem, for every
$x\in\Omega$, there exists $t(x)$, $0<t(x)<1$, such that

(3.24)
$A(x,\gamma_{1}(x))-A(x,\gamma_{2}(x))=(\gamma_{1}-\gamma_{2})(x)\frac{\partial
A(x,\>t)}{\partial t}\Big{|}_{t=c(x)},$

where $c(x)=\gamma_{2}(x)+t(x)\big{(}\gamma_{1}(x)-\gamma_{2}(x)\big{)}$,
which leads to

(3.25) $||E_{1}-E_{2}||_{L^{2}(\Omega)}\leq\Lambda
C_{1}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)}\Big{|}\Big{|}\frac{\partial
A(\cdot,\>t)}{\partial t}\Big{|}_{t=c(\cdot)}\Big{|}\Big{|}_{L^{2}(\Omega)},$

hence

(3.26) $||E_{1}-E_{2}||_{L^{2}(\Omega)}\leq
C_{4}||\gamma_{1}-\gamma_{2}||_{L^{2}(\Omega)},$

where $C_{4}=\Lambda\mathcal{E}_{2}C_{2}$. The remaining (3.11) follows by a
similar argument used to prove (3.8) in case $(a)$. ∎

###  3.2 Proof of the stability estimates

###### Proof of Theorem 2.3.

We write the difference in the data $F_{1}-F_{2}$ as

(3.27) $\begin{split}F_{1}-F_{2}&=\nabla\cdot\Big{(}A(\gamma_{1})E_{1}\times
B_{0}\Big{)}-\nabla\cdot\Big{(}A(\gamma_{2})E_{2}\times B_{0}\Big{)}\\\
&=\nabla\cdot\Big{(}(A(\gamma_{1})-A(\gamma_{2}))E_{1}\times
B_{0}\Big{)}+\nabla\cdot\Big{(}A(\gamma_{2})(E_{1}-E_{2})\times
B_{0}\Big{)}\end{split}$

and from the Lagrange Theorem, for every $x\in\Omega$, there exists $t(x)$,
$0<t(x)<1$, such that

(3.28)
$F_{1}-F_{2}=\nabla\cdot\Big{(}(\gamma_{1}-\gamma_{2})\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{1}\times
B_{0}\Big{)}+\nabla\cdot\Big{(}A(\gamma_{2})\>(E_{1}-E_{2})\times
B_{0}\Big{)},$

where $c(x)=\gamma_{2}(x)+t(x)\big{(}\gamma_{1}(x)-\gamma_{2}(x)\big{)}$.
Then, we can split $F_{1}-F_{2}$ into three contributions

(3.29) $F_{1}-F_{2}=I_{1}+I_{2}+I_{3},$

where

(3.30)
$\begin{split}&I_{1}=\nabla\cdot\Big{(}(\gamma_{1}-\gamma_{2})E_{1}\times
B_{0}\Big{)},\\\
&I_{2}=\nabla\cdot\Big{(}(\gamma_{1}-\gamma_{2})\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}-I\Big{)}E_{1}\times
B_{0}\Big{)},\\\ &I_{3}=\nabla\cdot\Big{(}A(\gamma_{2})(E_{1}-E_{2})\times
B_{0}\Big{)},\end{split}$

where $I$ denotes the $3\times 3$ identity matrix. To extract information
about $\gamma_{1}-\gamma_{2}$ in $\Omega$ we multiply the data difference
$F_{1}-F_{2}$ by $\gamma_{1}-\gamma_{2}$ and integrate over $\Omega$. We do
this by considering each contribution $I_{i}$, i = 1, 2, 3, to the data
separately. Starting with $I_{1}$, we obtain

(3.31)
$\begin{split}\int_{\Omega}(\gamma_{1}-\gamma_{2})I_{1}dx&=\int_{\Omega}(\gamma_{1}-\gamma_{2})^{2}\nabla\cdot(E_{1}\times
B_{0})\\\ &+\frac{1}{2}\nabla(\gamma_{1}-\gamma_{2})^{2}\cdot(E_{1}\times
B_{0})dx\\\
&=\frac{1}{2}\int_{\Omega}(\gamma_{1}-\gamma_{2})^{2}\nabla\cdot(E_{1}\times
B_{0})dx\\\
&=\frac{1}{2}||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}(\Omega)},\end{split}$

where in the second equality of (3.31) we performed integration by parts and
used the fact that $\gamma_{1}$ and $\gamma_{2}$ share the same boundary
values. In the last equality, we used the fact that $\nabla\cdot(E_{1}\times
B_{0})=1$.

Multiplying $I_{2}$ by $\gamma_{1}-\gamma_{2}$, we have

(3.32)
$\begin{split}\left|\int_{\Omega}(\gamma_{1}-\gamma_{2})I_{2}dx\right|&=\Big{|}\int_{\Omega}(\gamma_{1}-\gamma_{2})^{2}\nabla\cdot\left(\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}-I\Big{)}E_{1}\times
B_{0}\right)\\\
&+(\gamma_{1}-\gamma_{2})\nabla(\gamma_{1}-\gamma_{2})\cdot\left(\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}-I\Big{)}E_{1}\times
B_{0}\right)dx\Big{|}\\\
&=\Big{|}\int_{\Omega}(\gamma_{1}-\gamma_{2})\nabla(\gamma_{1}-\gamma_{2})\cdot\left(\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}-I\Big{)}E_{1}\times
B_{0}\right)dx\Big{|}\\\
&\leq\mathcal{C}C_{1}|\mathcal{E}_{1}-1|||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}(\Omega)},\end{split}$

where in the second equality of (3.32) we again performed integration by parts
and again recalled that $\gamma_{1}$ and $\gamma_{2}$ share the same boundary.
In the inequality, estimate (2.2), (2.14) and (3.6) have been used.

Finally, multiplyig $I_{3}$ by $\gamma_{1}-\gamma_{2}$, leads to

(3.33)
$\begin{split}\left|\int_{\Omega}(\gamma_{1}-\gamma_{2})I_{3}dx\right|&=\left|\int_{\Omega}\nabla(\gamma_{1}-\gamma_{2})\cdot\Big{(}A(\gamma_{2})(E_{1}-E_{2})\times
B_{0}\Big{)}dx\right|\\\ &\leq\mathcal{C}\Lambda
C_{3}||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}(\Omega)},\end{split}$

where the first equality in (3.33) is derived again by performing integration
by parts and noticing that $\gamma_{1}$ and $\gamma_{2}$ share the same
boundary values. In the inequality, (2.3), assumption (2.14) and (3.7) have
been used.

Hence, for $\mathcal{C}C_{1}(\mathcal{E}_{1}+1)+\mathcal{C}\Lambda
C_{3}<\frac{1}{2}$, we obtain

(3.34) $\int_{\Omega}(\gamma_{1}-\gamma_{2})(F_{1}-F_{2})dx\geq
C||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}(\Omega)},$

where $C=\frac{1}{2}-\mathcal{C}\Big{(}C_{1}(\mathcal{E}_{1}+1)+\Lambda
C_{3}\Big{)}>0$, which concludes the proof. ∎

###### Proof of Theorem 2.4.

Similarly to the proof of theorem 2.3, we write the data difference as

(3.35)
$\begin{split}\tilde{F}(\gamma_{1})-\tilde{F}(\gamma_{2})&=\nabla\cdot(A(\cdot,\gamma_{1})E_{1}\times
B_{0})-\nabla\cdot(A(\cdot,\gamma_{2})E_{2}\times B_{0})\\\
&=\nabla\cdot((A(\cdot,\gamma_{1})-A(\cdot,\gamma_{2}))E_{1}\times
B_{0})+\nabla\cdot(A(\cdot,\gamma_{2})(E_{1}-E_{2})\times B_{0}).\end{split}$

From the Lagrange Theorem, for every $x\in\Omega$, there exists $t(x)$,
$0<t(x)<1$, such that

(3.36)
$A(x,\gamma_{1})-A(x,\gamma_{2})=(\gamma_{1}-\gamma_{2})(x)\frac{\partial
A(x,t)}{\partial t}\Big{|}_{t=c(x)},$

where $c(x)=\gamma_{2}(x)+t(x)\big{(}\gamma_{1}(x)-\gamma_{2}(x)\big{)}$.
Therefore we obtain

(3.37)
$\tilde{F}_{1}-\tilde{F}_{2}=\nabla\cdot((\gamma_{1}-\gamma_{2})\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(x)}E_{1}\times
B_{0})+\nabla\cdot(A(\cdot,\>\gamma_{2})(E_{1}-E_{2})\times B_{0}).$

By splitting again $\tilde{F}_{1}-\tilde{F}_{2}$ as

(3.38) $\tilde{F}_{1}-\tilde{F}_{2}=I_{1}+I_{2}+I_{3},$

where

(3.39)
$\begin{split}&I_{1}=\nabla\cdot\Big{(}(\gamma_{1}-\gamma_{2})E_{1}\times
B_{0}\Big{)},\\\
&I_{2}=\nabla\cdot\left((\gamma_{1}-\gamma_{2})\Big{(}\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(\cdot)}-I\Big{)}E_{1}\times
B_{0}\right),\\\
&I_{3}=\nabla\cdot\Big{(}A(\cdot,\>\gamma_{2})(E_{1}-E_{2})\times
B_{0}\Big{)},\end{split}$

with a similar procedure as that of the proof of Theorem 2.3 we obtain the
following estimates

(3.40)
$\begin{split}\int_{\Omega}(\gamma_{1}-\gamma_{2})I_{1}dx&=\frac{1}{2}||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}(\Omega)}.\end{split}$
(3.41)
$\begin{split}\left|\int_{\Omega}(\gamma_{1}-\gamma_{2})I_{2}dx\right|&=\left|\int_{\Omega}(\gamma_{1}-\gamma_{2})\nabla(\gamma_{1}-\gamma_{2})\cdot\left(\Big{(}\frac{\partial
A(\cdot,\>t)}{\partial t}\Big{|}_{t=c(\cdot)}-I\Big{)}E_{1}\times
B_{0}\right)dx\right|\\\
&\leq\mathcal{C}C_{2}|\mathcal{E}_{2}-1|||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}}.\end{split}$
(3.42)
$\begin{split}\left|\int_{\Omega}(\gamma_{1}-\gamma_{2})I_{3}dx\right|&=\left|\int_{\Omega}\nabla(\gamma_{1}-\gamma_{2})\cdot(A(\cdot,\>\gamma_{2}(\cdot))\>(E_{1}-E_{2})\times
B_{0})dx\right|\\\ &\leq\mathcal{C}\Lambda
C_{4}||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}}\end{split}$

By combining (3.40) - (3.42) together and choosing the parameters such
that$(\mathcal{C}C_{2}(\mathcal{E}_{2}+1)+\mathcal{C}\Lambda
C_{4})<\frac{1}{2}$, we finally obtain

(3.43)
$\int_{\Omega}(\gamma_{1}-\gamma_{2})(\tilde{F}(\gamma_{1})-\tilde{F}(\gamma_{2}))dx\geq\tilde{C}||\gamma_{1}-\gamma_{2}||^{2}_{L^{2}(\Omega)},$

where
$\tilde{C}=\frac{1}{2}-(\mathcal{C}C_{2}(\mathcal{E}_{2}+1)+\mathcal{C}\Lambda
C_{4})>0$, which concludes the proof. ∎

##  4 Reconstruction of $\gamma$

###  4.1 The functional framework and the algorithm

To ensure well-posedness of the inverse problem we introduce a functional
framework, similar to that in [8], based assuming that $\gamma$ is known on
the boundary of $\Omega$ and by letting the true scalar function to be
$\gamma_{*}$ (the unknown conductivity to be reconstructed) in $\Omega$,
belonging to a bounded convex subset $\tilde{S}$ of $C^{1,\beta}(\Omega)$,
defined by

(4.1) $\tilde{S}=\\{\gamma:=\gamma_{0}+\alpha|\alpha\in S\\},$

where $\gamma_{0}$ satisfies (2.4), (2.5) and

(4.2) $S=\\{\alpha\in
C^{1,\beta}_{0}(\Omega)|\lambda^{-1}\leq\alpha+\gamma_{0}\leq\lambda,||\nabla\alpha||_{L^{2}(\Omega)}\leq\mathcal{C}||\alpha||_{L^{2}(\Omega)},||\alpha||_{L^{2}(\Omega)}\leq\tilde{k}\\},$

for some positive constant $\tilde{k}$.
For $A\in\mathcal{A}$, the true conductivity is denoted by
$A_{*}:=A(\gamma_{*})$ and the forward operator is given by

(4.3) $F(\gamma)=\nabla\cdot(A(\gamma)E_{A(\gamma)}\times B_{0}),$

where $E_{A}:=E_{A(\gamma)}$ solves

(4.4) $\Bigg{\\{}\begin{array}[]{llll}\nabla\times
E_{A}&=B_{0},&\text{in}\quad\Omega,\\\ \nabla\cdot
A(\gamma)E_{A}&=0,&\text{on}\quad\Omega,\\\
A(\gamma)E_{A}\cdot\nu&=0,&\text{on}\quad\partial\Omega\end{array}$

and the scalar function $\gamma$ is updated by solving a stationary advection-
diffusion equation as shown in the algorithm below.

The Algorithm (reconstruction)

  * •

We choose an initial conductivity $\gamma_{1}\in\tilde{S}$ and set $k=1$.

  * •

At step $k$, defining $A_{k}:=A(\gamma_{k})$, we solve the boundary value
problem

(4.5) $\Bigg{\\{}\begin{array}[]{llll}\nabla\times
E_{k}&=B_{0},&\text{in}\quad\Omega,\\\ \nabla\cdot
A_{k}E_{k}&=0,&\text{on}\quad\Omega,\\\
A_{k}E_{k}\cdot\nu&=0,&\text{on}\quad\partial\Omega.\end{array}$

to update the electric field $E_{k}$.

  * •

By solving the stationary advection-diffusion equation with the inflow
boundary condition, we calculate the updated conductivity
$A_{k+1/2}:=A(\gamma_{k+1})$ as follows

(4.6) $\Big{\\{}\begin{array}[]{llll}\nabla\cdot(A_{k+1/2}E_{k}\times
B_{0})&=\nabla\cdot(A_{*}E_{A_{*}}\times B_{0}),&\text{in}\quad\Omega,\\\
\gamma_{k+1/2}&=\gamma_{*},&\text{on}\quad\partial\Omega^{-},\end{array}$

where

(4.7)
$\partial\Omega^{-}=\left\\{x\in\partial\Omega|A(\gamma_{k}(x))E_{k}(x)\times
B_{0}\cdot\nu(x)<0\right\\}.$

  * •

We then let $\gamma_{k+1}=T[\gamma_{k+1/2}]$ where $T$ is the Hilbert
projection operator onto $\tilde{S}$, where the projection $T$ is needed as
$\gamma_{k}$ might not be in $\tilde{S}$. Finally, we set $k=k+1$ and go to
(4.5).

For $A\in\mathcal{A^{\prime}}$ the true conductivity is denoted by
$A_{*}:=A(\cdot,\gamma_{*})$ and the forward operator is given by

(4.8) $\tilde{F}(\gamma)=\nabla\cdot(A(\cdot,\gamma)E_{A(\cdot,\gamma)}\times
B_{0}),$

where $E_{A}:=E_{A(\cdot,\gamma)}$ solves

(4.9) $\Bigg{\\{}\begin{array}[]{llll}\nabla\times
E_{A}&=B_{0},&\text{in}\quad\Omega,\\\ \nabla\cdot
A(\cdot,\gamma)E_{A}&=0,&\text{on}\quad\Omega,\\\
A(\cdot,\gamma)E_{A}\cdot\nu&=0,&\text{on}\quad\partial\Omega\end{array}$

and the scalar function $\gamma$ together with the electric field $E$ are
updated as per Algorithm 1 above, where the inflow part of the boundary is in
this case defined by

(4.10)
$\partial\Omega^{-}=\left\\{x\in\partial\Omega|A(x,\gamma_{k}(x))E_{k}(x)\times
B_{0}\cdot\nu(x)<0\right\\}.$

###  4.2 Convergence analysis

Under a smallness condition on $\frac{dA(t)}{dt}$ and $\frac{\partial
A(\cdot,t)}{dt}$, on $[\lambda^{-1},\lambda]$ with respect to $I$, for
$A\in\mathcal{A}$ and $A\in\mathcal{A^{\prime}}$, respectively (see (4.12) and
(4.15), respectively), we obtain the following convergence results.

######  Theorem 4.1.

Suppose that the true conductivity $\gamma_{*}\in\tilde{S}$. Let
$A\in\mathcal{A}$ and, additionally, assume that

(4.11) $\frac{dA}{dt}\in C^{1}\left([\lambda^{-1},\lambda],Sym\right)$

and that there is a constant $\tilde{\mathcal{E}}_{1}$,
$0<\tilde{\mathcal{E}}_{1}<1$ such that

(4.12)
$\sup_{t\in[\lambda^{-1},\lambda]}\sup_{x\in\Omega}\left|\nabla\cdot\left[\left(\frac{dA}{dt}-I\right)E_{k}\times
B_{0}\right]\right|\leq\tilde{\mathcal{E}}_{1}.$

Assume that either

  1. 1.

$\gamma_{k}=\gamma_{*},\qquad\textnormal{on}\quad\partial\Omega,\quad\textnormal{for\>any}\>k,\quad
k\geq 1;$

or

  2. 2.

$\displaystyle\gamma_{k}=\gamma_{*},\qquad\textnormal{on}\quad\partial\Omega^{-}\quad\textnormal{and}\quad\sup_{t\in[\lambda^{-1},\lambda]}\frac{dA}{dt}E_{k}\times
B_{0}\cdot\nu\geq 0,\qquad\textnormal{on}\quad\partial\Omega^{+},$
$\displaystyle\textnormal{for\>any}\>k,\quad k\geq 1.$

Then, for an appropriate choice of $\mathcal{C}$, $\Lambda$, $C_{1}$,
$\tilde{C_{3}}$, $\mathcal{E}_{1}$ and $\tilde{\mathcal{E}_{1}}$, there exists
a constant $C$, $0<C<1$ such that

(4.13) $||\gamma_{k+1}-\gamma_{*}||_{L^{2}(\Omega)}\leq
C||\gamma_{k}-\gamma_{*}||_{L^{2}(\Omega)}.\\\ \\\ $

######  Theorem 4.2.

Suppose that the true conductivity $\gamma_{*}\in\tilde{S}$. Let
$A\in\mathcal{A^{\prime}}$, and, additionally, assume that

(4.14) $\frac{\partial A(\cdot,t)}{\partial t}\in
C^{1}(\overline{\Omega}\times[\lambda^{-1},\lambda],Sym)$

and there exists a constant $\tilde{\mathcal{E}_{2}}$,
$0<\tilde{\mathcal{E}_{2}}<1$ such that

(4.15)
$\underset{t\in[\lambda^{-1},\lambda]}{\sup}\sup_{x\in\Omega}\quad\Big{|}\nabla\cdot\Big{[}\Big{(}\frac{\partial
A}{\partial t}(\cdot,t)\Big{|}_{t=c(\cdot)}-I\Big{)}E_{k}\times
B_{0}\Big{]}\Big{|}\leq\tilde{\mathcal{E}_{2}}.$

Assume that either

  1. 1.

$\gamma_{k}=\gamma_{*},\qquad\textnormal{on}\quad\partial\Omega,\quad\textnormal{for\>any}\>k,\quad
k\geq 1;$

or

  2. 2.

$\displaystyle\gamma_{k}=\gamma_{*},\qquad\textnormal{on}\quad\partial\Omega^{-}\quad\textnormal{and}\quad\sup_{t\in[\lambda^{-1},\lambda]}\frac{\partial
A(\cdot,t)}{\partial t}E_{k}\times B_{0}\cdot\nu\geq
0,\qquad\textnormal{on}\quad\partial\Omega^{+},$
$\displaystyle\textnormal{for\>any}\>k,\quad k\geq 1.$

Then, for an appropriate choice of $\mathcal{C}$, $\Lambda$, $C_{2}$,
$\tilde{C_{4}}$ $\mathcal{E}_{2}$and $\tilde{\mathcal{E}_{2}}$, there exists a
constant $C$ $0<C<1$ such that

(4.16) $||\gamma_{k+1}-\gamma_{*}||_{L^{2}(\Omega)}\leq
C||\gamma_{k}-\gamma_{*}||_{L^{2}(\Omega)}.\\\ \\\ $

###### Proof of Theorem 4.1.

As $T$ projects into $\tilde{S}$, which is convex, we have that
$||\gamma_{k+1}-\gamma_{*}||_{L^{2}(\Omega)}\leq||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}$.
It is left to estimate $||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}$.

Subtracting $\nabla\cdot(A_{*}E_{k}\times B_{0})$ from both sides of (4.6) we
get

(4.17) $\nabla\cdot((A_{k+1/2}-A_{*})E_{k}\times
B_{0})=\nabla\cdot(A_{*}(E_{*}-E_{k})\times B_{0}).$

Multiplying by $\gamma_{k+1/2}-\gamma_{*}$ and integrating over $\Omega$
yields

(4.18)
$\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((A_{k+1/2}-A_{*})E_{k}\times
B_{0})=\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot(A_{*}(E_{*}-E_{k})\times
B_{0}).$

Using the property that

(4.19) $A\in C^{1}\left([\lambda^{-1},\lambda],Sym\right),$

from the Lagrange Theorem, for every $x\in\Omega$, there exists $t(x)$,
$0<t(x)<1$, such that

(4.20)
$A(\gamma_{k+1/2})-A(\gamma_{*})=(\gamma_{k+1/2}-\gamma_{*})\frac{dA}{dt}(t)\Big{|}_{t=c(x)},$

where
$c(x)=\gamma_{k+1/2}(x)+t(x)\big{(}\gamma_{*}(x)-\gamma_{k+1/2}(x)\big{)}$,
hence we get

(4.21)
$\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((\gamma_{k+1/2}-\gamma_{*})\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0})=\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot(A_{*}(E_{*}-E_{k})\times
B_{0}).$

The term on the left hand side of (4.21) can be estimated from below as
follows

(4.22)
$\begin{split}&\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((\gamma_{k+1/2}-\gamma_{*})\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0})dx\\\
=&\frac{1}{2}\int_{\partial\Omega}(\gamma_{k+1/2}-\gamma_{*})^{2}\cdot\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0}\Big{)}\cdot\nu ds(x)\\\
&+\frac{1}{2}\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})^{2}\nabla\cdot\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0}\Big{)}dx,\\\
\geq&\frac{1}{2}\int_{\partial\Omega}(\gamma_{k+1/2}-\gamma_{*})^{2}\cdot\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0}\Big{)}\cdot\nu ds(x)\\\
&+\frac{1}{2}||(\gamma_{k+1/2}-\gamma_{*})||^{2}_{L^{2}(\Omega)}\\\
&-\frac{1}{2}\int_{\Omega}|\gamma_{k+1/2}-\gamma_{*}|^{2}\quad\underset{\Omega}{\sup}\Big{|}\nabla\cdot\Big{(}\Big{(}\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}-I\Big{)}E_{k}\times
B_{0}\Big{)}\Big{|}dx.\end{split}$

In the second equality in (4.22) we performed integration by parts twice. In
either cases $1.$ or $2.$, (4.22), combined with (4.12), leads to

(4.23)
$\begin{split}&\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((\gamma_{k+1/2}-\gamma_{*})\frac{dA}{dt}(t)\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0})dx\\\
&\geq\frac{1}{2}(1-\tilde{\mathcal{E}_{1}})||(\gamma_{k+1/2}-\gamma_{*})||^{2}_{L^{2}(\Omega)}.\end{split}$

The term on the right hand side of (4.21) can be estimated from above as

(4.24)
$\begin{split}&\Big{|}\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot(A_{*}(E_{*}-E_{k})\times
B_{0})\Big{|}\\\
\leq&||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}||A_{*}(E_{*}-E_{k})\times
B_{0}||_{H^{1}(\Omega)},\\\
\leq&(\tilde{C_{3}}+\mathcal{C})(\Lambda+\mathcal{E}_{1}K)||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}||\gamma_{*}-\gamma_{k}||_{L^{2}(\Omega)},\end{split}$

where we combined estimates (2.2), (2.5), together with (2.3), (2.14) and
(3.8).
Choosing
$\frac{(\tilde{C_{3}}+\mathcal{C})(\Lambda+\mathcal{E}_{1}K)}{1-\tilde{\mathcal{E}_{1}}}\leq\frac{1}{2}$,
we finally derive

(4.25) $||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}\leq
C||\gamma_{k}-\gamma_{*}||_{L^{2}(\Omega)},$

where $0<C<1$. ∎

###### Proof of Theorem 4.2.

As this proof is very similar to the one above, we only point where the two
proofs slightly differ. In particular, we take care to point out what are the
a-priori constants that come into play. As above, the goal is to estimate
$||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}$. Subtracting
$\nabla\cdot(A_{*}E_{k}\times B_{0})$ from both sides of (4.6) leads to

(4.26) $\nabla\cdot((A_{k+1/2}-A_{*})E_{k}\times
B_{0})=\nabla\cdot(A_{*}(E_{*}-E_{k})\times B_{0}).$

Multiplying by $\gamma_{k+1/2}-\gamma_{*}$ and integrating over $\Omega$
yields

(4.27)
$\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((A_{k+1/2}-A_{*})E_{k}\times
B_{0})=\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot(A_{*}(E_{*}-E_{k})\times
B_{0}).$

Using the property that

(4.28) $A\in
C^{1}\left(\overline{\Omega}\times[\lambda^{-1},\lambda],Sym\right),$

from the Lagrange Theorem, for every $x\in\Omega$, there exists $t(x)$,
$0<t(x)<1$, such that

(4.29)
$A(x,\gamma_{k+1/2})-A(x,\gamma_{*})=(\gamma_{k+1/2}-\gamma_{*})(x)\frac{\partial
A(x,t)}{\partial t}\Big{|}_{t=c(x)},$

where
$c(x)=\gamma_{k+1/2}(x)+t(x)\big{(}\gamma_{*}(x)-\gamma_{k+1/2}(x)\big{)}$,
hence we have

(4.30)
$\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((\gamma_{k+1/2}-\gamma_{*})\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(\cdot)}E_{k}\times
B_{0})=\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot(A_{*}(E_{*}-E_{k})\times
B_{0}).$

Arguing as above, the left hand side of (4.30) can be estimated from below as

(4.31)
$\begin{split}&\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((\gamma_{k+1/2}-\gamma_{*})\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(\cdot)}E_{k}\times B_{0})dx\\\
&\geq\frac{1}{2}\int_{\partial\Omega}(\gamma_{k+1/2}-\gamma_{*})^{2}\cdot\Big{(}\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(\cdot)}E_{k}\times B_{0}\Big{)}\cdot\nu
ds(x)\\\ &+\frac{1}{2}||(\gamma_{k+1/2}-\gamma_{*})||^{2}_{L^{2}(\Omega)}\\\
&-\frac{1}{2}\int_{\Omega}|\gamma_{k+1/2}-\gamma_{*}|^{2}\quad\underset{\Omega}{\sup}\Big{|}\nabla\cdot\Big{(}\Big{(}\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(\cdot)}-I\Big{)}E_{k}\times
B_{0}\Big{)}\Big{|}dx.\end{split}$

and again, in either case, 1. or 2., (4.31), combined together with (4.15),
leads to

(4.32)
$\begin{split}&\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot((\gamma_{k+1/2}-\gamma_{*})\frac{\partial
A(\cdot,t)}{\partial t}\Big{|}_{t=c(\cdot)}E_{k}\times B_{0})dx\\\
&\geq\frac{1}{2}(1-\tilde{\mathcal{E}_{2}})||(\gamma_{k+1/2}-\gamma_{*})||^{2}_{L^{2}(\Omega)}.\\\
\end{split}$

The right hand side of (4.30) can be estimated from above as

(4.33)
$\begin{split}&\Big{|}\int_{\Omega}(\gamma_{k+1/2}-\gamma_{*})\nabla\cdot(A_{*}(E_{*}-E_{k})\times
B_{0})\Big{|}\\\
&\leq(\tilde{C_{4}}+\mathcal{C})(\Lambda+\mathcal{E}_{2}(1+K))||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}||\gamma_{*}-\gamma_{k}||_{L^{2}(\Omega)},\end{split}$

by combining estimates (2.5) and (2.10), and the use of (2.11), (2.14) and
(3.11).

Therefore, choosing
$\frac{(\tilde{C_{4}}+\mathcal{C})(\Lambda+\mathcal{E}_{2}(1+K))}{1-\tilde{\mathcal{E}_{2}}}\leq\frac{1}{2}$,
we derive

(4.34) $||\gamma_{k+1/2}-\gamma_{*}||_{L^{2}(\Omega)}\leq
C||\gamma_{k}-\gamma_{*}||_{L^{2}(\Omega)}.$

∎

##  5 Numerical Experiments

In this section we present several numerical experiments where we implement
the algorithm described in section 4. In each case, a scalar reference medium
$\gamma_{*}\in\tilde{S}$ (the true scalar function to be reconstructed) is
chosen, along with a specific choice of a $3\times 3$ matrix function $A$
belonging either to $\mathcal{A}$ or $\mathcal{A^{\prime}}$. We start by
considering a number of examples in which $A\in\mathcal{A}$ only (Examples 1 -
4). The more general case of $A\in\mathcal{A^{\prime}}$ is exploited in
Examples 5 and 6.

We then compute the synthetic acoustic source data

(5.1) $F(\gamma_{*})=\nabla\cdot(A_{\gamma_{*}}E_{*}\times B_{0}),$

where $E_{*}$ is the electric field solution to (4.5) corresponding to
$\gamma_{*}$. For simplicity, in our examples we always choose reference
conductivity $\gamma_{0}=1$. We note that in the particular examples here, we
found that the projection step of the algorithm (Step 4) was not necessary for
convergence of the iterates, and in two of the examples below, $\gamma_{*}$ is
not in $C^{1}$ (Examples 4 and 6, where $\gamma_{*}$ is piecewise affine and
piecewise constant, respectively). This is to test the reconstruction from non
smooth data. Again, it will be part of future work to extend our theoretical
framework of sections 2 \- 4 to conductivities with lower regularity
assumptions and a more general anisotropic structure.

To compute solutions to both (4.5) and (4.6), we use the Python suite FEniCS.
As described in the proof of Proposition 3.2, we replace (4.5) with the
Neumann problem (3.12) and solve numerically for $u_{k+1}=u$ using a standard
variational formulation and piecewise linear finite elements. Recall then that
$E_{k+1}$ is given by

$E_{k+1}=\nabla u_{k+1}+\tilde{E},$

where $\tilde{E}=0.5[-y,x,0]^{T}$. We note that while (3.12) is always a
linear conductivity equation, (4.6) changes more drastically for different
choices of $A$. To compute solutions to this generally nonlinear transport
equation, we used two different approaches; a discontinuous Galekin method and
a built-in FEniCS nonlinear solver. We give more details about these in the
first two examples.

Figure 1: Reference conductivity $\gamma_{*}$ at $z=0$

Figure 2: Reconstructed conductivity at $z=0$

Figure 3:  $A$ given by (5.2). The resulting transport equation is linear. Ran
for 10 iterations on a mesh of size 150 x 150 x 1.

Example 1. For our first example we choose $A$ to be

(5.2) $A(\gamma)\ \ =\ \ \begin{bmatrix}\gamma&0&0\\\ 0&\gamma&0\\\
0&0&1\end{bmatrix}$

and reference medium $\gamma_{*}$ to be the Gaussian curve independent of $z$:

$\gamma_{*}=\exp{\left(-\frac{(x-0.5)^{2}}{0.02}-\frac{(y-0.5)^{2}}{0.02}\right)}+1.$

In this case the transport equation (4.6) for $\gamma_{k+1}$ becomes

$\begin{cases}\nabla\cdot(\gamma_{k+1}(E_{k}\times
B_{0}))=\nabla\cdot(\gamma_{*}(E_{*}\times B_{0})),&\text{in }\Omega\\\
\gamma_{k+1}=\gamma_{*},\hfill&\text{ on }\partial\Omega^{-},\end{cases}$

which is 2D linear isotropic, as was studied previously in [8]. To solve this
linear transport equation, we use a discontinuous Galerkin (DG) method with
upwinding, which we now describe briefly. Given a mesh $T_{h}$ of $\Omega$, we
let $V$ be a finite dimensional space containing functions not necessarily
continuous across mesh elements. We define the ‘upwinding’ term on the
boundary of a triangle by

$\bar{E}_{k}=0.5((E_{k}\times B_{0})\cdot\vec{n}+|(E_{k}\times
B_{0})\cdot\vec{n}|)$

and denote the jump of a function on an internal edge $E$ by

$[[f]]=f^{+}-f^{-},$

where $f^{+}$ ($f^{-}$) is its restriction to the edge from the outflow
(inflow) direction, respectively. Then the variational formulation for the
numerical transport problem is:

find $\gamma_{k+1}=\gamma\in V$ such that, for all $v\in V$,

$\int_{\partial\Omega}v\gamma(E_{k}\times
B_{0})\cdot\vec{n}\,ds+\sum_{E}\int_{E}[[v]][[\gamma\bar{E}_{k}]]\,dS-\int_{\Omega}\nabla
v\cdot\gamma(E_{k}\times B_{0})\,dx=\int_{\Omega}F(\gamma^{*})v\,dx,$

where $F(\gamma^{*})$ is the source data (5.1). Note that the upwinding term
provides numerical stability. We refer to [14] for more details on DG methods
and upwinding. Figure 3 shows the true $\gamma_{*}$, plotted alongside its
numerical reconstruction using the above algorithm.

Figure 4: Reference conductivity $\gamma_{*}$ at $z=0$

Figure 5: Reconstructed conductivity at $z=0$

Figure 6:  $A$ given by (5.3), resulting in transport equation (5.4). Ran for
10 iterations on a mesh of size 75 x 75 x 10.

Example 2. In this next example we add nonlinearity in the dependence of A on
$\gamma$. To solve the transport equation (4.6) in this and all of the
examples that follow, instead of DG, we use the Fenics nonlinear solver with
higher order Lagrange elements. We choose here

(5.3) $A(\gamma)\ \ =\ \ \begin{bmatrix}0.4(\gamma+1)^{2}&0.01&0\\\
0.01&3\gamma&0\\\ 0&0&\gamma\end{bmatrix}$

and reference conductivity $\gamma_{*}$ to be a sum of Gaussian curves given
by

$\gamma_{*}=\exp{\left(-\frac{(x-0.65)^{2}}{0.02}-\frac{(y-0.65)^{2}}{0.02}\right)}+0.5*\exp{\left(-\frac{(x-0.25)^{2}}{0.05}-\frac{(y-0.25)^{2}}{0.05}\right)}.$

With the definitions above, the transport problem (4.6) becomes the nonlinear
equation for $\gamma=\gamma_{k+1}$:

(5.4)
$a_{1}\gamma^{2}+a_{2}\gamma+a_{3}\gamma\gamma_{x}+a_{4}\gamma_{x}-a_{5}\gamma_{y}+c=F(\gamma_{*}),\qquad\text{
in }\quad\Omega,$

with inflow condition $\gamma=\gamma_{*}$ on $\partial\Omega^{-}$, where the
spatially varying coefficients are functions of $E=[E_{1},E_{2},E_{3}]^{T}$,
given by $a_{1}=0.4E_{2,x}$, $a_{2}=0.8E_{2,x}-3E_{1,y}$,
$a_{3}=a_{4}=0.8E_{2}$, $a_{5}=3E_{1}$ and $c=0.4E_{2,x}$. We show the
reconstruction in Figure 6.

Figure 7: Reference conductivity $\gamma_{*}$ at $z=0$

Figure 8: Reconstructed conductivity at $z=0$

Figure 9: A given by (5.5), yielding transport equation (5.6). Ran for 10
iterations on a mesh of size 75 x 75 x 10.

Example 3. In this next example we add nonlinearity on the off diagonals. Let

(5.5) $A(\gamma)\ \ =\ \
\begin{bmatrix}0.4(\gamma+1)^{2}&0.01\gamma(1-\gamma)&0\\\
0.01\gamma(1-\gamma)&3\gamma&0\\\ 0&0&\gamma\end{bmatrix}$

and let $\gamma_{*}$ be given by

$\gamma_{*}(x,y)=\cos{(75(x-0.5)^{2}+75(y-0.5)^{2})}\exp{\left(-\frac{(x-0.5)^{2}}{2}-\frac{(y-0.5)^{2}}{2}\right)}+1.$

In this case the transport equation (4.6) for $\gamma_{k+1}=\gamma$ becomes:

(5.6)
$a_{1}\gamma^{2}+a_{2}\gamma\gamma_{y}+a_{3}\gamma_{y}+a_{4}\gamma+a_{5}\gamma_{x}+a_{6}\gamma\gamma_{x}+c=F(\gamma_{*}),\qquad\text{in}\quad\Omega,$

with $\gamma=\gamma_{*}$ on $\partial\Omega^{-}$, where the coefficients are
given by $a_{1}=0.4E_{2,x}+0.01E_{1,x}-0.01E_{2,y}$, $a_{2}=0.898E_{2}$,
$a_{3}=0.01E_{2}-3E_{1}$, $a_{4}=0.8E_{2,x}-0.01E_{1,x}+0.01E_{2,y}-3E_{1,y}$,
$a_{5}=0.8E_{2}-0.01E_{1}$, $a_{6}=0.02E_{1}$, and $c=0.4E_{2,x}$.
Reconstruction results are in Figure 9.

Figure 10: Reference conductivity $\gamma_{*}$ at $z=0$

Figure 11: Reconstructed conductivity at $z=0$

Figure 12: A given by (5.7). Ran for 10 iterations on a mesh of size 75 x 75 x
10.

Example 4. Here we choose a piecewise affine $\gamma_{*}$ to test the
reconstruction of non smooth conductivity. Let

(5.7) $A(\gamma)\ \ =\ \
\begin{bmatrix}0.4(\gamma+1)^{2}&\frac{1}{\gamma+20}&0\\\
\frac{1}{\gamma+20}&3\gamma&0\\\ 0&0&\gamma\end{bmatrix}$

and define $\gamma_{*}$ by

$\gamma_{*}(x,y)=\begin{cases}1+5(x-0.3)&,\ \ \ 0.3\leq x\leq 0.5\\\
2-5(x-0.5)&,\ \ \ 0.5\leq x\leq 0.7\\\ 1&,\ \ \ \text{else}\end{cases}$

With the definitions above, the transport problem (4.6) becomes: find
$\gamma=\gamma_{k+1}$ such that

(5.8)
$(E_{2,x}-E_{1,y})\gamma+(E_{2})\gamma_{x}-\left(\frac{E_{2}}{(\gamma+20)^{2}}+E_{1}\right)\gamma_{y}\\\
+\left(\frac{1}{\gamma+20}\right)\left(E_{2,y}-E_{1,x}+\frac{E_{1,x}}{\gamma+20}\right)=F(\gamma^{*}),\qquad\text{in}\quad\Omega,$

with $\gamma=\gamma_{*}$ on $\partial\Omega^{-}$. Numerical reconstructions
are show in Figure 12.

Figure 13: Reference conductivity $\gamma_{*}$ at $z=0$

Figure 14: Reconstructed conductivity at $z=0$

Figure 15: A given by (5.9). Ran for 10 iterations on a mesh of size 100 x 100
x 10.

Example 5. Next we choose $A$ to be spatially varying as well as dependent on
$\gamma$, as follows,

(5.9) $A(\vec{x},\gamma)\ \ =\ \
\begin{bmatrix}0.4(\gamma+1)^{2}&0.25(x^{2}+y^{2})\gamma&0\\\
0.25(x^{2}+y^{2})\gamma&3\gamma&0\\\ 0&0&\gamma\end{bmatrix}$

and we let $\gamma_{*}$ be the trigonometric function

$\gamma_{*}(x,y)=\sin(10x)\sin(5y)\sin(7(1-x))\sin(y-1)+1.$

Numerical results are given in Figure 15.

Example 6. Finally, we present an example in which $A$ is spatially varying as
well as dependent on $\gamma$ and $\gamma_{*}$ is z-dependent . We let

(5.10) $A(\vec{x},\gamma)\ \ =\ \
\begin{bmatrix}0.4(\gamma+1)^{2}&0.25((x-0.5)^{2}+(y-0.5)^{2})\gamma&0\\\
0.25((x-0.5)^{2}+(y-0.5)^{2})\gamma&3\gamma&0\\\ 0&0&\gamma\end{bmatrix}$

and we let $\gamma_{*}$ be a piecewise constant corresponding to a spherical
inclusion, that is,

$\gamma_{*}(x,y,z)=\begin{cases}2\ ,&(x-0.5)^{2}+(y-0.5)^{2}+(z-0.5)^{2}\leq
0.4\\\ 1\ ,&\text{else}\end{cases}.$

Slices of the three dimensional reconstruction are shown in Figures 16 \- 20.

Figure 16: Reconstruction for (5.10) at $z=0$.

Figure 17: Reconstruction for (5.10) at $z=0.282$.

Figure 18: Reconstruction for (5.10) at $z=0.513$.

Figure 19: Reconstruction for (5.10) at $z=0.718$.

Figure 20: Reconstruction for $A$ given by the $z$-dependent (5.10) for
$z=0.897$. Ran for 10 iterations on a mesh of size 40 x 40 x 40.

##  6 Conclusions

In this work we studied issues of stability and reconstruction of the
anisotropic conductivity $\sigma$ of a biological medium
$\Omega\subset\mathbb{R}^{3}$ by the hybrid inverse problem of Magneto-
Acoustic Tomography with Magnetic Induction MAT-MI. More specifically, we
considered a class of conductivities which correspond to a one-parameter
family of symmetric and uniformly positive matrix-valued functions $t\mapsto
A(x,t)$, which are a-priori known to depend nonlinearly on
$t\in[\lambda^{-1},\lambda]$. This gives rise to the family of anisotropic
conductivities $A(x,\gamma(x))$, $x\in\Omega$, for which the goal is to stably
reconstruct the scalar function $\gamma$ in $\Omega$. We showed that if $A$
belongs to certain classes of admissible anisotropic structures, then a
Lipschitz type stability estimate of the scalar function $\gamma$ from the
data given by and internal functional, holds true. In particular, the argument
for our theoretical framework requires that $\gamma$ and $A$ belong to
$C^{1,\beta}$. Our stability estimates extend the results in [8] to the case
where $\sigma$ depends nonlinearly on $\gamma$, hence allowing us to consider
a more realistic type of anisotropic structures. Furthermore, we showed that
the convergence of the reconstruction algorithm introduced in [8] extends to
this nonlinear case, and demonstrated its effectiveness in several numerical
experiments.

Several questions remain, including, as mentioned in [8], the reconstruction
of full anisotropy, which we expect will require more measurements at hand. It
will also be interesting to investigate more precisely to what extent the
regularity assumptions considered in this paper are needed for the convergence
of the reconstruction algorithm in practice; and the related question of when
one needs to actually regularize the iterates by invoking the projections into
the convex set $\tilde{S}$. These questions are the subject of future work.

## Acknowledgments

The research conducted by N. Donlon and R. Gaburro in this publication was
funded by the Irish Research Council under the Grant number: GOIPG/2021/527.
R. Gaburro was partially supported by Science Foundation Ireland under Grant
number 16/RC/3918. S. Moskow was partially supported by NSF grant DMS-2008441.

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