An adaptive ﬁnite elementﬁnite difference domain decomposition method for applications in microwave imaging

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An adaptive ﬁnite element/ﬁnite difference domain

decomposition method for applications in

microwave imaging

L. Beilina *Eric Lindstr¨

om †‡

Abstract

A new domain decomposition method for Maxwell’s equations in conductive me-

dia is presented. Using this method reconstruction algorithms are developed for de-

termination of dielectric permittivity function using time-dependent scattered data of

electric ﬁeld. All reconstruction algorithms are based on optimization approach to ﬁnd

stationary point of the Lagrangian. Adaptive reconstruction algorithms and space-

mesh reﬁnement indicators are also presented. Our computational tests show qual-

itative reconstruction of dielectric permittivity function using anatomically realistic

breast phantom.

Keywords: Maxwell’s equations; conductive media; microwave imaging; coefﬁcient

inverse problem; adaptive ﬁnite element method; ﬁnite difference method; domain decom-

position

MSC: 65M06; 65M32; 65M55; 65M60

1 Introduction

In this work are presented reconstructions algorithms for the problem of determination of

the spatially distributed dielectric permittivity function in conductive media using scattered

time-dependent data of the electric ﬁeld at the boundary of investigated domain. Such

*Department of Mathematical Sciences, Chalmers University of Technology and University of Gothen-

burg, SE-42196 Gothenburg, Sweden, e-mail: larisa@chalmers.se

†Department of Mathematical Sciences, Chalmers University of Technology and University of Gothen-

burg, SE-42196 Gothenburg, Sweden, e-mail: erilinds@chalmers.se

‡Journal version of this paper is published in Electronics 2022, 11, 1359.

https://doi.org/10.3390/electronics/11091359

arXiv:2210.14680v1 [math.NA] 26 Oct 2022

problems are called Coefﬁcient Inverse Problems (CIPs). A CIP for a system of time-

dependent Maxwell’s equations for electric ﬁeld is a problem about the reconstruction of

unknown spatially distributed coefﬁcients of this system from boundary measurements.

One of the most important application of algorithms of this paper is microwave imag-

ing including microwave medical imaging and imaging of improvised explosive devices

(IEDs). Potential application of algorithms developed in this work are in breast cancer

detection. In numerical examples of current paper we will focus on microwave medical

imaging of realistic breast phantom provided by online repository [59]. In this work we de-

velop simpliﬁed version of reconstruction algorithms which allow determine the dielectric

permittivity function under the condition that the effective conductivity function is known.

Currently we are working on the development of similar algorithms for determination of

both spatially distributed functions, dielectric permittivity and conductivity, and we are

planning report about obtained results in a near future.

Microwave medical imaging is non-invasive imaging. Thus, it is very attractive addi-

tion to the existing imaging technologies like X-ray mammography, ultrasound and MRI

imaging. It makes use of the capability of microwaves to differentiate among tissues based

on the contrast in their dielectric properties.

In [30] were reported different malign-to-normal tissues contrasts, revealing that ma-

lign tumors have a higher water/liquid content, and thus, higher relative permittivity and

conductivity values, than normal tissues. The challenge is to accurately estimate the rel-

ative permittivity of the internal structures using the information from the backscattered

electromagnetic waves of frequencies around 1 GHz collected at several detectors.

Since the 90-s quantitative reconstruction algorithms based on the solution of CIPs

for Maxwell’s system have been developed to provide images of the complex permittivity

function, see [17] for 2D techniques, [15, 18, 31, 38] for 3D techniques in the frequency

domain and [49, 56] for time domain (TD) techniques.

In all these works microwave medical imaging remained the research ﬁeld and had

little clinical acceptance [37] since the computations are inefﬁcient, take too long time,

and produce low contrast values for the inside inclusions. In all the above cited works lo-

cal gradient-based mathematical algorithms use frequency-dependent measurements which

often produce low contrast values of inclusions and miss small cancerous inclusions. More-

over, computations in these algorithms are done often in MATLAB, sometimes requiring

around 40 hours for solution of inverse problem.

It is well known that CIPs are ill-posed problems [2, 32, 53, 55]. Development of

non-local numerical methods is a main challenge in solution of a such problems. In

works [6, 7, 51, 52] was developed and numerically veriﬁed new non-local approximately

globally convergent method for reconstruction of dielectric permittivity function. The two-

stage global adaptive optimization method was developed in [6] for reconstruction of the

dielectric permittivity function. The two-stage numerical procedure of [6] was veriﬁed in

several works [7,51,52] on experimental data collected by the microwave scattering facility.

The experimental and numerical tests of above cited works show that developed meth-

ods provide accurate imaging of all three components of interest in imaging of targets:

shapes, locations and refractive indices of non-conductive media. In [38], see also ref-

erences therein, authors show reconstruction of complex dielectric permittivity function

using convexiﬁcation method and frequency-dependent data. Potential applications of all

above cited works are in the detection and characterization of improvised explosive devices

(IEDs).

The algorithms of the current work can efﬁciently and accurately reconstruct the di-

electric permittivity function for one concrete frequency using single measurement data

generated by a plane wave.

A such plane wave can be generated by a horn antenna as it was done in experimental

works [7, 51, 52]. We are aware that conventional measurement conﬁguration for detection

of breast cancer consists of antennas placed on the breast skin [1, 18, 19, 37, 49]. In this

work we use another measurement set-up: we assume that the breast is placed in a coupling

media and then the one component of a time-dependent electric plane wave is initialized at

the boundary of this media. Then scattered data is collected at the transmitted boundary.

This data is used in reconstruction algorithms developed in this work. Such experimental

set-up allows avoid multiply measurements and overdetermination since we are working

with data resulted from a single measurement. An additional advantage is that in the case

of single measurement data one can use the method of Carleman estimates [33] to prove

the uniqueness of reconstruction of dielectric permittivity function.

For numerical solution of Maxwell’s equations we have developed ﬁnite element/ﬁnite

difference domain decomposition method ( FE/FD DDM).

This approach combines the ﬂexibility of the ﬁnite elements and the efﬁciency of the

ﬁnite differences in terms of speed and memory usage as well as ﬁts the best for recon-

struction algorithms of this paper. We are unaware of other works which use similar set-up

for solution of CIP for time-dependent Maxwell’s equations in conductive media solved

via FE/FD DDM, and this is the ﬁrst work on this topic.

An outline of the work is as follows: in section 2 we present the mathematical model

and in section 3 we describe the structure of domain decomposition. Section 4 presents

reconstruction algorithms including formulation of inverse problem, derivation of ﬁnite

element and ﬁnite difference schemes together with optimization approach for solution

of inverse problem. Section 5 shows numerical examples of reconstruction of dielectric

permittivity function of anatomically realistic breast phantom at frequency 6 GHz of online

repository [59]. Finally, section 6 discusses obtained results and future research.

2 The mathematical model

Our basic model is given in terms of the electric ﬁeld E(x,t)=(E1,E2,E3)(x,t),x∈R3

changing in the time interval t∈(0,T)under the assumption that the dimensionless relative

magnetic permeability of the medium is µr≡1. We consider the Cauchy problem for the

Maxwell equations for electric ﬁeld E(x,t), further assuming that that the electric volume

charges are equal zero, to get the model equation for x∈R3,t∈(0,T].

c2εr

∂2E

∂t2+∇×∇×E=−µ0σ∂E

∂t,

∇·(εE) = 0,

E(·,0) = f0,∂E

∂t(·,0) = f1.

(1)

Here, εr(x) = ε(x)/ε0is the dimensionless relative dielectric permittivity and σ(x)is the ef-

fective conductivity function, ε0,µ0are the permittivity and permeability of the free space,

respectively, and c=1/√ε0µ0is the speed of light in free space.

We are not able numerically solve the problem (1) in the unbounded domain, and thus

we introduce a convex bounded domain Ω⊂R3with boundary ∂Ω. For numerical solu-

tion of the problem (1), a domain decomposition ﬁnite element/ﬁnite difference method is

developed and summarized in Algorithm 1 of section 3.

A domain decomposition means that we divide the computational domain Ωinto two

subregions, ΩFEM and ΩFDM such that Ω=ΩFEM ∪ΩFDM with ΩFEM ⊂Ω, see Fig-

ure 2. Moreover, we will additionally decompose the domain ΩFEM =ΩIN ∪ΩOUT with

ΩIN ⊂ΩFEM such that functions εr(x)and σ(x)of equation (1) should be determined only

in ΩIN, see Figure 2. When solving the inverse problem IP this assumption allows stable

computation of the unknown functions εr(x)and σ(x)even if they have large discontinu-

ities in ΩFEM.

The communication between ΩFEM and ΩFDM is arranged using a mesh overlapping

through a two-element thick layer around ΩFEM, see elements in blue color in Figure 1-

a),b). This layer consists of triangles in R2or tetrahedrons in R3for ΩFEM, and of squares

in R2or cubes in R3for ΩFDM.

The key idea with such a domain decomposition is to apply different numerical methods

in different computational domains. For the numerical solution of (1) in ΩFDM we use

the ﬁnite difference method on a structured mesh. In ΩFEM, we use ﬁnite elements on a

sequence of unstructured meshes Kh={K}, with elements Kconsisting of tetrahedron’s in

R3satisfying minimal angle condition [34].

We assume in this paper that for some known constants d1>1,d2>0, the functions

εr(x)and σ(x)of equation (1) satisfy

εr(x)∈[1,d1],σ(x)∈[0,d2],for x∈ΩIN,

εr(x) = 1,σ(x) = 0 for x∈ΩFDM,εr(x),σ(x)∈C2R3.(2)

Turning to the boundary conditions at ∂Ω, we use the fact that (2) and (1) imply that

since εr(x) = 1,σ(x) = 0 for x∈ΩFDM ∪ΩOUT,then a well known transformation

∇×∇×E=∇(∇·E)−∇·(∇E)(3)

makes the equations (1) independent on each other in ΩFDM, and thus, in ΩFDM we need to

solve the equation

∂2E

∂t2−∆E=0,(x,t)∈ΩFDM ×(0,T].(4)

We write ∂Ω=∂Ω1∪∂Ω2∪∂Ω3, meaning that ∂Ω1and ∂Ω2are the top and bottom sides

of the domain Ω, while ∂Ω3is the rest of the boundary. Because of (4), it seems natural to

impose ﬁrst order absorbing boundary condition for the wave equation [22],

∂E

∂n+∂E

∂t=0,(x,t)∈∂Ω×(0,T].(5)

Here, we denote the outer normal derivative of electrical ﬁeld on ∂Ωby ∂·

∂n, where ndenotes

the unit outer normal vector on ∂Ω.

It is well known that for stable implementation of the ﬁnite element solution of Maxwell’s

equation divergence-free edge elements are the most satisfactory from a theoretical point

of view [40, 43]. However, the edge elements are less attractive for solution of time-

dependent problems since a linear system of equations should be solved at every time

iteration. In contrast, P1 elements can be efﬁciently used in a fully explicit ﬁnite element

scheme with lumped mass matrix [20, 29]. It is also well known that numerical solution of

Maxwell equations using nodal ﬁnite elements can be resulted in unstable spurious solu-

tions [41, 46]. There are a number of techniques which are available to remove them, see,

for example, [26–28, 42, 46].

In the domain decomposition method of this work we use stabilized P1 FE method for

the numerical solution of (1) in ΩFEM. Efﬁciency of usage an explicit P1 ﬁnite element

scheme is evident for solution of CIPs. In many algorithms which solve electromagnetic

CIPs a qualitative collection of experimental measurements is necessary on the boundary

of the computational domain to determine the dielectric permittivity function inside it. In

this case the numerical solution of time-dependent Maxwell’s equations are required in the

entire space R3, see for example [6, 7, 11, 51, 52], and it is efﬁcient to consider Maxwell’s

equations with constant dielectric permittivity function in a neighborhood of the boundary

of the computational domain. An explicit P1 ﬁnite element scheme with σ=0 in (1) is

numerically tested for solution of time-dependent Maxwell’s system in 2D and 3D in [3].

Convergence analysis of this scheme is presented in [4] and CFL condition is derived in [5].

The scheme of [3] is used for solution of different CIPs for determination of dielectric per-

mittivity function in non-conductive media in time-dependent Maxwell’s equations using

simulated and experimentally generated data, see [7, 11, 51, 52].

The stabilized model problem considered in this paper is:

c2εr∂2E

∂t2+∇(∇·E)−4E−ε0∇(∇·(εrE)) = −µ0σ∂E

∂tin Ω×(0,T),

E(·,0) = f0,and ∂E

∂t(·,0) = f1in Ω,

∂E

∂n=−∂E

∂ton ∂Ω×(0,T),

(6)

with functions εr,σsatisfying conditions (2).

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摘要：

Anadaptiveniteelement/nitedifferencedomaindecompositionmethodforapplicationsinmicrowaveimagingL.Beilina*EricLindstr¨omAbstractAnewdomaindecompositionmethodforMaxwell'sequationsinconductiveme-diaispresented.Usingthismethodreconstructionalgorithmsaredevelopedforde-terminationofdielectricpermittivi...

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An adaptive ﬁnite elementﬁnite difference domain decomposition method for applications in microwave imaging

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