A computational method for the inverse transmission

Home

Search

Collections

Journals

About

Contact us

My IOPscience

A computational method for the inverse transmission eigenvalue problem

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 Inverse Problems 29 104010 (http://iopscience.iop.org/0266-5611/29/10/104010) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 123.13.205.185 This content was downloaded on 15/10/2013 at 11:52

Please note that terms and conditions apply.

IOP PUBLISHING

INVERSE PROBLEMS

doi:10.1088/0266-5611/29/10/104010

Inverse Problems 29 (2013) 104010 (14pp)

A computational method for the inverse transmission eigenvalue problem Drossos Gintides and Nikolaos Pallikarakis Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece E-mail: [email protected] and [email protected]

Received 8 March 2013, in final form 22 July 2013 Published 17 September 2013 Online at stacks.iop.org/IP/29/104010 Abstract In this work, we consider the inverse transmission eigenvalue problem to determine the refractive index from transmission eigenvalues. We adopt a weak formulation of the problem and provide a Galerkin scheme in H02 (D) to compute transmission eigenvalues. Using a proper operator representation of the problem, we show convergence of the method. Next, we define the inverse transmission problem and show that numerically the problem can be considered as a discrete inverse quadratic eigenvalue problem. First, we investigate the case of a spherically symmetric piecewise constant refractive index and confirm our results with analytic computations. Then, we show that a relative small number of eigenvalues are sufficient for simple cases of a few layers by just minimizing the total error between measured and computed eigenvalues to reconstruct the refractive index. Finally, we propose a computational method based on a Newton-type algorithm for reconstructions of a general piecewise constant refractive index for any domain from transmission eigenvalues. The algorithm can be performed without having knowledge of the exact position of the eigenvalues in the spectrum. (Some figures may appear in colour only in the online journal)

1. Introduction The interior transmission problem is a special eigenvalue problem that appears in cases where a continuation of an exterior field in the interior domain where inhomogeneities are present is not possible uniquely. Applications of this problem appear in inverse scattering problems for inhomogeneous acoustic, electromagnetic or elastic media when sampling type methods are used [12]. We assume that the domain D is a bounded and simply connected set in Rn with smooth boundary and ν is the outward unit normal to ∂D. The interior transmission problem is a boundary value problem of finding two fields w and v such that w + k2 n(x)w = 0,

x ∈ D,

0266-5611/13/104010+14$33.00 © 2013 IOP Publishing Ltd

(1.1) Printed in the UK & the USA

1

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

v + k2 v = 0, w = v,

x ∈ D,

(1.2)

x ∈ ∂D,

(1.3)

∂w ∂v = , x ∈ ∂D. (1.4) ∂ν ∂ν The complex values of k for which the problem has a nontrivial solution (w, v) are the transmission eigenvalues. It is well known that the real eigenvalues can be determined from far-field measurements (see e.g. [4, 5, 14, 15, 17, 27–29, 31]). Existence of transmission eigenvalues was an open problem for many years since there is no standard theory available for non-self-adjoint eigenvalue problems. For variable index of refraction, a proof that there exists a countably infinite set of real and positive transmission eigenvalues was recently given in [11]. From analytic Fredholm theory, it was shown that transmission eigenvalues form at most a discrete set with infinity as the only possible accumulation point. The proof applies to inhomogeneous, isotropic and anisotropic media for both Helmholtz and Maxwell’s equations including the case of media with cavities. The only assumption is that the index of refraction is less than or greater than the index of refraction of the background medium (see also [7, 10, 13, 24, 30]). The most interesting question related to problem (1.1)–(1.4) is what can we say about n(x), if the transmission eigenvalues are known. It is observed that these eigenvalues carry information about the index of refraction (see for example [3, 6]). In the case where D is the ball of radius b > 0 and n(x) is radially symmetric, it has been recently established [7] that the transmission eigenvalues determine n(x) uniquely. A rather natural question is whether n(x) can be determined only from the knowledge of a subset of the spectrum. One such set is the set of all transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. This special inverse spectral problem has been studied in [27–29] and recently in [1] where some uniqueness results were established. The assumptions for n(x) are n ∈ C1 [0, b],

n ∈ L2 [0, b];

n(x) > 0,

x ∈ [0, b].

(1.5)

We would like to mention here that for domains with arbitrary geometry and refractive index, the inverse spectral transmission problem is not faced yet. There is an extensive literature concerning methods for solving inverse spectral problems, for example, in vibration of discrete systems and Sturm Liouville problems [20]. Most of the problems in this category are based on self-adjoint formulation of the problems. The purpose of this paper is to solve computationally the inverse transmission problem. In our consideration, we include both real and complex transmission eigenvalues. We mention that complex eigenvalues have been only recently investigated [8, 26] and cannot be detected using far-field measurements or other methods up to now. Our work aims to provide numerical evidence that both real and complex eigenvalues carry information about the refractive index. We show that the first eigenvalues contain information about the refractive index. In simple cases, that is with a few layers of piecewise constant refractive index, we can have exact reconstructions. First, we investigate the discrete version of the direct problem using the standard variational formulation for fourth-order problems. We introduce a Galerkin-type numerical method to approximate the transmission eigenvalues. Then, we investigate the case of a spherically symmetric piecewise constant refractive index and confirm our results analytically from separation of variables. For a small number of eigenvalues and simple cases, we just minimize the total error between computed and given eigenvalues. This approach needs the exact position of the eigenvalues, measured and computed, in the spectrum. To overcome this difficulty, we introduce a computational method based on a Newton-type algorithm. The 2

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

algorithm is tested for spherically symmetric piecewise constant refractive index with a few layers. In all examples, the case of having complex eigenvalues is also considered in the numerical inversions. Finally, we present the form of the algorithm for general domains for a piecewise constant refractive index. 2. Discrete transmission eigenvalue problem 2.1. Operator representation Let D be a bounded and simply connected subregion of R2 with n ∈ L∞ (D). We assume that n(x) and 1/(n(x) − 1) > 0 are positive real-valued functions defined in D. The transmission eigenvalue problem can be transformed into a fourth-order equation for u := w −v ∈ H02 (D) : 1 ( + k2 )u = 0, in D (2.1) ( + k2 n) n−1 together with the homogeneous boundary conditions ∂u = 0, in ∂D, (2.2) u=0 and ∂ν where ν denotes the outward normal vector to ∂D. The results of this section hold true for the same equations in R3 . In the variational form, problem (2.1)–(2.2) is equivalent to finding a function u ∈ H02 (D) such that  1 (u + k2 u)(φ + k2 nφ) dx = 0, ∀ φ ∈ H02 (D). (2.3) n − 1 D We define the sesquilinear forms in H02 (D) × H02 (D):   1 u, φ a(u, φ) := n−1 D     n 1 u, φ u, φ a1 (u, φ) := − − n−1 n−1 D D   n u, φ a2 (u, φ) := n−1 D ∀u, φ ∈ H02 (D), where (·, ·)D denotes the L2 (D) inner product. For n(x) − 1  δ > 0, a(u, φ) is coercive and a2 (u, φ) is non-negative. Using the Riesz representation theorem, we define the following bounded linear operators T : H02 (D) → H02 (D), T1 : H02 (D) → H02 (D), T2 : H02 (D) → H02 (D) by a(u, φ) = (Tu, φ)H 2 (D) a1 (u, φ) = (T1 u, φ)H 2 (D) a2 (u, φ) = (T2 u, φ)H 2 (D) . Setting k2 := τ , we can write (2.3) as a quadratic pencil operator problem [12]: Tu − τ T1 u + τ 2 T2 u = 0.

(2.4)

By definition, T is positive definite and self-adjoint. Also, as a consequence of the compact embedding of H02 (D) in L2 (D), we have that T1 and T2 are compact operators. Moreover, T1 and T2 are self-adjoint. Since T is coercive, we have that T −1 is bounded, and the transmission eigenvalue problem can be written equivalently in the following form: u − τ K1 u + τ 2 K2 u = 0,

(2.5) 3

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

where K1 and K2 are self-adjoint, compact operators and K2 is non-negative, defined as [12]: K1 = T −1/2 T1 T −1/2

and

K2 = T −1/2 T2 T −1/2 .

(2.6)

We use for simplicity in (2.5) the symbol u instead of T 1/2 u following similar notation in the literature. Note that if U is a separable Hilbert space and A is a bounded, positive definite ∞ and self-adjoint operator on U, we define the operators A±1/2 := 0 λ±1/2 dEλ , where dEλ is ±1/2 are also bounded, positive the spectral measure associated with A. It is well known that A definite and self-adjoint operators on U, A−1/2 A1/2 = I and A1/2 A1/2 = A. Now, the transmission problem can be written as a non-self-adjoint linear eigenvalue problem   1 ξ := (K − ξ I)U = 0, U = u, τ K21/2 u , (2.7) τ for the non-self-adjoint compact operator K : H02 (D) × H02 (D) → H02 (D) × H02 (D), given by   −K21/2 K1 . K := K21/2 0 This form is completely equivalent to the quadratic form (2.5). From here, one can see that the set of transmission eigenvalues is at most discrete with +∞ as the only one accumulation point. 2.2. Discrete representation for the direct eigenvalue problem We now present the numerical method to solve the interior transmission problem. Some other numerical methods for the computation of transmission eigenvalues have been considered in [16, 23, 32]. These works are based on mixed finite element methods and iterative methods. We adopted a simple Galerkin-type method, based on the weak formulation of the problem. The main difficulties of our approximation method are that the problem is non-self-adjoint and that we have to use Sobolev spaces for the discrete representation of the problem. We assume that {φi }∞ i=1 is a set of eigenfunctions of the problem Lφi = μi φi

in D

(2.8)

∂φi =0 on ∂D, (2.9) ∂ν where L is a fourth-order elliptic operator. We can use any elliptic operator L. In our work, we adopt the bilaplacian operator for which the eigenpairs can be easily computed and the eigenfunctions form a Hilbert basis in H02 (D). For fourth-order eigenvalue problems, there are many approximation methods, e.g., the Weinstein–Aronjszajn method [22], based on a similar simple Hilbert basis. Assume now that {φi }∞ i=1 is such a set and any transmission eigenfunction uk can be expanded in this system as φi = 0,

uk =

∞ 

ci φi

(2.10)

i=1

which is a convergent series in H02 (D). We approximate uk by uk(N )

=

(N )  i=1

4

ci φi .

(2.11)

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

We enter uk(N ) in (2.3) and we use as test functions the eigenfunctions φi , i = 1, . . . , N. So, the approximate nonlinear eigenvalue problem is written in matrix form as [A(N ) − (k (N ) )2 B(N ) + (k (N ) )4C (N ) ] c = 0, where A

(N )

 := D

B

(N )

1 φi φ j dx n(x) − 1

 := − D

C

(N )

 := D

(2.12)

n(x) φi φ j dx + n(x) − 1

 D

1 φi φ j dx n(x) − 1

 (2.13)

n(x) φi φ j dx n(x) − 1

are N × N matrices and c = (c1 , c2 , . . . , cN ) , i, j = 1, . . . , N. Equation (2.12) is a typical quadratic eigenvalue problem [34] and it is completely analogous to operator eigenvalue problem (2.4). More precisely A(N ) , B(N ) and C (N ) are selfadjoint and A(N ) and C (N ) are positive definite. These properties of the matrices follow directly from the properties of the corresponding operators in (2.4). Now, we can transform (2.12) into a linear eigenvalue problem for a block matrix: (K(N ) − ξ (N ) I(N ) )U (N ) = 0,

(2.14)

where

 1/2  (N )  1 U (N ) = c, τ (N ) K2(N ) c , ξ := (N ) , τ the block matrix is defined as 1/2

 K1(N ) − K2(N ) (N ) K :=  (N ) 1/2 0 K2

(2.15)

and K1(N ) := (A(N ) )−1/2 B(N ) (A(N ) )−1/2

(2.16)

K2(N ) := (A(N ) )−1/2C (N ) (A(N ) )−1/2 .

(2.17)

We underline that any positive semi-definite matrix has a unique positive semi-definite square root [25]. In the following, we study the existence and convergence of the eigenvalues of the discrete problem. The following proposition shows the existence of eigenvalues of (2.12). Proposition 1. Assume n(x) > 1 for x ∈ D. Then there exist 2N eigenvalues of problem (2.12). Proof. This result comes directly from linearized problem (2.14), which is a standard eigenvalue problem for a square 2N × 2N matrix. Thus, there exist 2N eigenvalues of (2.14).  In order to prove that the eigenvalues of the discrete problem converge to the corresponding eigenvalues of the original problem, we have to prove that τ (N ) →τ (or equivalently ξ (N ) → ξ ). But, the main difficulty of our problem is that it is non-self-adjoint and we cannot apply convergence results for compact and self-adjoint eigenvalue problems. To avoid this difficulty, we use some abstract results for convergence in Banach spaces, [2, 18] where the main tools are based on compactness arguments and convergence behavior of isolated eigenvalues. 5

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

Let X be a complex Banach space with || || norm and {Xn } be a sequence of finitedimensional subspaces of X parameterized by n, which will be identified with the dimension. We introduce the following framework [18]: n : X → X are linear projectors with range Xn , A : X → X is a linear bounded operator, the linear operators Bn : X → X, with range in Xn , are supposed to approximate A. An is then defined as the restriction of Bn to Xn . Bn will be called the Galerkin approximation of A if Bn = n A and then An := n A|Xn : Xn → Xn . Following [2], we introduce the necessary framework for convergence of compact operators. Let H be a complex Banach space and K : H → H a linear compact operator. We assume that μ is a nonzero eigenvalue of K with algebraic multiplicity m and that K(N ) : H → H is a sequence of linear operators which converge in K in norm as N → ∞. Let  be a circle in the complex plane centered at μ which lies in the resolvent set ρ(K) and no other eigenvalues of K are contained in . Then, it is proved in [2] that for N sufficiently large,  ⊂ ρ(K(N ) ) and counting according to algebraic multiplicity, there are m eigenvalues of K(N ) in . If we denote these eigenvalues by μ1,N , μ2,N , . . . , μm,N and if   is another circle centered at μ with an arbitrarily small radius, we have that μ1,N , μ2,N , . . . , μm,N are all contained in   for N sufficiently large. So, we see that limN→∞ μ j,N = μ f or j = 1, . . . , m. Now, we can prove the following proposition, which gives us the desired convergence result. Proposition 2. Eigenvalues of linear matrix problem (2.14) converge to the corresponding eigenvalues of operator problem (2.7) for N → ∞. Proof. We define as X := H02 (D), as XN := H02 (D),N an N-dimensional subspace of H02 (D) and the linear orthogonal projectors N : H02 (D) → H02 (D) with range H02 (D),N . We introduce the linear bounded operators: T (N ) := N T |XN : XN → XN T1(N ) := N T1 |XN : XN → XN T2(N ) := N T2 |XN : XN → XN . The matrices given by (2.13) expressing our Galerkin approximation correspond to the linear operators defined above. Also, we define K1(N ) := (T (N ) )−1/2 T1(N ) (T (N ) )−1/2 ,

(2.18)

K2(N ) := (T (N ) )−1/2 T2(N ) (T (N ) )−1/2 with corresponding matrices defined in (2.16) and (2.17). Let H := K : H → H is compact. Moreover, we set 1/2

 K1(N ) − K2(N ) (N ) K :=  (N ) 1/2 K2 0

(2.19) H02 (D)

×

H02 (D)

then

with corresponding block matrix defined in (2.15). K(N ) represents a sequence of operators which approximate operator K and have the same eigenvalues with the block matrices (2.15). Therefore, we can prove that the eigenvalues of K(N ) converge to those of K if we infer that K(N ) → K in norm as N → ∞. From the definition of the operator norm for a block operator we have         K − K(N )   K1 − K (N )  + 2K 1/2 − K (N ) 1/2 . 1 2 2 6

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

From the form of K1 and K1(N ) as products of operators given by (2.6) and (2.18), respectively, we have     K1 − K (N )  = T −1/2 T1 T −1/2 − (T (N ) )−1/2 T (N ) (T (N ) )−1/2  1

1

−1/2

(N ) −1/2

−1/2

 ||(T − (T ) )T1 T    +T1 − T1(N ) ||(T (N ) )−1/2 ||2 .

|| + ||(T (N ) )−1/2 ||||T1 (T −1/2 − (T (N ) )−1/2 )||

The sequence (T (N ) )−1/2 is uniformly bounded and pointwise convergent, T1 is compact and self-adjoint and T −1/2 is self-adjoint. From the result that multiplying by a compact operator on the right converts a pointwise convergent sequence of bounded operators into a norm convergent one [21, p 108], we infer that ||(T −1/2 − (T (N ) )−1/2 )T1 T −1/2 || → 0 and ||(T (N ) )−1/2 ||||T1 (T −1/2 − (T (N ) )−1/2 )|| = ||(T (N ) )−1/2 ||||(T −1/2 − (T (N ) )−1/2 )T1 || → 0. Since T1 is compact and (T (N ) )−1/2 is uniformly bounded, we deduce directly that ||(T1 − T1(N ) )||||(T (N ) )−1/2 ||2 → 0. So, we have ||K1 − K1(N ) || → 0. Using the same arguments for the operators K21/2 and (K2(N ) )1/2 , having similar structure to K1 and K1(N ) , respectively, we deduce that ||K21/2 − (K2(N ) )1/2 || → 0. So, the desired convergence result follows.  3. Numerical results for the discrete inverse eigenvalue problem—a computational method We now present numerical results for the discrete inverse eigenvalue problem. The direct quadratic eigenvalue problem is given n(x) to determine eigenvalues and eigenvectors. The inverse quadratic eigenvalue problem is given spectral data to determine n(x). We use the Galerkin method, as described above, and pose the inverse quadratic eigenvalue problem for piecewise constant index of refraction n(x). We first investigate the case of a spherically symmetric domain D. We also propose a Newton scheme for the general piecewise constant index problem. In all cases, since there does not exist a general uniqueness theory for the inverse spectral problem, we assume that we have situations where all the original and known transmission eigenvalues are well separated and the computed eigenvalues are very close to the corresponding original eigenvalues. We mention that in the next subsections we examine examples of problems where this assumption has also been checked analytically. 3.1. Circular domain Let D be a circular domain of radius r with constant refractive index n(x) = n. It is well known that we can recover n from the knowledge of only the first transmission eigenvalue, [7]. Theorem 3. The constant index of refraction n is uniquely determined from knowledge of the corresponding smallest transmission eigenvalue k1,n > 0 provided that it is known a priori that either n > 1 or 0 < n < 1. We use this result to reconstruct the refractive index for unit disc. We can analytically compute the eigenvalues and eigenfunctions of the clamped plate and the transmission eigenvalues. The first transmission eigenvalue is the lowest positive value for which [14, 17]  √  J (kr) Jm (k√nr) = 0, m = 0, 1, . . . (3.1) det m Jm (kr) Jm (k nr) where Jm are the Bessel functions of the first kind. This relation can be derived from separation of variables for Helmholtz equations (1.1) and (1.2). Using root-finding software, we compute the lowest real transmission eigenvalues k0 with refractive index in the range 2  n  20. 7

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

Table 1. Reconstructions for constant index of refraction.

Original n

First eigenvalue k0

Error |k0 − k0(N) |

Estimated n

3 6 10 12 20

4.159 24 1.849 72 1.296 30 1.166 12 0.881 54

0.0046 8.66 × 10−4 1.21 × 10−4 4.07 × 10−5 8.24 × 10−5

3.03 6.01 10.01 12.01 20.01

(N ) Now, we construct the basis {φi }i=1 from problem (2.8)–(2.9) with L =  . In polar coordinates, the eigenfunctions ui for one μ are linear combinations of Ji (μr) cos iθ, Ji (μr) sin iθ , Ii (μr) cos iθ, Ii (μr) sin iθ, where Ii are hyperbolic Bessel functions. The eigenvalues can be computed from the relation:    Ji (kr) Ji (kr) det = 0, i = 0, 1, . . . (3.2)  Ii (kr) Ii (kr) We constructed a basis with 12 eigenfunctions, {φi }12 i=1 and computed the 12 × 12 matrices (N ) A , B(N ) and C (N ) for r = 1. We used the Matlab function polyeig with step 0.01 to solve direct problem (2.12) for different values of n in the interval [2, 20], and the results are shown in table 1. Using only the lowest transmission eigenvalue and a few eigenfunctions of the clamped plate, we can estimate n very well by minimizing |k0(N ) − k0 | by simply considering k0(N ) = k0(N ) (n). Some plots of the error |k0(N ) − k0 | versus the index n are shown in figure 1. We see that the error is minimized for estimated index very close to the original one which corresponds to the analytically known first transmission eigenvalue. We also tested the method adding a small error to the transmission eigenvalue and the reconstructions were also accurate.

3.2. Spherically stratified domain We choose D to be a disc of radius r = R, but we now assume that the refractive index is a piecewise constant function. As a first simple case, we use two layers: n , x ∈ D1 n(x) = 1 n2 , x ∈ D2 . We used the first four transmission eigenvalues for the reconstructions. The input transmission eigenvalues were analytically computed from the equation: ⎛ ⎞ √ √ Jm (kR) 0 Jm (k n2 R) Nm (k n2 R) √ √   ⎜J  (kr)|r=R 0 Jm (k n2 r)|r=R Nm (k n2 r)|r=R ⎟ m ⎟=0 √ √ √ det ⎜ (3.3) ⎝ 0 Jm (k n1 r1 ) Jm (k n2 r1 ) Nm (k n2 r1 ) ⎠ √ √ √    0 Jm (k n1 r)|r=r1 Jm (k n2 r)|r=r1 Nm (k n2 r)|r=r1 for m = 0, 1, . . ., where Nm are Neumann functions. This relation is completely analogous with (3.1). The eigenfunctions of Helmholtz equations (1.1) and (1.2) are linear combinations of Bessel and Neumann functions since the domain is spherically stratified. Using the rootfinding software, we computed both real and complex transmission eigenvalues for R = 1, inner radius r1 in the range 0.5  r1  1 and refractive indices in the range 2  n1 , n2  13. The main idea for solving the inverse problem was the same as for the constant index case. We assumed that in all cases we know the position of each eigenvalue in the spectrum and we minimized the error li=1 |ki(N ) − ki | considering that the computed eigenvalues are functions of (n1 , n2 , r1 ). We examined two different problems: first, having the knowledge of the lowest four real transmission eigenvalues and afterwards using both real and complex transmission eigenvalues. 8

3.5

3.5

3

3

2.5

2.5

2

1.5

2

1.5

1

1

0.5

0.5

0 0

5

10 15 refractive index n

20

0 0

25

4.5

4.5

4

4

3.5

3.5

3

3 error |k0−kN | 0

error |k0−kN | 0

D Gintides and N Pallikarakis

error |k0−kN | 0

error |k0−kN | 0

Inverse Problems 29 (2013) 104010

2.5 2 1.5

10 15 refractive index n

20

25

2.5 2 1.5

1

1

0.5

0.5

0

5

0

5

10 15 refractive index n

20

25

0

0

5

10 15 refractive index n

20

25

Figure 1. Plots of the |k0 − k0(N) | versus n for original n = 3, n = 6, n = 12 and n = 20, respectively.

3.2.1. Reconstructions using only real transmission eigenvalues. We have reconstructions using as test data the first four real transmission eigenvalues, for example, in the range 0.5  r1  1 and 5  n1 , n2  13. In this case, the refractive index is not close enough to n(x) = 1. From numerical computations, we have that complex eigenvalues with small modulus do not appear and consequently we can use as first eigenvalues only real eigenvalues. We used the same basis {φi }12 i=1 as in constant index case and we computed the A(N ) , B(N ) and C (N ) matrices for 0.5  r1  1 with step 0.1. We solved the direct problem using Matlab function polyeig for 5  n1 , n2  13 with step 0.1. Minimizing the error  l (N ) 2 i=1 |ki − ki | for l = 4, we reconstruct n(x) with relatively good accuracy, as we can see in table 2. We mention here that we used the interior radius as an unknown parameter. So, the method can cover cases where the exact size of the layer is not known. There are cases where the reconstruction of the unknown index is not successful. For example, the index (13, 5, 0.9) is reconstructed as (11.4, 11.7, 0.5). This problem is circumvented if we use more transmission eigenvalues. Indeed, if we use as test data the first five transmission eigenvalues instead of four, the index is reconstructed as (13, 7, 0.9). Nevertheless, the numerical method we presented above can be useful in many applications such as non destructive testing of materials because it can completely estimate the unknown index using only a few transmission eigenvalues. 3.2.2. Reconstructions using both real and complex transmission eigenvalues. In the case when n(x) is closer to 1, more complex transmission eigenvalues arise [8]. Note that since 9

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

Table 2. Reconstructions for piecewise constant index of refraction using only real transmission eigenvalues.

(n1 , n2 , r1 )

Analytic t.e. {k0 , k1 , k2 , k3 }

Approximate {k˜0 , k˜1 , k˜2 , k˜3 }

(n˜1 , n˜2 , r˜1 )

13 5 0.5 5 8 0.6 10 8 0.7 13 11 0.8 6 13 0.9

1.4957, 1.7336, 2.1694, 2.6974 1.7889, 2.2483, 2.6654, 3.0329 1.3716, 1.7304, 2.1086, 2.4934 1.1487, 1.4884, 1.8248, 2.1547 1.5885, 2.0239, 2.5120, 3.0009

1.4976, 1.7893, 2.1860, 2.6434 1.7681, 2.2483, 2.6769, 3.0431 1.3720, 1.7314, 2.1105, 2.4968 1.1456, 1.4888, 1.8266, 2.1584 1.5833, 2.0320, 2.5183, 2.9944

10.7 5.8 0.6 5.0 8.4 0.6 10.1 7.9 0.7 13.0 11.2 0.8 6.3 9.6 0.8

Table 3. Reconstructions for piecewise constant index of refraction using both real and complex transmission eigenvalues.

(n1 , n2 , r1 )

Analytic t.e. {k0 , k1 , k2 , k3 }

Approximate {k˜0 , k˜1 , k˜2 , k˜3 }

(n˜1 , n˜2 , r˜1 )

2 4 0.5

2.4317+0.6964i, 3.5768+0.5709i, 3.9082, 4.1101 2.2966+0.7428i, 2.8845, 3.1834, 3.2932 2.4257+0.6292i, 3.7545+0.6136i, 4.9642+0.4386i, 5.0402 2.2202+0.4778i, 3.0247, 3.6685, 3.7963 2.0212, 2.4177, 2.7382+0.4470i, 2.9295

2.4254+0.6665i, 3.5084+0.5890i, 3.9795, 4.0556 2.2828+0.7410i, 2.8600, 3.1933, 3.2845 2.3811+0.6396i, 3.6552+0.5786i, 4.8170+0.3986i, 5.0275 2.2060+0.5009i, 3.0621, 3.6779, 3.8051 2.0209, 2.4362, 2.7238+0.4662i, 2.9351

2.0 4.5 0.5

5 3 0.6 2 4 0.7 3 6 0.8 6 2 0.9

5.1 3.0 0.6 2.1 4.0 0.7 3.0 6.0 0.8 6.0 2.9 0.9

n(x) has to be real, the complex transmission eigenvalues must appear in complex conjugate pairs. Using the root-finding software and contour plots, we compute the complex transmission eigenvalues from the equations derived from the separation of variables (3.3). for 2  n1 , n2  6 with We used the same basis {φi }12 i=1 and solved the inverse problem l (N ) 2 step 0.1 and 0.5  r1  1 with step 0.1. Minimizing the error i=1 (|Re ki − Re ki | + (N ) 2 |Im ki − Im ki | ), we reconstruct the index n(x). As we can see in table 3, we have reconstructions of n(x) with good accuracy in the case which the index is close to 1. Note that in this case if we use only real transmission eigenvalues, the reconstructions are not satisfying. That is the reason why we treated this case individually. This result is important because from this numerical method, we can see that both real and complex eigenvalues carry information about the refractive index. 3.3. The inversion problem for a general spherically stratified domain We now propose an algorithm for estimating the unknown index of refraction for domains with more than two layers. The main advantage of this method, beside the fact that it can be used for more general domains, is that we do not have to pair the analytically known eigenvalues with the numerically estimated correctly according to their position on the spectrum in the complex plane. We suppose that D is a spherically stratified domain with k-layers such that D = ∪ki=1 Di and {∂Di }ki=1 are concentric circles. We also suppose that the exact geometry of the subdomains is known. The unknown piecewise constant index of refraction n(x) is given by ⎧ ⎪ ⎨n1 , x ∈ D1 . n(x) = ... ⎪ ⎩ nk , x ∈ Dk 10

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

We assume again that n(x) > 1. The transmission eigenvalues are the zeros of the determinant of a 2k × 2k matrix, analogous to (3.3), which can be obtained using separation of variables. We can solve corresponding inverse quadratic eigenvalue problem (2.12) using a Newton method [19]. Some previous works concerning the reconstruction of the refractive index with optimization methods using the lowest eigenvalue are considered in [9, 33]. We can write the N × N matrices A(N ) , B(N ) and C (N ) in the following form:

B(N )

k 

1 Al n −1 l=1 l  k   nl 1 (1) (2) B + B = nl − 1 l nl − 1 l l=1

A(N ) =

(3.4)

(3.5)

k 

nl Cl , (3.6) n −1 l=1 l     where Al = Dl φi φ j dx, Bl(1) = Dl φi φ j dx, Bl(2) = Dl φi φ j dx and Cl = Dl φi φ j dx, for i, j = 1, . . . , N. From the analysis of the previous paragraphs, we have that {Al }kl=1 , {(Bl(1) + Bl(2) )}kl=1 , {Cl }kl=1 are symmetric and {Al }kl=1 are positive definite. Also if we set al := 1/(nl −1) then (3.3)–(3.6) can be written as C

(N )

=

A(N ) =

k 

al Al

(3.7)

l=1

B(N ) =

k 

Bl(1) +

l=1

C (N ) =

k 

k    al Bl(1) + Bl(2)

(3.8)

l=1

Cl +

l=1

k 

al Cl .

(3.9)

l=1

Now, the inverse problem has the following form: given a set of transmission eigenvalues S = {μi }ki=1 , find scalars {al }kl=1 which are such that the pencil P(λ) = λ4C (N ) + λ2 B(N ) + A(N ) has spectrum σ (A(N ) , B(N ) , C (N ) ) = S. We used a modification of an algorithm designed for higher degree matrix polynomial inverse eigenvalue problems in [19], adapted for the special case of our quadratic problem. We denote a = (a1 , a2 , . . . , ak ) the set of the unknown coefficients. The main idea of this Newton-based iterative method is solving the nonlinear system f (a) := ( f1 (a), . . . , fk (a)) = (0, . . . , 0) , where

 k

k  k k k    (1)     Cl + al Cl + μ2i Bl + al Bl(1) + Bl(2) + al Al fi (a) = det μ4i l=1

l=1

l=1

l=1

l=1

rather than minimizing a cost functional like g(a) =

k 

|λi (a) − μi |

i=1

which was our first approach for the problem. With the previous method, we had to pair λi (a) with μi correctly in each iterative step. With the new method, this problem is circumvented. This result is crucial in applications because we can estimate the unknown n(x) using eigenvalues for which we do not know the exact position in the spectrum of the transmission problem. This could happen if our eigenvalues were derived from scattering data in a specific interval of wavenumbers. 11

Inverse Problems 29 (2013) 104010

3.3.1. The algorithm. Input

D Gintides and N Pallikarakis

The Newton method

• the set of {Al }kl=1 , {Bl }kl=1 , {Cl }kl=1 of N × N matrices. • an initial estimate of a(0) = (a1(0) , a2(0) , . . . , ak(0) ) of the unknown {al }kl=1 set. • the set S = {μi }ki=1 of transmission eigenvalues. Output at convergence, the method produces a vector of the unknown {al } which are such σ (A(N ) , B(N ) , C (N ) ) = S. The iteration (1) choose a starting value a(0) for the vector of the unknown coefficients (2) for m = 0, 1, . . . (a) compute the Jacobian J(a(m) ) and the function f (a(m) ) by the algorithm below (b) solve the system: J(a(m) )ξ (m) = − f (a(m) ) for ξ (m) (c) compute the new estimate of the coefficient vector a(m+1) = ξ (m) + a(m) (d) stop if ||ξ (m) || is sufficiently small end loop (2) (m-loop). Computing the Jacobian Input • the set of {Al }kl=1 , {Bl }kl=1 , {Cl }kl=1 of N × N matrices. • the k-vector of a(m) of the coefficients resulting at the mth iteration step of the Newton method. • the set S = {μi }ki=1 of transmission eigenvalues. Output • the function f (a(m) ) • the Jacobian matrix J(a(m) ) with ilth component ∂ fi (a(m) )/∂al . The algorithm (1) for each i = 1, 2, . . . , k

 k   k k (1) 2 + (a) compute the N × N matrix H = μ4i l=1 Cl + l=1 al Cl + μi l=1 Bl  (1)   k k (2) B + a + B a A l=1 l l=1 l l l l (b) use LU or QR factorization to triangularize H and then compute f (a(m) ) as the product of the diagonal elements of the triangle (c) for each l = 1, 2, . . . , k 1. compute the matrix:  2  k  (1) k N × N  k (1) k (2)  + C + D = μ4i j=1 j j=1, j=l a jC j +μi j=1 B j + j=1, j=l a j B j +B j k j=1, j=l a j A j 2. use the QZ (generalized Schur decomposition) algorithm to find matrices Q and R with determinant unity which simultaneously triangularize the pair  Al + μ2i Bl(1) + Bl(2) + μ4i Cl , D 3. denote as (αi , βi )Ni=1 the pairs of the diagonal elements of the QZ triangular matrices 4. determine the number of nonzero αi 5. relabel (αi , βi ) so that αr+2 =  · · · = αN = 0   N αr+1 = r N (m) 6. set [J(a(m) )]il = i=r+1 βi i=1 αi j=1, j=i al α j + β j end loop (c) (l-loop) end loop (1) (i-loop).

12

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

Remark 4. Note that this algorithm requires the same number k of known transmission eigenvalues and layers. In the case where the number of layers is less than the number of transmission eigenvalues (which is possible because we probably need many transmission eigenvalues for k > 2 layers), the Jacobian matrix of step (c)-6 is not square and then the system (2)-b can be solved using the generalized inverse of the Jacobian matrix, or using a Tikhonov-type inversion method. We have tested the algorithm for the simple case of spherically stratified domain with two layers. We used a set of 8 × 8 square matrices A, B and C in the form (3.7), (3.8) and (3.9), respectively. The set S of target eigenvalues was computed numerically from the pencil P(λ) = λ4C + λ2 B + A. The case of unit disc with n1 = 5, n2 = 8 and inner radius r1 = 0.6, given as initial estimate for the indices, the mean value 6.5 was reconstructed as n1 = 4.999, n2 = 8.000 after 5 steps with tolerance 10−12 . The case of unit disc with n1 = 12, n2 = 6 and inner radius r1 = 0.8, given as initial estimate for the indices, the mean value 9 was reconstructed as n1 = 11, 999, n2 = 6.000 after 7 steps with tolerance 10−12 . In both examples, we used real and complex eigenvalues. We also tested the algorithm adding a small error in the eigenvalues with order of magnitude 10−3 and the results were also satisfying. 3.3.2. Generalization of the method for piecewise constant index. We can apply the previous method for any bounded and simply connected domain D ⊂ R2 , D = ∪m i=1 Di , Di ∩ D j = ∅, i = j, where the index of refraction is piecewise constant: ⎧ ⎪ ⎨n1 , x ∈ D1 . n(x) = ... ⎪ ⎩ nm , x ∈ Dm The final scheme and the algorithm are the same, the only difference is that the integrals in basic matrices given in (3.4)–(3.6) are defined on the corresponding domains D j . An error and stability analysis of the numerical methods presented here is a worthwhile future project. Also, the numerical implementation of the Newton method for the piecewise constant index in general domains is under investigation. Acknowledgments The authors would like to thank the referees for their fruitful remarks and suggestions. This work is supported by the PEBE 2010 grant from the National Technical University of Athens. References [1] Aktosun T, Gintides D and Papanikolaou V G 2011 The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation Inverse Problems 27 115004 [2] Babuˇska I and Osborn J 1991 Eigenvalue problems Handbook of Numerical Analysis vol 2 (Amsterdam: Elsevier) pp 641–787 [3] Cakoni F, Cayoren M and Colton D 2008 Transmission eigenvalues and the nondestructive testing of dielectrics Inverse Problems 24 065016 [4] Cakoni F, Colton D and Haddar H 2009 The computation of lower bounds for the norm of the index of refraction in an anisotropic media J. Integral Eqns Appl. 21 203–27 [5] Cakoni F, Colton D and Haddar H 2010 On the determination of Dirichlet and transmission eigenvalues from far field data C. R. Math. 348 379–83 [6] Cakoni F, Colton D and Monk P 2007 On the use of transmission eigenvalues to estimate the index of refraction from far field data Inverse Problems 23 507–22 13

Inverse Problems 29 (2013) 104010

D Gintides and N Pallikarakis

[7] Cakoni F, Colton D and Gintides D 2010 The transmission eigenvalue problem Advanced Topics in Scattering and Biomedical Engineering: Proc. 9th Int. Workshop on Mathematical Methods in Scattering Theory and Biomedical Engineering ed A Charalambopoulos, D I Fotiadis and D Polyzos (Singapore: World Scientific) pp 368–80 [8] Cakoni F, Colton D and Gintides D 2010 The interior transmission eigenvalue problem SIAM J. Math. Anal. 42 2912–21 [9] Cakoni F, Colton D, Monk P and Sun J 2010 The inverse electromagnetic scattering problem for anisotropic media Inverse Problems 26 074004 [10] Cakoni F and Gintides D 2010 New results on transmission eigenvalues Inverse Problems Imaging 4 39–48 [11] Cakoni F, Gintides D and Haddar H 2010 The existence of an infinite discrete set of transmission eigenvalues SIAM J. Math. Anal. 42 237–55 [12] Cakoni F and Haddar H 2012 Transmission eigenvalues in inverse scattering theory Inverse Problems and Applications: Inside Out II (MSRI Publications 60) ed G Uhlmann (Cambridge: Cambridge University Press) pp 527–78 [13] Cakoni F and Haddar H 2009 On the existence of transmission eigenvalues in an inhomogeneous medium Appl. Anal. 88 475–93 [14] Colton D and Kress R 1998 Inverse Acoustic and Electromagnetic Scattering Theory 2nd edn (Berlin: Springer) [15] Colton D and Monk P 1988 The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium Q. J. Mech. Appl. Math. 41 97–125 [16] Colton D, Monk P and Sun J 2010 Analytical and computational methods for transmission eigenvalues Inverse Problems 26 045011 [17] Colton D, P¨aiv¨arinta L and Sylvester J 2007 The interior transmission problem Inverse Problems Imaging 1 13–28 [18] Descloux J, Nassif N and Rapaz J 1978 On spectral approximation: part 1. The problem of convergence RAIRO Anal. Numer. 12 97–112 [19] Elhay S and Ram Y M 2002 An affine inverse eigenvalue problem Inverse Problems 18 455–66 [20] Gladwell G M L 2004 Inverse Problems in Vibration (Solid Mechanics and Its Applications) 2nd edn (New York: Academic) [21] Gohberg I, Goldberg S and Kaashoek A M 2000 Basic Classes of Linear Operators (Basel: Birkh¨auser) [22] Gould S H 1996 Variational Methods for Eigenvalue Problems (New York: Dover) [23] Ji X, Sun J and Turner T 2012 Algorithm 922: a mixed finite element method for Helmholtz transmission eigenvalues ACM Trans. Math. Softw. 38 1–8 [24] Kirsch A 2009 On the existence of transmission eigenvalues Inverse Problems Imaging 3 155–72 [25] Lancaster P and Tismenetsky M 1985 The Theory of Matrices 2nd edn (London: Academic) [26] Leung Y and Colton D 2012 Complex transmission eigenvalues for spherically stratified media Inverse Problems 28 075005 [27] McLaughlin J R and Polyakov P L 1994 On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues J. Differ. Eqns 107 351–82 [28] McLaughlin J R, Polyakov P L and Sacks P E 1994 Reconstruction of a spherically symmetric speed of sound SIAM J. Appl. Math. 54 1203–23 [29] McLaughlin J R, Sacks P E and Somasundaram M 1997 Inverse scattering in acoustic media using interior transmission eigenvalues Inverse Problems in Wave Propagation ed G Chavent, G Papanicolaou, P Sacks and W Symes (Berlin: Springer) pp 357–74 [30] P¨aiv¨arinta L and Sylvester J 2008 Transmission eigenvalues SIAM J. Math. Anal. 40 738–53 [31] Rynne B P and Sleeman B D 1992 The interior transmission problem and inverse scattering from inhomogeneous media SIAM J. Math. Anal. 22 1755–62 [32] Sun J 2011 Iterative methods for transmission eigenvalues SIAM J. Numer. Anal. 49 1860–74 [33] Sun J 2011 Estimation of transmission eigenvalues and the index of refraction from Cauchy data Inverse Problems 27 015009 [34] Tisseur F and Meerbergen K 2001 The quadratic eigenvalue problem SIAM Rev. 43 235–86

14