Jing An?,Jie Shen

2SchoolofMathematicsandComputerScience,GuizhouNormalUniversity,Guiyang 550001,P.R.China

3Department of Mathematics Purdue University,West Lafayette,IN 47907,USA

Received 2 February 2014;Accepted 1 March 2014

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

Jing An1,2,?,Jie Shen1,31School of Mathematical Science,Xiamen University,Xiamen 361005,P.R.China

2SchoolofMathematicsandComputerScience,GuizhouNormalUniversity,Guiyang 550001,P.R.China

3Department of Mathematics Purdue University,West Lafayette,IN 47907,USA

Received 2 February 2014;Accepted 1 March 2014

.An important step in estimating the index of refraction of electromagnetic scattering problems is to compute the associated transmission eigenvalue problem.We develop in this paper efficient and accurate spectral methods for computing the transmission eigenvalues associated to the electromagnetic scattering problems.We present ample numerical results to show that our methods are very effective for computing transmission eigenvalues(particularly for computing the smallest eigenvalue),and together with the linear sampling method,provide an efficient way to estimate the index of refraction of a non-absorbing inhomogeneous medium.

AMS subject classifications:78A45,65N35,35J05,41A58

spectral method,index of refraction,transmission eigenvalue,electromagnetic scattering,inverse problem.

1 Introduction

The inverse electromagnetic scattering problem plays an important role in many applications,and is notoriously difficult.Recently a new method using transmission eigenvalues to estimate the index of refraction of a non-absorbing inhomogeneous medium is proposed in[4,5,9].The method consists of several steps.First,the support of the scattering obstacle can be recovered by using the measured scattering data and the linear sampling method[12],and the transmission eigenvalues can be identi fied from the far field data.Then,the bounds for smallest and largest eigenvalues of the(matrix)index of refraction can be obtained in terms of the support of the scattering obstacle and the first transmission eigenvalue of the anisotropic media[3].Finally,reconstructions of the electric permittivity(if it is a scalar constant)or an estimate of the eigenvalues of the matrix in the case of anisotropic permittivity can be obtained[5].

The effectiveness of the above method rests on having an efficient and robust algorithm for computing transmission eigenvalues for a scalar permittivity.In this paper,we develop efficient spectral methods for computing the transmission eigenvalues in circular and rectangular domains.In particular,for circular domains with stratified media,our method reduces the problem to a sequence of one-dimensional transmission eigenvalue problems that can be solved efficiently and accurately by a spectral-element methods.An error estimate for convergence of the transmission eigenvalues is also provided in this case.

We present ample numerical results to show that our methods are very effective,particularly for computing the few smallest eigenvalues.By using this together with the linear sampling method,we can effectively estimate the(matrix)index of refraction of a non-absorbing inhomogeneous medium.

The organization of the paper is as follows:In§2,we describe the general approach introduced in[4,5,9]for estimation of the index of refraction of a non-absorbing inhomogeneous medium.In§3,we derive a weak formulation,construct a Fourier-spectralelement method and derive error estimates for transmission eigenvalues in circular domains.In§4,we describe a spectral method for computing transmission eigenvalues in rectangular domains.We present some numerical results to validate our numerical algorithms in§5.

2 Description of the general approach

In this section,we describe briefly the method introduced in[4,5,9]for estimation of the index of refraction of a non-absorbing inhomogeneous medium.We first show how to obtain transmission eigenvalues from far field data,and then describe an algorithm to reconstruct/estimate the index of refraction.

2.1 Transmission eigenvalue problem

LetD?Rd(d=2,3)be a bounded,simply connected open set with a piecewise smooth boundary?D.We assume thatthe domainDis the supportofan anisotropic dielectric object,and the incident field is a time-harmonic electromagnetic plane wave with frequencyω.Then,the scattering by the anisotropic medium leads to the following problem for the interior electric and magnetic fieldsEint,Hint,and the scattered electric and magnetic fieldEext,Hext[5]:

whereνis the unit outward normal to?DandN(x)is a 3×3 symmetric matrix representing the electric permittivity.We write

whereEiandHiare the incident electric and magnetic fields given by

withdbeing a unit vector giving the direction of propagation andpthe polarization vector.The scattered electric and magnetic fieldsEsandHssatisfy the Silver-M¨uller radiation condition

In this paper,we considerthe two-dimensionalscattering problem from an orthotropic medium.More precisely,we assume that the scatterer is an infinitely long dielectric cylinder with axis in the z-direction,and denote byDthe cross section of the cylinder in(x1,x2)plane.We assume that the dielectric cylinder is orthotropic,i.e.the matrixN(x)takes the form

Assuming thatN-1exists,and expressing the electric fields in terms of magnetic fields,Eqs.(2.1)-(2.6)now lead to the following transmission problem forωandv[2]:

wherevsis the scattered field andviis the given incident field.In the case of plane waves,the incident field is given byvi:=eikx·dwith|x|=1.Moreover

and the radiation condition(2.10)holds uniformly with respect to?x=x/|x|.The existence of a unique solution to(2.7)-(2.10)can be established by using a variational procedure[6,13].

We recall that the scattererDcan be determined with the linear sampling method by solving the far field equation

andH(1)denotes the Hankel function of the first kind of order zero.Indeed,it is shown in[6]t0h at the far field operatorF:L2(Ω)→L2(Ω)de fined by

is injective with dense range,provided thatkis not a transmission eigenvalue,i.e.,a value ofkfor which the(homogeneous)interior transmission problem

has a nontrivial solutionω,v∈H1(D).Due to the lack of injectivity and the denseness of the range of the far field operatorF,theL2-norm of the(regularized)solution to

can be expected to be large whenkis a transmission eigenvalue[2].This provides a method to determine the transmission eigenvalues using the far field data.

2.2 Estimation of the index of refraction

We denote the smallest real transmission eigenvalue of(2.11)-(2.14)byk1(D,A).Sincek1(D,A)can be computed from far field data[5]as shown above,we now describe an algorithm similar to that used in[15]to estimate the index of refraction.by usingk1(D,A).We shall restrict our attention to the caseA=I/n(x),more general case can be treated similarly as in[5].Denoting

we can rewrite Eqs.(2.11)-(2.14)as

The suitable functional spaces to analyze this problem are(see[2]and[7]for details)

which can be written in the following variational form

We note that the eigenvalue problem(2.20)can be written as an operator equation

whereτ=k2.Here the bounded linear operatorsAτ:eH0(D)→eH0(D),?Aτ:eH0(D)→eH0(D)andB:eH0(D)→eH0(D)are the operators defined using the Riesz representation theorem associated with the sesquilinear form Aτ,?Aτand B which are defined by(see[7]for more details)

respectively.

Now we consider the following generalized eigenvalue problem:

From(2.21),a transmission eigenvalue is the root of

Theorem 2.1.Letk1(D,n)be the first transmission eigenvalue for(2.15)-(2.18)and letαandβbe positive constants.Denote byk1(D,n?)andk1(D,n?)the first transmission eigenvalue for(2.15)-(2.18)forn(x)≡n?andn(x)≡n?respectively.Then,we have the following results:

Proof.The proof is similar to that of Theorem 3.3 in[5],so we shall only sketch the proof for the case ofn?≥n(x)≥n?≥α＞1.

Therefore,for an arbitraryτ＞0,we have

whereλ1(τ,D,n?),λ1(τ,D,n(x))andλ1(τ,D,n?)are the first transmission eigenvalues of(2.22)corresponding to the index of refractionn?,n(x)andn?,respectively.Now using(2.25)forτ1=k21(D,n?),we haveλ1(τ1,D,n(x))-τ1≥ 0.Again using(2.25)forτ2=k21(D,n?),we haveλ1(τ2,D,n(x))-τ2≤ 0.Then by continuity of the mappingτ→λ1(τ,D,n(x)),there is an eigenvalue corresponding to(D,n(x))betweenk1(D,n?)andk1(D,n?).

To complete the proof,we need to show that this is the first eigenvalue for(D,n(x)).Indeed,ifk1(D,n(x))＜k1(D,n?),then from(2.25),we haveλ1(τ3,D,n(x))-τ3≤ 0 forτ3=k21(D,n(x)).On the other hand,it is shown in[8]that forτ0＞ 0 sufficiently small,we haveλ1(τ0,D,n(x))-τ0≥0,which means that there is a transmission eigenvalue for(D,n?)less than the first one,a contradiction to the assumption.

LetμD:L∞(D)→Rwhich maps a given index of refractionn(x)onto the smallest transmission eigenvalue ofD,namely,μD(n)=k1(D,n).Assuming thatk1(D,n)is obtained from farfield data,we seek a constantn0minimizing the difference betweenμD(n)andk1(D,n).whenn＞1(the case of 0＜n＜1 can be treated the same way),Theorem(2.1)shows that the transmission eigenvalues fornbeing a constant are monotonically decreasing with respect ton.Sincek1(D,n)is a continuous function ofn,we can estimaten0using the following algorithm(cf.[15])such that the computed lowest transmission eigenvalueμ(D)coincides with the valuek1(D,n)obtained from the far field data:

AlgorithmNn0=algorithmN(k1(D,n),tol)

(i)estimate an intervalaandb,such thatk1(D)lie betweenk1(D,a)andk1(D,b)

(ii)computek1(D,a)andk1(D,b)

while abs(a-b)＞tol

We observe that a key component in the above algorithm is to have an efficient and robust numerical method for computing the smallest real transmission eigenvalue.Next,we will develop efficient and accurate spectral methods for computing the transmission eigenvalues.

3 An efficient spectral method for transmission eigenvalues in circular domains

We consider in this section the domainDbeing a disk with radiusR,and assume that the index of reflectionnis stratified along the radial directionr,namely,n=n0(r).

3.1 Separation of variables

We start by employ a classical technique,separation of variables,to reduce the problem to a sequence of one-dimensional problems.

The equation(2.11)-(2.14)can be restated as

Applying the polar transformationx=rcosθ,y=rsinθto(3.1)-(3.4),and denoting

we obtain that for allθ∈[0,2π),

Substituting these expansions in(3.5)-(3.8),we obtain a sequence of one-dimensional problems for each Fourier modem.

·Casem=0:

Next,we show a simple result linking the transmission eigenvalues of the original problem(3.1)-(3.4)to that of the one-dimensional problems(3.9)-(3.11)or(3.12)-(3.14).

Proposition 3.1.LetD={(x,y):x2+y2＜R}.Then,

1.any transmission eigenvaluek,ofthe one-dimensionalproblems(3.9)-(3.11)or(3.12)-(3.14),is a transmission eigenvalue of the problem(3.1)-(3.4);

2.for any transmission eigenvaluekof the the problem(3.1)-(3.4),there exists at least onemsuch thatkis a transmission eigenvalue of(3.9)-(3.11)or(3.12)-(3.14).

3.2 Weak formulation

Then,(3.9)-(3.11)become

Hence,a weak formulation for(3.15)-(3.17)is: find?wem=wm0+h,vem=vm0+hwithwm0,vm0∈Xandh∈Xb,k∈C,such that?w?m0,v?m0∈Xandh∈Xb,we have

3.3 An efficient spectral-element method

In many applications,n0(r)is a piecewise smooth function.In order to deal with the piecewise smoothness,we shall use a spectral-element method.

LetIi=(ti-1,ti),1≤i≤Mwith-1=t0＜t1＜···＜tM=1.LetPNbe the set of polynomials of degree less than or equal toN,and define the spectral-element approximation toX:

To deal withXb,we define

We now construct a set of basis functions forXNwhich will consist of interior basis functions and interface basis functions.

Next,we define the following interface basis functions

Then,it is clear that

Hence,we look for

wherehMis defined in(3.21).

Let us denote

Then,the vector contains all the unknowns is

whereAandBare corresponding ”stiffness” and ”mass” matrices,which are sparse ifn0(r)is piecewise constant,and their non zero entries can be explicitly computed using the properties of Legendre polynomials.Hence,the above one-dimensional eigenvalue problem can be efficiently solved by using a standard procedure[10].

The system form=0,(3.12)-(3.14),can be treated similarly.

3.4 Error analysis

We establish below an error estimate for the transmission eigenvalues in terms of the errors for the corresponding eigenfunctions.In particular,we show that the convergence rate of the eigenvalue is twice of that of the eigenfunctions in the energy norm,as in the case of usual eigenvalue problems.

For any positive weight functionω,we denote the weightedL2-norm and weightedH1semi-norm by

In order to describe errors more precisely,we also define two related pseudo norms

Using a argument similar to the proof of Theorem 2.2 in[1],we can prove the following result:

Similar results can be derived for the casem=0.We omit the detail for brevity.

4 Rectangular domains

4.1 Weak formulation

LetX=H1(D).Writingw=w0+hb,v=v0+hbwithw0,v0∈H10(D)andhb∈Xb,whereXbis the complement ofH10(D)inX,namelyX=H10(D)⊕Xb.

4.2 Legendre-Galerkin method

LetXN=P2N,X0N=XN∩H10(D)andXbNbe the complement ofX0NinXN,namelyXN=X0N⊕XbN.Then,the Legendre-Galerkin approximation of(4.1)-(4.3)is:

We write

5 Numerical results and discussions

We now present the results of numerical experiments which exhibit the stability and accuracy of our new algorithm.

5.1 Approximation of transmission eigenvalues

5.1.1 Circular domain

We now perform a sequence of tests to study the convergence behavior of our algorithm in the case of circular domains.

In some applications,it is also useful to compute complex transmission eigenvalues.Our method is obviously not restricted to real eigenvalues.As an example,we list in Table 2 approximations of the first pair complex transmission eigenvalue.

Table 1:First real transmission eigenvalues corresponding to different m’s on a disk with R=1/2 and n=4.

Table 2:The first pair complex transmission eigenvalues corresponding to different m’s on a disk with R=1/2 and n=4.

Table 3:First real transmission eigenvalues corresponding to different m’s on a disk with n being a piecewise constant on two intervals.

Table 4:First real transmission eigenvalues corresponding to different m’s on a disk with n being a piecewise constant on four intervals.

We observe that in all cases,we can obtain 10-digit accuracy withN=10.Notice also that the smallest real eigenvalue(for allm)occurs atm=0 in all these cases.

5.1.2 Rectangular domain

We takeD=(-1/2,1/2)2andnbeing a constant.Approximations of the first real transmission eigenvalue for differentnare reported in Table 5.For comparison,the corresponding numerical results of[2]are listed in the last row.We observe that our method achieves much higher accuracy than those reported in[2].

Table 5:first real transmission eigenvalue for different n and D=[-1/2,1/2]2.

5.2 Estimation of index of refraction

LetDbe a given domain andAbe the matrix of the index of refraction,and letkapp1(D,A)be the approximate first real transmission eigenvalue estimated from the far field data.Our goal is to use the AlgorithmN in Section 2 to find a constantasuch that the isotropic matrixB=aIwill lead to the same first real eigenvalue,i.e.,k1(D,B)=kapp1(D,A).

Table 6:Numerical results for estimation of scalar a.

Table 7:Numerical results in[5]for estimation of scalar a.

We consider the following four differentA’s used in[5]:An isotropic case with

and three anisotropic cases

Note thatA2rcan be obtained by rotatingA2by 1 radian.Thus,A2andA2rhave the same eigenvalues.

From Theorem 2.1 and the continuity of eigenvalues,we expect thatashould lie between the upper and lower eigenvalues ofA.The numerical results are reported in Table 6.We observe that in the isotropic case,the predictedareconstructs the diagonal value of theAiso,while in the anisotropic case,the predictedalies between the eigenvalues of the matrixA.

As a comparison,the corresponding numerical results in[5]are listed in Table 7.We observe that our method leads to much more accurate results.

5.3 Summary

A key step in estimating the index of refraction is to compute efficiently and accurately the first real transmission eigenvalues of the associated transmission eigenvalue problems.We developed in this paper efficient spectral methods for computing the transmission eigenvalue problems,and derived,in the case of circular domain,an error estimate for the transmission eigenvalues in terms of the error for the corresponding eigenfunctions.We presented numerical results to show that our methods can efficiently compute very accurate approximations of transmission eigenvalues,and can be effectively used,along with the linear sampling method[12],to estimate the index of refraction for electromagnetic scattering problems.

Acknowledgments

This work is Supported in part by NSF of China grants 91130002 and 11371298.

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?Corresponding author.Email addresses:aj154@163.com(J.An),shen7@purdue.edu(J.Shen)