Inverse Scattering for Materials with a Conductive Boundary Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with: O. Bondarenko and A. Kleefeld March 2016 Inverse Scattering for Conductive Boundary 1 / 36 Formulation of the Problem Inverse Scattering for Conductive Boundary 2 / 36 The Direct Scattering Problem We consider the time-harmonic scattering problem in R2 and R3 1. D ⊂ Rm be a bounded open region with ∂D-Smooth 2. The function n ∈ L∞ (D) is the refractive index 3. The function η ∈ L∞ (∂D) is conductivity parameter Inverse Scattering for Conductive Boundary 3 / 36 The Direct Scattering Problem We consider the scattering by an inhomogeneous material with a 1 (R3 ) conductive boundary. The total field u = u s + u i ∈ Hloc satisfies: − u =u + in R3 \ D ∆u + k 2 n(x)u = 0 in D on ∂D − + + ∂ν u = ∂ν u + ηu ∂r u − iku = o r −1 as r → ∞. s and ∆u + k 2 u = 0 s Let u i = eikx·d then it is known that the scattering problem is well-posed for that radiating scattered fields u s (x, d; k) which depends on the incident direction d and the wave number k. Inverse Scattering for Conductive Boundary 4 / 36 The Inverse Scattering Problem The Inverse Problem Given the Cauchy Data (u s , ∂ν u s ) on the collection curve C for all d on the unit sphere and a range of wave numbers k ∈ [kmin , kmax ] can we obtain information about the scatterer D and it’s material parameters n and η? F. Cakoni, David Colton and Peter Monk, “The inverse electromagnetic scattering problem for a partially coated dielectric”, J. Comput. Appl. Math. 204, 256-267 (2007). O. Bondarenko and X. Liu “The factorization method for inverse obstacle scattering with conductive boundary condition”, Inverse Problems 29 (2013) 095021. Inverse Scattering for Conductive Boundary 5 / 36 The Far Field Pattern The scattered field, has the following asymptotic expansion eik|x| u (x, d; k) = 4π|x| s ∞ u (x̂, d; k) + O 1 |x| as |x| → ∞. Using Green’s Representation theorem one can show that the far field pattern is given by Z u ∞ (x̂, d; k) = u s (y , d; k)∂ν e−ik x̂·y − ∂ν u s (y , d; k)e−ik x̂·y ds(y ) C where x̂, d ∈ S = {x ∈ R3 : |x| = 1}. Inverse Scattering for Conductive Boundary 6 / 36 Reconstruction via the Far Field Operator We now define the far field operator as F : L2 (S) 7−→ L2 (S) Z (F g )(x̂) := u ∞ (x̂, d; k)g (d) ds(d) S The inverse problem now reads: given the far field operator F for a range of wave numbers k ∈ [kmin , kmax ] obtain information about the scatterer D and it’s material parameters n and η. Inverse Scattering for Conductive Boundary 7 / 36 Solution to the Inverse Scattering Problem Inverse Scattering for Conductive Boundary 8 / 36 Reconstruction Methods 1. Optimization Methods: Solve nonlinear model using iterative scheme to reconstruct the material parameters, which requires a priori information, and can be computationally expensive. For matrix valued coefficient the inverse problem lacks uniqueness. 2. Qualitative Methods: Using nonlinear model to reconstruct limited information with no a priori information, such as the support of the region in a computationally simple manner (i.e. solving a linear integral equations). Inverse Scattering for Conductive Boundary 9 / 36 Reconstruction of D by the LSM Let Φ∞ (x̂, z) = e−ikz·x̂ be the far field pattern for the fundamental solution of the Helmholtz equation. The Linear Sampling Method (LSM) The Far Field equation in R3 is given by (fix the wave number k) (F g )(x̂) = Φ∞ (x̂, z) for a z ∈ R3 , then the regularized solution of the far-field equation gz,δ satisfies I kgz,δ kL2 (S) is bounded as δ → 0 provided that z ∈ D, kgz,δ kL2 (S) is unbounded as δ → 0 provided that z ∈ / D. ∞ Assuming that F gz,δ − Φ (·, z) 2 → 0 as δ → 0. I L (S) Inverse Scattering for Conductive Boundary 10 / 36 Numerical Reconstruction i.e. plot z 7→ 1/kgz,δ k F. Cakoni, David Colton and Peter Monk, “The inverse electromagnetic scattering problem for a partially coated dielectric”, J. Comput. Appl. Math. 204, 256-267 (2007). Inverse Scattering for Conductive Boundary 11 / 36 Reconstruction of D by the FM Provided the far field operator has a suitable factorization, we define the positive self-adjoint compact operator F] := < F + = F : L2 (S) 7−→ L2 (S). The Factorization Method (FM) Let the pair (λi , ψi ) ∈ R+ × L2 (S) be an orthonormal eigensystem of F] . Then z ∈ D ⇐⇒ W (z) = ∞ X |(Φ∞ (x̂, z), ψi )|2 i=1 Inverse Scattering for Conductive Boundary λi < ∞. 12 / 36 Numerical Reconstruction i.e. plot z 7→ 1/W (z) O. Bondarenko and X. Liu “The factorization method for inverse obstacle scattering with conductive boundary condition”, Inverse Problems 29 (2013) 095021. Inverse Scattering for Conductive Boundary 13 / 36 TE-Problem for a Material with a Conductive Boundary O. Bondarenko, I. Harris, and A. Kleefeld “The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary” Submitted to Applicable Analysis - (arXiv:1510.01762). Inverse Scattering for Conductive Boundary 14 / 36 Motivation I The Transmission eigenvalue problem is non-selfadjoint and non-linear I The linear sampling method fails if k is a transmission eigenvalue I The Transmission eigenvalues can be used to determine/estimate the material properties. Inverse Scattering for Conductive Boundary 15 / 36 Injectivity of the Far Field Operator The associated Herglotz function is of the form Z vg (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 , vg ) that satisfies ∆w + k 2 nw = 0 in D ∆vg + k vg = 0 in D w = vg and ∂ν w = ∂ν vg + ηvg on ∂D 2 is given by (w , vg ) = (0, 0) Inverse Scattering for Conductive Boundary 16 / 36 Reconstructing the Real TEs Recall that the far field equation is given by (F gz )(x̂) = Φ∞ (x̂, z). Determination of TEs from scattering data: Let z ∈ D and gz,δ regularized solution to the Far Field equation I if k is not a TE then kgz,δ kL2 (S) is bounded as δ → 0 I if k is a TE then kgz,δ kL2 (S) is unbounded as δ → 0. Since the Transmission eigenvalues can be reconstructed from the data the next question one can ask is what information do they hold about the coefficients. Inverse Scattering for Conductive Boundary 17 / 36 Numerical Examples i.e. plot k 7→ ||gz,δ || Figure: Plot of the average ||gz,δ ||L2 (S) for 25 points z ∈ D I. Harris, F. Cakoni, and J. Sun ”Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids” Inverse Problems, 30 035016 (2014). Inverse Scattering for Conductive Boundary 18 / 36 The Transmission Eigenvalues Values of k ∈ C where nontrivial (w , v ) ∈ L2 (D) × L2 (D) such that w − v ∈ H 2 (D) ∩ H01 (D) satisfies ∆w + k 2 nw = 0 w −v =0 and ∆v + k 2 v = 0 and ∂ν w − ∂ν v = ηv in D on ∂D. Where we assume that n > 0 and η > 0. Equivalently there exists u = w − v ∈ H 2 (D) ∩ H01 (D) satisfies 1 (∆ + k 2 n)u = 0 n−1 η and ∂ν u = − 2 (∆ + k 2 n)u k (n − 1) (∆ + k 2 ) u=0 Inverse Scattering for Conductive Boundary in D on ∂D. 19 / 36 The Variational Formulation By Green’s 2nd Theorem 1 (∆ + k 2 n)u = 0 n−1 η (∆ + k 2 n)u and ∂ν u = − 2 k (n − 1) (∆ + k 2 ) u=0 in D on ∂D has the variational form Z 2 Z k 1 ∂ν u ∂ν ϕ ds + (∆u + k 2 nu)(∆ϕ + k 2 ϕ) dx = 0, η n−1 ∂D D for all ϕ ∈ H 2 (D) ∩ H01 (D). Inverse Scattering for Conductive Boundary 20 / 36 The transmission eigenvalue problem can be written as Ak u − k 2 Bu = 0 Z (Ak u, ϕ)H 2 (D) = D where 1 (∆u + k 2 u)(∆ϕ + k 2 ϕ) + k 4 uϕ dx n−1 Z 2 k + ∂ν u ∂ν ϕ ds η ∂D and Z ∇u · ∇ϕ dx (Bu, ϕ)H 2 (D) = D Assume that either n > 1 a.e. in D and that η > 0 on ∂D then 1. the operator B is positive, compact, and self-adjoint. 2. the mapping k 7→ Ak is continuous from (0, ∞) to the set of self-adjoint coercive operators on H 2 (D) ∩ H01 (D). Inverse Scattering for Conductive Boundary 21 / 36 The Key Ingredient F. Cakoni and H. Haddar “On the existence of transmission eigenvalues in an inhomogenous medium”, Applicable Analysis 88, no 4, 475-493 (2009). We recall this key result in the following lemma. Lemma If k 7→ Ak is continuous from (0, ∞) to the set of self-adjoint positive definite bounded operators and B is a self-adjoint non-negative compact operator. 1. Ak0 − k02 B is a positive operator, 2. Ak1 − k12 B is non-positive on a m dimensional subspace then each of the equations λj (k) − k 2 = 0 for j = 1, . . . , m has at least one solution in [k0 , k1 ] where λj (k) is such that Ak − λj (k)B has a non-trivial kernel. Inverse Scattering for Conductive Boundary 22 / 36 Notice that we have the following estimates Ak u − k 2 Bu, u 2 ≥ αkuk2H 2 (D) − k 2 k∇uk2L2 (D) H (D) ≥ αkuk2H 2 (D) − k 2 kuk2H 2 (D) ≥ (α − k 2 )kuk2H 2 (D) , where α is the coercivity constant for the operator Ak (that is independent of k). Hence, for all k 2 sufficiently small we have that Inverse Scattering for Conductive Boundary Ak u − k 2 Bu, u H 2 (D) > 0. 23 / 36 Let xi ∈ D where Bi = B(xi , ε) := {x ∈ Rm : |x − xi | < ε} with ε > 0, sufficiently small. Also assume n(x) ≥ nmin and find ∆ + kε2 1 ∆ + kε2 nmin ui = 0 nmin − 1 in Bi where ui ∈ H02 (Bi ). Extend ui to H02 (D) by zero, then one can then show that Akε u − kε2 Bu, u 2 ≤ 0, for all u ∈ Span{u1 , u2 , · · · }. H (D) Theorem (O. Bondarenko, I. H, and A. Kleefeld) Assume that either n > 1 or 0 < n < 1 a.e. in D and η > 0, then there exists infinitely many real transmission eigenvalues. Moreover the set of real transmission eigenvalues is discrete. Inverse Scattering for Conductive Boundary 24 / 36 It is known that for a every fixed k ∈ (0, ∞) there exists an increasing sequence λj (k; n, η) of positive generalized eigenvalues Ak u − λj (k)Bu = 0 that satisfy (Ak u, u)H 2 (D) U∈Uj u∈U\{0} (Bu, u)H 2 (D) λj (k; n, η) = min max where Uj is the set of all j-dimensional subspaces of H02 (D) ∩ H01 (D). Recall that the real transmission eigenvalues satisfy λj (k; n, η) − k 2 (n, η) = 0. Inverse Scattering for Conductive Boundary 25 / 36 Monotonicity Results Theorem (O. Bondarenko, I. H, and A. Kleefeld) Assume that 0 < n with 0 < η and that kj is a transmission eigenvalue such that λj (k) − k 2 = 0, 1. if 1 < n, then kj is decreasing with respect to n. 2. if 1 < n, then kj is decreasing with respect to η. 3. if 0 < n < 1, then kj is increasing with respect to n. 4. if 0 < n < 1, then kj is increasing with respect to η. Inverse Scattering for Conductive Boundary 26 / 36 An Inverse Spectral Result Corollary (O. Bondarenko, I. H, and A. Kleefeld) 1. If it is known that n > 1 or 0 < n < 1 is a constant refractive index with η known and fixed, then n is uniquely determined by the first transmission eigenvalue. 2. If n > 1 or 0 < n < 1 is known and fixed with η a constant, then the first transmission eigenvalue uniquely determines η. Since the Transmission eigenvalues can be reconstructed from the data this uniqueness result can be used for non-destructive testing. Inverse Scattering for Conductive Boundary 27 / 36 Let D =Unit Sphere with n and η constant. Using separation of variables we have that k is a transmission eigenvalue if and only if √ −jm (k) jm (k n) √ √ = 0. det 0 (k) − ηj (k) 0 (k n) −kjm k njm m Here, jm denotes the spherical Bessel function of the first kind. η 1 2 3 10 1. 2.798 386 2.458 714 2.204 525 1.743 402 2. 3.029 807 2.914 716 2.809 294 2.467 800 3. 3.141 593 3.141 593 3.141 593 3.138 749 4. 3.601 813 3.514 484 3.435 429 3.141 593 Table: The first four interior transmission eigenvalues for a unit sphere using the index of refraction n = 4 and various choices of η. Inverse Scattering for Conductive Boundary 28 / 36 Let n and η be constants, therefore by Green’s representation w (x) = SLk √n (∂ν w )(x) − DLk √n (w )(x) , v (x) = SLk (∂ν v )(x) − DLk (v )(x) , x ∈ D, x ∈D, where Z SLk (f )(x) = Φk (x, y )f (y ) ds(y ) , x ∈D, Z∂D DLk (f )(x) = ∂ν(y ) Φk (x, y )f (y ) ds(y ) , x ∈D, ∂D are the single and double layer potentials that are defined for points in the domain D, respectively. Inverse Scattering for Conductive Boundary 29 / 36 Boundary Integral Operators Z Sk (f )(x) = Φk (x, y )f (y ) ds(y ) , x ∈ ∂D , Z∂D Dk (f )(x) = ∂ν(y ) Φk (x, y )f (y ) ds(y ) , x ∈ ∂D , ∂D are the single and double layer boundary integral operators, respectively. Z Kk (f )(x) = ∂ν(x) Φk (x, y )f (y ) ds(y ) , x ∈ ∂D , ∂D Z Tk (f )(x) = ∂ν(x) ∂ν(y ) Φk (x, y )f (y ) ds(y ) , x ∈ ∂D , ∂D are the normal derivative of the single and double layer boundary integral operators, respectively. Inverse Scattering for Conductive Boundary 30 / 36 Boundary Integral Operators Using the notation ψ1 = ∂ν v (x) and ψ2 = v (x), x ∈ ∂D, ! Sk √n − Sk −Dk √n + Dk + ηSk √n ψ1 0 = . ψ2 0 Kk √n − Kk −Tk √n + Tk + η Kk √n − 21 I The boundary integral operator Z depending on the wave number k is given by ! −Dk √n + Dk + ηSk √n Sk √n − Sk Z(k) = . Kk √n − Kk −Tk √n + Tk + η Kk √n − 12 I Abstractly k is a transmission eigenvalue provided that Z(k) has a non-trivial kernel. Inverse Scattering for Conductive Boundary 31 / 36 Numerical Results For a cushion-shaped obstacle with index of refractions n = 1/2 and n = 4 for η = 1/2, η = 1, and η = 3. (n, η) (1/2,1/2) (1/2,1) (1/2,3) (4,1/2) (4,1) (4,3) 1. 1.481 1.889 2.482 2.754 2.678 2.391 2. 1.754 2.245 2.947 2.987 2.930 2.723 3. 2.080 2.713 3.640 3.460 3.404 3.196 W.J. Beyn “An integral method for solving nonlinear eigenvalue problems”, Linear Algebra Appl. 436 (2012) 3839-3863. Inverse Scattering for Conductive Boundary 32 / 36 Numerical Results The monotonicity of the first interior transmission eigenvalues for the peanut-shaped obstacle using n = 4 for increasing η. Inverse Scattering for Conductive Boundary 33 / 36 Convergence as η → 0+ We conjecture that as η → 0+ the transmission eigenvalues converge to the transmission eigenvalues for η = 0. η 1 1/2 1/4 1/8 1/16 1/32 1/64 ERROR 1.118· 10−1 5.468· 10−2 2.695· 10−2 1.337· 10−2 6.659· 10−3 3.323· 10−3 1.660· 10−3 EOC 1.032 1.021 1.011 1.006 1.003 1.001 ERROR 9.063· 10−2 4.535· 10−2 2.264· 10−2 1.130· 10−2 5.647· 10−3 2.822· 10−3 1.411· 10−3 EOC 0.999 1.002 1.003 1.001 1.001 1.000 Table: The estimated order of convergence for the 2nd and 4th transmission eigenvalue for a unit sphere using n = 4 as η → 0+ . Inverse Scattering for Conductive Boundary 34 / 36 Some Answered Questions: I Discreteness of the transmission eigenvalues I Existence of transmission eigenvalues I Monotonicity of the transmission eigenvalues I Reconstruction from far field pattern Some ‘Unanswered’ Questions: I Existence of Complex eigenvalues I Inverse Spectral Analysis I Convergence Analysis as η → 0+ I Asymptotic Analysis as η → 0+ Inverse Scattering for Conductive Boundary 35 / 36 Figure: Questions? Inverse Scattering for Conductive Boundary 36 / 36