INTRODUCTION
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Inverse Scattering Theory: Transmission
Eigenvalues and Non-destructive Testing
Isaac Harris
Texas A & M University College Station, Texas 77843-3368
iharris@math.tamu.edu
Joint work with: F. Cakoni, H. Haddar and J. Sun
Research Funded by: NSF Grant DMS-1106972
and University of Delaware Graduate Student Fellowship
October 2015
INTRODUCTION
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Overview
1
2
3
INTRODUCTION
TE-PROBLEM FOR MATERIALS WITH A CAVITY
TE-PROBLEM FOR PERIODIC MEDIA
INTRODUCTION
TE-PROBLEM FOR MATERIALS WITH A CAVITY
TE-PROBLEM FOR PERIODIC MEDIA
The Direct Scattering Problem
We consider the time-harmonic inverse acoustic (in R3 ) or
electromagnetic (in R2 ) scattering problem
1
D ⊂ Rm be a bounded open region with ∂D-Lipshitz
2
The matrix A ∈ L∞ (D, Rm×m ) is symmetric-positive definite
3
The function n ∈ L∞ (D) such that n(x) > 0 for a.e. x ∈ D
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The Direct Scattering Problem
The boundary value problem
We consider the scattering by an anisotropic material where the
scattered field us and the total field u = us + ui satisfies:
∆u s + k 2 u s = 0
∇ · A(x)∇u + k 2 n(x)u = 0
∂u
∂ s
u = u s + u i and
=
(u + u i )
∂νA
∂ν
s
m−1
∂u
s
lim r 2
− iku = 0.
r →∞
∂r
in
Rm \ D
in
D
on
∂D
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Far-Field Operator
Let ui = eikx·d then it is known that the radiating scattered fields
us (x, d; k) depends on the incident direction d and the wave
number k, has the following asymptotic expansion
us (x, d; k) =
e ik|x|
|x|
m−1
2
u∞ (x̂, d; k) + O
1
|x|
as |x| → ∞
We now define the far-field operator as F : L2 (S) 7−→ L2 (S)
Z
(Fg )(x̂) :=
u∞ (x̂, d; k)g (d) ds(d)
S
where S = {x ∈ Rm : |x| = 1} is the unit circle or sphere.
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The Inverse Scattering Problem
The Inverse Problem
Given the far-field pattern u∞ (x̂, d; k) for all x̂ , d ∈ S and a range
of wave numbers k ∈ [kmin , kmax ] can we obtain information about
the scatterer D and it’s material parameters A and n?
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The Inverse Scattering Problem
The Inverse Problem
Given the far-field pattern u∞ (x̂, d; k) for all x̂ , d ∈ S and a range
of wave numbers k ∈ [kmin , kmax ] 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).
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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 2014.
2
Factorization Method
A. Kirsch and N. Grinberg
The Factorization Method for Inverse Problems.
Oxford University Press, Oxford 2008.
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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, 2014.
Q. Chen, H. Haddar, A. Lechleiter, P. Monk
A Sampling Method for Inverse Scattering in the Time Domain.
Inverse Problems, 26, 085001, IOPscience, 2010.
H. Heck, G. Nakamura and H. Wang
Linear sampling method for identifying cavities in a heat conductor.
IInverse Problems, 28, 075014, IOPscience, 2012.
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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∞ (x̂, z) = γexp(−ikz · x̂) with γ constant.
The Far Field equation in Rm is given by (fix the wave number k)
Z
u∞ (x̂, d; k)gz (d) ds(d) = Φ∞ (x̂, z) for a z ∈ Rm .
S
then the regularized solution of the far-field equation gzα satisfies
if z ∈ D then kgzα kL2 (S) is bounded as α → 0
if z ∈ Rm \ D then kgzα kL2 (S) is unbounded as α → 0.
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Numerical Reconstruction i.e. plot z 7→ 1/kgzα k
Figure: Reconstruction of a square anisotropic scatterer via the
factorization method and the generalized linear sampling method.
Dashed line: exact boundaries of the scatterer.
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Numerical Reconstruction i.e. plot z 7→ 1/kgzα k
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), 513535.
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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
∇ · A∇w + k 2 nw = 0
2
∆vg + k vg = 0
∂vg
∂w
w = vg and
=
∂νA
∂ν
is given by (w , vg ) = (0, 0)
in
D
in
D
on
∂D
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Transmission Eigenvalues
Transmission eigenvalues are related to specific frequencies
where there is an incident field ui 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.
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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) 035016.
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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
D0
in
D \ D0
∆v + k v = 0
in
D
w −v =0
∂w
∂v
−
=0
∂νA ∂ν
on
∂D
on
∂D
2
∇ · A∇w + k nw = 0
2
with continuity of the Cauchy data across ∂D0 for w .
This is a Non-Selfadjoint Eigenvalue Problem!
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Complex TEs
Let A = 0.2I, n = 1, D = BR and D0 = B where = 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 :-(
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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, 145-162 (2010).
F. Cakoni and A. Kirsch
On the interior transmission eigenvalue problem.
Int. Jour. Comp. Sci. Math. 2010.
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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, 145-162 (2010).
F. Cakoni and A. Kirsch
On the interior transmission eigenvalue problem.
Int. Jour. Comp. Sci. Math. 2010.
Theorem (IH-Cakoni-Sun 2014)
Assume that A(x) − I and n(x) − 1 have different signs in D \ D0 , then there
exists at least one Real TE, provided the cavity D0 is sufficiently small.
Moreover the set of Real TEs is at most discrete.
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Tricks of the Trade
Let u ∈ H01 (D) and find v := vu such that for all ϕ ∈ H 1 D \ D 0
Z
(A − I)∇v · ∇ϕ − k 2 (n − 1)v ϕ dx =
D\D 0
Z
2
Z
A∇u · ∇ϕ − k nuϕ dx +
D\D 0
ϕ Tk u ds.
∂D0
Tk is the DtN mapping for Helmholtz equation in D0 . Now define
Z
Z
(Lk u, ϕ)H 1 (D\D 0 ) =
∇vu · ∇ϕ − k 2 vu ϕ dx +
ϕ Tk vu ds,
D\D 0
∂D0
then the TE-Problem is equivalent to Lk u = 0 for u nontrivial.
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Lemma
Note that Lk : H01 (D \ D 0 ) 7−→ H01 (D \ D 0 ) is well defined for k 2
not a Dirichlet eigenvalue of −∆ in D0 .
1
Lk is a self-adjoint operator for k ∈ R.
2
Lk − L0 is a compact operator.
3
∞
The mapping k 7→ Lk is analytic for k 2 ∈ C+ \ λj (D0 ) j=1
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Lemma
Note that Lk : H01 (D \ D 0 ) 7−→ H01 (D \ D 0 ) is well defined for k 2
not a Dirichlet eigenvalue of −∆ in D0 .
1
Lk is a self-adjoint operator for k ∈ R.
2
Lk − L0 is a compact operator.
3
∞
The mapping k 7→ Lk is analytic for k 2 ∈ C+ \ λj (D0 ) j=1
Define the auxiliary self-adjoint compact operator
Tk := (L0 )−1/2 (Lk − L0 )(L0 )−1/2
Therefore k is a TE provided that 1 is an eigenvalue of Tk , 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.
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An Inverse Spectral Result
Theorem (IH-Cakoni-Sun 2014)
Assume that A(x) − I and n(x) − 1 have different signs and let k1
denote the first TE and assume D0 ⊆ D1 then we have that
k1 (D0 ) ≤ k1 (D1 )
Figure: Since the mapping Area(D0 ) 7−→ k1 is increasing this gives that
the 1st TE can be used to approximate the area of the void(s).
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Reconstructing the Real TEs
(IH-Cakoni-Sun 2014) and (Cakoni-Colton-Haddar 2010)
Recall that the far-field equation is given by
Z
u∞ (x̂, d; k)gz (d) ds(d) = Φ∞ (x̂, z).
S
Determination of TEs from scattering data:
Let z ∈ D and gzα regularized solution to the Far Field equation
if k is not a TE then kgzα kL2 (S) is bounded as α → 0
if k is a TE then kgzα kL2 (S) is unbounded as α → 0.
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Numerical Examples i.e. plot k 7→ kgzα k
For A = 0.2I and n = 1, with void B and domain BR , where we
take = 0.1 and R = 1 we have that separation of variables gives
that k = 2.4887, 5.2669 are transmission eigenvalues.
Figure: Plot of the k ∈ [2, 6] v.s. ||gzα ||L2 (S)
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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
k1 (Disk)
k1 (Square)
k1 (Ellipse)
1.77
1.78
1.78
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)
k1
1.77
1.80
1.78
1.80
1.78
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Reconstruction of Void Size
Inversion Algorithm
Construct a polynomial P(t) s.t. P(Area(Br )) ≈ k1 (Br )
Plot k 7→ kgzα kL2 (S) to reconstruct k1 (D0 )
Solve: P(t) = k1 (D0 )
Then use Area(Br ) to approximate Area(D0 )
Table: Reconstruction of Area from Measurements
D
D0
|Br |
|D0 |
Percent Error
Disk R = 1
Disk
Square
Ellipse
Square
0.0328
0.0303
0.0613
0.0749
0.0314
0.0300
0.0628
0.1256
4.46%
1.00%
2.39%
40.37%
[−1, 1] × [−1, 1]
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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 1025 - 1049, Volume 9, Issue 4, 2015
(arXiv:1410.37297).
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Homogenization for the TEs
Homogenization is used to study composite periodic media. We
are interested in the limiting case as → 0 for the TE-Problem.
Find non-trivial k , (v , w ) ∈ R+ × X (D) such that:
∇ · A(x/)∇w + k2 n(x/)w = 0
k2 v
∆v +
=0
∂v
∂w
=
w = v and
∂νA
∂ν
in
D
in
D
on
∂D
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What is Homogenization
The matrix A(y) ∈ L∞ (Y , Rm×m ) is Y -periodic
symmetric-positive definite and the function n(y) ∈ L∞ (Y ) is a
positive Y -periodic function.
We have that as → 0
n := n(x/) → nh :=
1
|Y|
Z
n(y) dy weakly in L∞
Y
A := A(x/) → Ah in the sense of H-convergence
i.e. for u * u in H 1 (D) then A ∇u * Ah ∇u in [L2 (D)]m
We have that Ah is a constant symmetric matrix
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The case of an Isotropic media(i.e. A = I )
For this case (w , v ) ∈ L2 (D) × L2 (D) with u = v − w ∈ H02 (D),
where we have that the difference u satisfies for n := n(x/)
0 = ∆ + k 2 n
1
∆ + k 2 u in D.
n − 1
The equivalent variational form is given by
Z
1
∆u + k2 u ∆ϕ + k2 n ϕ dx = 0 for all ϕ ∈ H02 (D).
n − 1
D
We can rewriting the variational form to the following form
A,k (u , ϕ) − k2 B(u , ϕ) = 0 for all ϕ ∈ H02 (D).
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Proof: Sketch
Assume that nmin ≤ n(y) ≤ nmax for all y ∈ Y , let u be the
eigenfunction corresponding to the eigenvalue k with
ku kH 1 (D) = 1 then for all ≥ 0
αk∆u k2L2 (D) ≤ A,k (u , u ) = k2 B(u , u ) ≤ k2 ku k2H 1 (D)
provided that nmin > 1 or 0 < nmax < 1.
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Proof: Sketch
Assume that nmin ≤ n(y) ≤ nmax for all y ∈ Y , let u be the
eigenfunction corresponding to the eigenvalue k with
ku kH 1 (D) = 1 then for all ≥ 0
αk∆u k2L2 (D) ≤ A,k (u , u ) = k2 B(u , u ) ≤ k2 ku k2H 1 (D)
provided that nmin > 1 or 0 < nmax < 1.
Lemma (Cakoni-Haddar-IH. 2015)
Assume that either nmin > 1 or 0 < nmax < 1. There exists an
infinite sequence of real transmission eigenvalues k, j for j ∈ N
such that
kj (nmax , D) ≤ k, j < kj (nmin , D)
if
nmin > 1
kj (nmin , D) ≤ k, j < kj (nmax , D)
if
0 < nmax < 1
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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 nmin
> 1 or 0 < nmax < 1, then there is a
subsequence of k , (v , w ) ∈ R+ × L2 (D) × L2 (D) that
converges weakly to (v, w) ∈ L2 (D) × L2 (D) and k → k that
solves:
∆w + k 2 nh w = 0 and ∆v + k 2 v = 0
∂w
∂v
w = v and
=
∂ν
∂ν
provided that k is bounded
in
D
on
∂D
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Reconstructing Material Properties A = I
We can reconstruct the effective material property nh from finding
a n0 such that k1 (n0 ) = k1 (n ) where
∆w + k 2 n0 w = 0 and ∆v + k 2 v = 0
∂w
∂v
w = v and
=
∂ν
∂ν
in
D
on
∂D.
Now let
n(x/) = sin2 (2πx1 /) + 2
Table: Reconstruction from scattering data
k1 (n )
nh
n0
0.1
5.046
2.5
2.5188
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Convergence for the TEs A 6= 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
of
a subsequence
+
1
k , (v , w ) ∈ R × H (D) × H 1 (D) that converges weakly to
(v, w) ∈ H1 (D) × H1 (D) and k → k that solves:
∇ · Ah ∇w + k 2 nh w = 0 and ∆v + k 2 v = 0
∂w
∂v
w = v and
=
∂νAh
∂ν
in
D
on
∂D
provided that k is bounded. Recall that A(x/) → Ah in the sense
of H-convergence as → 0.
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Reconstructing Material Properties n = 1
We can reconstruct the effective material property Ah = ah I by
finding an a0 such that k1 (a0 ) = k1 (A ) where.
Now let
a0 ∆w + k 2 w = 0 and ∆v + k 2 v = 0 in D
∂w
∂v
=
on ∂D
w = v and a0
∂ν
∂ν
A(x/) =
1
3
2
sin (2πx2 /) + 1
0
0
cos2 (2πx1 /) + 1
Table: Reconstruction from scattering data
k1 (A )
ah
a0
0.1
7.349
0.5
0.4851
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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 1025 - 1049, Volume 9, Issue 4, 2015
(arXiv:1410.37297).
In the paper we have also considered:
The convergence of the interior transmission problem for the cases where
A = I and A 6= I
Construct a bulk corrector to prove strong convergence for A 6= I
Numerical test for the order of convergence
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A Current Project:
I. Harris
The interior transmission eigenvalue problem for an inhomogeneous media
with a conductive boundary.
Inverse Problems and Imaging (Submitted) - (arXiv:1510.01762)
Find k ∈ C and nontrivial (w , v ) ∈ L2 (D) × L2 (D) such that
v − w ∈ H 2 (D) ∩ H01 (D) satisfies
∆w + k 2 nw = 0
w −v =0
and
and
∆v + k 2 v = 0
∂w
∂v
−
= µv
∂ν
∂ν
in D
on ∂D.
Discreteness of the transmission eigenvalues
Existence of transmission eigenvalues
Monotonicity of the transmission eigenvalues
Determination of transmission eigenvalues from scattering data
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Thanks for your Attention:
Figure: Questions?
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