Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing

Transcription

1 Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing Isaac Harris Texas A & M University College Station, Texas iharris@math.tamu.edu Joint work with: F. Cakoni, H. Haddar and J. Sun Research Funded by: NSF Grant DMS and University of Delaware Graduate Student Fellowship October 2015

2 Overview 1 INTRODUCTION 2 TE-PROBLEM FOR MATERIALS WITH A CAVITY 3 TE-PROBLEM FOR PERIODIC MEDIA

3 The Direct Scattering Problem We consider the time-harmonic inverse acoustic (in R 3 ) or electromagnetic (in R 2 ) scattering problem 1 D R m be a bounded open region with D-Lipshitz 2 The matrix A L (D, R m m ) is symmetric-positive definite 3 The function n L (D) such that n(x) > 0 for a.e. x D

4 The Direct Scattering Problem The boundary value problem We consider the scattering by an anisotropic material where the scattered field u s and the total field u = u s + u i satisfies: u s + k 2 u s = 0 in R m \ D A(x) u + k 2 n(x)u = 0 in D u = u s + u i u and = ν A ν (us + u i ) on D ( ) lim r m 1 u s 2 iku s = 0. r r

5 Far-Field Operator Let u i = e ikx d then it is known that the radiating scattered fields u s (x, d; k) depends on the incident direction d and the wave number k, has the following asymptotic expansion u s (x, d; k) = eik x x m 1 2 { ( )} 1 u (ˆx, d; k) + O x as x We now define the far-field operator as F : L 2 (S) L 2 (S) (Fg)(ˆx) := u (ˆx, d; k)g(d) ds(d) S where S = {x R m : x = 1} is the unit circle or sphere.

6 The Inverse Scattering Problem The Inverse Problem Given the far-field pattern u (ˆx, d; k) for all ˆx, d S and a range of wave numbers k [k min, k max ] can we obtain information about the scatterer D and it s material parameters A and n?

7 The Inverse Scattering Problem The Inverse Problem Given the far-field pattern u (ˆx, d; k) for all ˆx, d S and a range of wave numbers k [k min, k max ] can we obtain information about the scatterer D and it s material parameters A and n? Reconstruction Methods 1 Optimization Methods: Reconstruct all material parameters which require a priori information, and can be computationally expensive. (our problem suffers from lack of uniqueness) 2 Qualitative Methods: Reconstruct limited information, such as the support of a defective region in a computationally simple manner(i.e. solving a linear integral equation).

8 Qualitative Methods for Inverse Scattering 1 Linear Sampling Method F. Cakoni, D. Colton and P. Monk The linear Sampling Method in Inverse Electromagnetic Scattering CBMS Series, SIAM Publications 80, (2011). F. Cakoni and D. Colton A Qualitative Approach to Inverse Scattering Theory Applied Mathematical Sciences, Vol 188, Springer, Berlin Factorization Method A. Kirsch and N. Grinberg The Factorization Method for Inverse Problems. Oxford University Press, Oxford 2008.

9 Qualitative Methods in the Time Domain H. Haddar, A. Lechleiter, S. Marmorat An improved time domain linear sampling method for Robin and Neumann obstacles. Applicable Analysis, 93(2), 2014, Taylor & Francis, Q. Chen, H. Haddar, A. Lechleiter, P. Monk A Sampling Method for Inverse Scattering in the Time Domain. Inverse Problems, 26, , IOPscience, H. Heck, G. Nakamura and H. Wang Linear sampling method for identifying cavities in a heat conductor. IInverse Problems, 28, , IOPscience, 2012.

10 The Linear Sampling method The linear sampling method is based on solving the far-field equation, therefore assume that F is injective with dense range. Let Φ (ˆx, z) = γexp( ikz ˆx) with γ constant. The Far Field equation in R m is given by (fix the wave number k) u (ˆx, d; k)g z (d) ds(d) = Φ (ˆx, z) for a z R m. S then the regularized solution of the far-field equation gz α satisfies if z D then gz α L 2 (S) is bounded as α 0 if z R m \ D then gz α L 2 (S) is unbounded as α 0.

11 Numerical Reconstruction i.e. plot z 1/ g α z Figure: Reconstruction of a square anisotropic scatterer via the factorization method and the generalized linear sampling method. Dashed line: exact boundaries of the scatterer.

12 Numerical Reconstruction i.e. plot z 1/ g α z Figure: Reconstruction of 2 circular scatterers via the linear sampling method for Maxwell s Equations. T. Arens and A. Lechleiter, Indicator Functions for Shape Reconstruction Related to the Linear Sampling Method SIAM J. Imaging Sci., 8(1),

13 Injectivity of the Far Field Operator The associated Herglotz function is of the form v g (x) := e ikx d g(d) ds(d). S The far-field operator F is injective with dense range if and only if the only pair (w, v g ) that satisfies is given by (w, v g ) = (0, 0) A w + k 2 nw = 0 in D v g + k 2 v g = 0 in D w = v g and w = v g ν A ν on D

14 Transmission Eigenvalues Transmission eigenvalues are related to specific frequencies where there is an incident field u i that does not scatter They are also connected to the injectivity(and the density of the range) of the far-field operator F The Transmission eigenvalues can be used to determine/estimate the material properties.

15 TE-Problem for a Material with a Cavity I. Harris, F. Cakoni and J. Sun Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids Inverse Problems 30 (2014)

16 TE-Problem for anisotropic materials with a cavity Definition of the TE-Problem The transmission eigenvalues are values of k C to which there exists nontrivial (w, v) H 1 (D) H 1 (D) such that w + k 2 w = 0 in D 0 A w + k 2 nw = 0 in D \ D 0 v + k 2 v = 0 in D w v = 0 on D w v = 0 ν A ν on D with continuity of the Cauchy data across D 0 for w. This is a Non-Selfadjoint Eigenvalue Problem!

17 Complex TEs Let A = 0.2I, n = 1, D = B R and D 0 = B ɛ where ɛ = 0.1 and R = 1, then using separation of variables we have that two complex TEs are given by k 3.84 ± 0.58i Figure: I told you this wasn t a self-adjoint problem :-(

18 Existence and Discreteness Result F. Cakoni, D. Colton and H. Haddar The interior transmission problem for regions with cavities. SIAM J. Math. Analysis 42, no 1, (2010). F. Cakoni and A. Kirsch On the interior transmission eigenvalue problem. Int. Jour. Comp. Sci. Math

19 Existence and Discreteness Result F. Cakoni, D. Colton and H. Haddar The interior transmission problem for regions with cavities. SIAM J. Math. Analysis 42, no 1, (2010). F. Cakoni and A. Kirsch On the interior transmission eigenvalue problem. Int. Jour. Comp. Sci. Math Theorem (IH-Cakoni-Sun 2014) Assume that A(x) I and n(x) 1 have different signs in D \ D 0, then there exists at least one Real TE, provided the cavity D 0 is sufficiently small. Moreover the set of Real TEs is at most discrete.

20 Tricks of the Trade Let u H 1 0 (D) and find v := v u such that for all ϕ H 1 ( D \ D 0 ) (A I) v ϕ k 2 (n 1)vϕ dx = D\D 0 A u ϕ k 2 nuϕ dx + ϕ T k u ds. D\D 0 D 0 T k is the DtN mapping for Helmholtz equation in D 0. Now define (L k u, ϕ) H1 (D\D 0) = v u ϕ k 2 v u ϕ dx + ϕ T k v u ds, D\D 0 D 0 then the TE-Problem is equivalent to L k u = 0 for u nontrivial.

21 Lemma Note that L k : H 1 0 (D \ D 0) H 1 0 (D \ D 0) is well defined for k 2 not a Dirichlet eigenvalue of in D 0. 1 L k is a self-adjoint operator for k R. 2 L k L 0 is a compact operator. 3 The mapping k L k is analytic for k 2 C + \ { λ j (D 0 ) } j=1

22 Lemma Note that L k : H 1 0 (D \ D 0) H 1 0 (D \ D 0) is well defined for k 2 not a Dirichlet eigenvalue of in D 0. 1 L k is a self-adjoint operator for k R. 2 L k L 0 is a compact operator. 3 The mapping k L k is analytic for k 2 C + \ { λ j (D 0 ) } j=1 Define the auxiliary self-adjoint compact operator T k := (L 0 ) 1/2 (L k L 0 )(L 0 ) 1/2 Therefore k is a TE provided that 1 is an eigenvalue of T k, the existence result can be obtained by using the Rayleigh quotient and the min-max principle. Discreteness follows from appealing to the analytic Fredholm theorem.

23 An Inverse Spectral Result Theorem (IH-Cakoni-Sun 2014) Assume that A(x) I and n(x) 1 have different signs and let k 1 denote the first TE and assume D 0 D 1 then we have that k 1 (D 0 ) k 1 (D 1 ) Figure: Since the mapping Area(D 0 ) k 1 is increasing this gives that the 1st TE can be used to approximate the area of the void(s).

24 Reconstructing the Real TEs (IH-Cakoni-Sun 2014) and (Cakoni-Colton-Haddar 2010) Recall that the far-field equation is given by u (ˆx, d; k)g z (d) ds(d) = Φ (ˆx, z). S Determination of TEs from scattering data: Let z D and gz α regularized solution to the Far Field equation if k is not a TE then gz α L 2 (S) is bounded as α 0 if k is a TE then g α z L 2 (S) is unbounded as α 0.

25 Numerical Examples i.e. plot k g α z For A = 0.2I and n = 1, with void B ɛ and domain B R, where we take ɛ = 0.1 and R = 1 we have that separation of variables gives that k = , are transmission eigenvalues. Figure: Plot of the k [2, 6] v.s. g α z L2 (S)

26 Dependance on the shape and position Question: Does the shape of the void affect the 1st TEs? Table: Dependance of first transmission eigenvalue w.r.t. shape k 1 (Disk) k 1 (Square) k 1 (Ellipse) Question: Does the location of the void change the 1st TEs? Table: Dependence of first transmission eigenvalue w.r.t. position location (0, 0) (0.6, 0) (0.3, 0.7) (-0.2, 0.4) (0.6, 0.6) k

27 Reconstruction of Void Size Inversion Algorithm Construct a polynomial P(t) s.t. P(Area(B r )) k 1 (B r ) Plot k gz α L 2 (S) to reconstruct k 1 (D 0 ) Solve: P(t) = k 1 (D 0 ) Then use Area(B r ) to approximate Area(D 0 ) Table: Reconstruction of Area from Measurements D D 0 B r D 0 Percent Error Disk R = 1 Disk % Square % [ 1, 1] [ 1, 1] Ellipse % Square %

28 TE-Problem for a Periodic Media F. Cakoni, H. Haddar and I. Harris Homogenization approach for the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging , Volume 9, Issue 4, 2015 (arxiv: ).

29 Homogenization for the TEs Homogenization is used to study composite periodic media. We are interested in the limiting case as ɛ 0 for the TE-Problem. Find non-trivial { k ɛ, (v ɛ, w ɛ ) } R + X (D) such that: A(x/ɛ) w ɛ + k 2 ɛ n(x/ɛ)w ɛ = 0 in D v ɛ + kɛ 2 v ɛ = 0 in D w ɛ = v ɛ and w ɛ = v ɛ on D ν ν Aɛ

30 What is Homogenization The matrix A(y) L (Y, R m m ) is Y -periodic symmetric-positive definite and the function n(y) L (Y ) is a positive Y -periodic function. We have that as ɛ 0 n ɛ := n(x/ɛ) n h := 1 Y Y n(y) dy weakly in L A ɛ := A(x/ɛ) A h in the sense of H-convergence (i.e. for u ɛ u in H 1 (D) then A ɛ u ɛ A h u in [L 2 (D)] m ) We have that A h is a constant symmetric matrix

31 The case of an Isotropic media(i.e. A = I ) For this case (w ɛ, v ɛ ) L 2 (D) L 2 (D) with u ɛ = v ɛ w ɛ H 2 0 (D), where we have that the difference u ɛ satisfies for n ɛ := n(x/ɛ) 0 = ( + k 2 ) 1 ( n ɛ + k 2 ) u ɛ in D. n ɛ 1 The equivalent variational form is given by 1 ( uɛ + k 2 )( ɛ u ɛ ϕ + k 2 n ɛ 1 ɛ n ɛ ϕ ) dx = 0 for all ϕ H0 2 (D). D We can rewriting the variational form to the following form A ɛ,kɛ (u ɛ, ϕ) k 2 ɛ B(u ɛ, ϕ) = 0 for all ϕ H 2 0 (D).

32 Proof: Sketch Assume that n min n(y) n max for all y Y, let u ɛ be the eigenfunction corresponding to the eigenvalue k ɛ with u ɛ H 1 (D) = 1 then for all ɛ 0 α u ɛ 2 L 2 (D) A ɛ,k ɛ (u ɛ, u ɛ ) = k 2 ɛ B(u ɛ, u ɛ ) k 2 ɛ u ɛ 2 H 1 (D) provided that n min > 1 or 0 < n max < 1.

33 Proof: Sketch Assume that n min n(y) n max for all y Y, let u ɛ be the eigenfunction corresponding to the eigenvalue k ɛ with u ɛ H 1 (D) = 1 then for all ɛ 0 α u ɛ 2 L 2 (D) A ɛ,k ɛ (u ɛ, u ɛ ) = k 2 ɛ B(u ɛ, u ɛ ) k 2 ɛ u ɛ 2 H 1 (D) provided that n min > 1 or 0 < n max < 1. Lemma (Cakoni-Haddar-IH. 2015) Assume that either n min > 1 or 0 < n max < 1. There exists an infinite sequence of real transmission eigenvalues k ɛ, j for j N such that k j (n max, D) k ɛ, j < k j (n min, D) if n min > 1 k j (n min, D) k ɛ, j < k j (n max, D) if 0 < n max < 1

34 Convergence for the TEs A = I The following convergence result then follows from the previous analysis along with elliptic regularity and basic results of homogenization. Theorem (Cakoni-Haddar-IH. 2015) Assume that n min > 1 or 0 < n max < 1, then there is a subsequence of { k ɛ, (v ɛ, w ɛ ) } R + L 2 (D) L 2 (D) that converges weakly to (v, w) L 2 (D) L 2 (D) and k ɛ k that solves: w + k 2 n h w = 0 and v + k 2 v = 0 in D w = v and w ν = v on D ν provided that k ɛ is bounded

35 Reconstructing Material Properties A = I We can reconstruct the effective material property n h from finding a n 0 such that k 1 (n 0 ) = k 1 (n ɛ ) where Now let w + k 2 n 0 w = 0 and v + k 2 v = 0 in D w = v and w ν = v ν on D. n(x/ɛ) = sin 2 (2πx 1 /ɛ) + 2 Table: Reconstruction from scattering data ɛ k 1 (n ɛ ) n h n

36 Convergence for the TEs A I Theorem (Cakoni-Haddar-IH. 2015) Assume that A(y) I and n(y) 1 have different sign in Y. Assume that n(y) = 1 and A(y) I is positive(or negative) definite. Then there is a subsequence of { kɛ, (v ɛ, w ɛ ) } R + H 1 (D) H 1 (D) that converges weakly to (v, w) H 1 (D) H 1 (D) and k ɛ k that solves: A h w + k 2 n h w = 0 and v + k 2 v = 0 in D w w = v and = v on D ν ν Ah provided that k ɛ is bounded. Recall that A(x/ɛ) A h in the sense of H-convergence as ɛ 0.

37 Reconstructing Material Properties n = 1 We can reconstruct the effective material property A h = a h I by finding an a 0 such that k 1 (a 0 ) = k 1 (A ɛ ) where. Now let a 0 w + k 2 w = 0 and v + k 2 v = 0 w w = v and a 0 ν = v ν A(x/ɛ) = 1 3 in D on D ( sin 2 (2πx 2 /ɛ) ) 0 cos 2 (2πx 1 /ɛ) + 1 Table: Reconstruction from scattering data ɛ k 1 (A ɛ ) a h a

38 For more details: F. Cakoni, H. Haddar and I. Harris Homogenization approach for the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging , Volume 9, Issue 4, 2015 (arxiv: ). In the paper we have also considered: The convergence of the interior transmission problem for the cases where A = I and A I Construct a bulk corrector to prove strong convergence for A I Numerical test for the order of convergence

39 A Current Project: I. Harris The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary. Inverse Problems and Imaging (Submitted) - (arxiv: ) Find k C and nontrivial (w, v) L 2 (D) L 2 (D) such that v w H 2 (D) H 1 0 (D) satisfies w + k 2 nw = 0 and v + k 2 v = 0 in D w v = 0 and w ν v ν = µv Discreteness of the transmission eigenvalues Existence of transmission eigenvalues Monotonicity of the transmission eigenvalues on D. Determination of transmission eigenvalues from scattering data

40 Thanks for your Attention: Figure: Questions?