One-Dimensional Inverse Scattering: Reconstruction of Permittivity
The Factorization Method for Inverse Scattering Problemsaipc.tamu.edu/speakers/kirsch.pdfD. Colton,...
Transcript of The Factorization Method for Inverse Scattering Problemsaipc.tamu.edu/speakers/kirsch.pdfD. Colton,...
Karlsruhe Institute of Technology
KIT – University of the State of Baden-Wurttemberg and National Large-scale Research Center of the Helmholtz Association www.kit.edu
The Factorization Method forInverse Scattering ProblemsAndreas Kirsch AIP 2011, College Station
Department of Mathematics
Karlsruhe Institute of TechnologyOutline of the Talk
Introduction
The Direct Scattering Problem
The Inverse Scattering Problem
Reconstruction Techniques
Factorization of the Far Field Operator and Range Identity
Explicit Form of Characteristic Function
Connections to Linear Sampling Method and Time Reversal
Some Numerical Simulations
Final Remarks
The Factorization Method for Inverse Scattering Problems 2/30
Karlsruhe Institute of TechnologyIntroduction
uinc
us
D
General situation in scattering theory:
A given (“incident”) wave uinc
is disturbed by a medium Dand produces a scattered field us
Total field: u = uinc + us
The Factorization Method for Inverse Scattering Problems 3/30
Karlsruhe Institute of TechnologyIntroductionNecessary specification of:
model of wave propagation (scalar wave equation, Maxwell’sequations, Helmholtz equation, Navier’s equation, ...)incident wave uinc
form of scattering medium (impenetrable, penetrable, index ofrefraction)
In this talk fields uinc ,us,u : Rd → C are scalar functions (d = 2 ord = 3). Incident field uinc satisfies Helmholtz equation (reducedwave equation)
∆uinc(x) + k2uinc(x) = 0 in Rd
with wave number k > 0.Scattering medium is given by index of refraction n ∈ L∞(Rd) withn = 1 outside of bounded domain D ⊂ Rd . Total field u satisfies
∆u(x) + k2n(x) u(x) = 0 in Rd .
The Factorization Method for Inverse Scattering Problems 4/30
Karlsruhe Institute of TechnologyIntroduction
This model appears
as TM-mode for the time harmonic Maxwell system with d = 2and wave number k = ω
√ε0µ0 with frequency ω > 0 and index
of refraction n(x) = 1ε0
[ε(x) + i σ(x)/ω
]in acoustic wave progation with d = 2 or d = 3 and wavenumber k = ω/c0 where ω > 0 is frequency and c0 is speed ofsound in vacuum and index of refractionn(x) =
c20
c(x)2
[1 + i γ(x)/ω
]where γ is a damping coefficient.
This models penetrable media. Also possible: impenetrable media:
Helmholtz equation ∆u + k2u = 0 in Rd \ D and boundaryconditions as, e.g., u = 0 on ∂D or ∂u/∂ν = 0 on ∂D or∂u/∂ν + iλu = 0 on ∂D.
In addition, a radiation condition for the scattered field us = u − uinc
is necessary:
The Factorization Method for Inverse Scattering Problems 5/30
Karlsruhe Institute of TechnologyThe Direct Scattering ProblemGiven: wave number k > 0, incident field uinc , and refractive indexn ∈ L∞(Rd), determine u ∈ H1
loc(Rd) as a solution of
∆u(x) + k2n(x) u(x) = 0 in Rd ,
us = u − uinc satisfies Sommerfeld’s radiation condition (SRC):
∂us(x)
∂r− ik us(x) = O
(r−(d+1)/2) , r = |x | → ∞ ,
uniformly with respect to x = x/|x | ∈ Sd−1.Examples of incident waves:
(a) Plane wave of direction θ ∈ Sd−1: uinc(x) = eik θ·x , x ∈ Rd .
(b) Spherical wave with source point y ∈ Rd :
uinc(x) = Φ(x , y) :=
i4 H(1)
0 (k |x − y |) , d = 2,
exp(ik |x − y |)4π|x − y | , d = 3,
x 6= y .
The Factorization Method for Inverse Scattering Problems 6/30
Karlsruhe Institute of TechnologyThe Direct Scattering ProblemWriting equation for scattered wave yields
∆us + k2n us = −∆uinc − k2n uinc = −k2(n − 1)uinc in Rd ,
and us satisfies the SRC. This is special case of
∆v + k2n v = −f in Rd , and v satisfies SRC.
First uniqueness: Assume f = 0. Then:
o(1) =
∫|x |=R
∣∣∣∣∂v∂r− ikv
∣∣∣∣2 ds
=
∫|x |=R
∣∣∣∣∂v∂r
∣∣∣∣2 + k2|v |2ds − 2k Im∫
|x |=R
v∂v∂r
ds
The Factorization Method for Inverse Scattering Problems 7/30
Karlsruhe Institute of TechnologyThe Direct Scattering Problem
Green’s theorem yields∫|x |=R
v∂v∂r
ds =
∫∫|x |<R
[|∇v |2 + v ∆v
]dx =
∫∫|x |<R
[|∇v |2 − k2n|v |2
]dx
and thus
o(1) =
∫|x |=R
∣∣∣∣∂v∂r
∣∣∣∣2 + k2|v |2ds + 2k3∫∫D
(Im n)︸ ︷︷ ︸≥ 0
|v |2 dx
Therefore,∫|x |=R |v |
2ds −→ 0 as R →∞. Rellich’s Lemma yieldsv = 0 outside ball. Unique continuation yields v = 0 everywhere.
The Factorization Method for Inverse Scattering Problems 8/30
Karlsruhe Institute of TechnologyThe Direct Scattering ProblemExistence by integral equation approach:
Theorem: v ∈ H1loc(R
d) is solution of
∆v + k2n v = −f in Rd , and v satisfies SRC
if, and only if, v ∈ L2(D) solves Lippmann-Schwinger equation
v(x) =
∫∫D
[k2q(y) v(y) + f (y)
]Φ(x , y) dy , x ∈ D ,
where q = n − 1 is contrast. Asymptotic behavior of Φ(x , y) as|x | → ∞ yields:
v(x) = γdexp(ik |x |)|x |(d−1)/2
[v∞(x) + O(1/|x |)
], |x | → ∞ ,
uniformly with respect to x := x/|x | ∈ Sd−1 with far field pattern v∞.The Factorization Method for Inverse Scattering Problems 9/30
Karlsruhe Institute of TechnologyThe Inverse Scattering ProblemIf, in particular, us = us(x , θ) is scattered field corresponding toincident plane wave uinc(x) = exp(ik θ · x) then
us(x , θ) = γdexp(ik |x |)|x |(d−1)/2
[u∞(x , θ) + O(1/|x |)
], |x | → ∞ ,
with far field pattern u∞ = u∞(x , θ).Inverse scattering problem: u∞(x , θ) is known for all x , θ ∈ Sd−1,medium n or only its support D has to be determined.Example: Which domain D ⊂ R2 corresponds to the following farfields u∞(φ, θ), φ, θ ∈ [0,2π]?
Re u∞ Im u∞ Re u∞ Im u∞The Factorization Method for Inverse Scattering Problems 10/30
Karlsruhe Institute of TechnologyThe Inverse Scattering ProblemLeft example simple:Theorem of Karp: If u∞(x , θ
)= ψ(x · θ) for all x , θ ∈ Sd−1, then D
is a ball in Rd−1.This follows from uniqueness of inverse problem:Theorem (Nachman, Novikov, Ramm (all 1988, d = 3), Bukhgeim(2007, d = 2)) The far field patterns u∞(x , θ) determine nuniquely; that is, if nj ↔ u∞j (x , θ) for j = 1,2, then:
u∞1 (x , θ) = u∞2 (x , θ) for all x , θ ∈ Sd−1 =⇒ n1 = n2 .
Some literature on time harmonic (inverse) scattering theory:F. Cakoni, D. Colton: Qualitative Methods in Inverse ScatteringTheory. An Introduction. Springer, 2006.D. Colton, R. Kress: Inverse Acoustic and ElectromagneticScattering Theory. 2nd edition, Springer, 1998.A. Kirsch: Introduction to the Mathematical Theory of InverseProblems. Springer 1996, 2011.
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Karlsruhe Institute of TechnologyReconstruction Techniques(A) Linearization, e.g. Born approximation: Let n = 1 + q
∆u + k2(1 + q)u = 0 , u = uinc + us , us SRC
∆u + k2u = −k2qu i.e. ∆us + k2us = −k2qu ≈ −k2quinc
usB(x , θ) = k2
∫∫D
q(y) uinc(y , θ) Φ(x , y) dy , x ∈ Rd ,
u∞B (x , θ) = k2∫∫
Dq(y) uinc(y , θ) e−ik x ·ydy
= k2∫∫
Dq(y) eik(θ−x)·ydy = k2 q
(k(x − θ)
).
x , θ ∈ Sd−1; that is, k(x − θ) ∈ {z ∈ Rd : |z| ≤ 2k}.The Factorization Method for Inverse Scattering Problems 12/30
Karlsruhe Institute of TechnologyReconstruction Techniques
(B) Iterative methods to determine contrast function n: Definemapping T : L∞(D)→ L2(Sd−1 × Sd−1), n 7→ u∞. Applyiterative method to solve T (n) = f for n where f = f (x , θ) is given(measured) far field pattern.
Possible methods: Newton-type methods, gradient-type methods,second order methods.
Derivative: T ′(n)h = v∞ where v is radiating solution of∆v + k2nv = −k2hu. Derivative T ′(n) is compact and one-to-one!
Advantages: Very general, accurate, incorporation of a prioriinformation possible.
Disadvantages: “Expensive”, only local convergence is expected, norigorous convergence result known.
The Factorization Method for Inverse Scattering Problems 13/30
Karlsruhe Institute of TechnologyThe Far Field Operator(C) Factorization Method determines only support of q = 1− n!Values of n ∈ L∞(Rd) do not have to be known in advance.
Define far field operator F : L2(Sd−1)→ L2(Sd−1) by
(Fg)(x) =
∫Sd−1
u∞(x , θ) g(θ) ds(θ) , x ∈ Sd−1 .
Properties of F :F is compact.If n is real-valued then F is normal; that is, F ∗F = F F ∗, andeven: S := I + ik
2π F is unitary (=scattering matrix).F is one-to-one if k2 is not an interior transmission eigenvalue;
that is, ∆u + k2n u = 0 , ∆w + k2w = 0 in D ,
u = w on ∂D , ∂u/∂ν = ∂w/∂ν on ∂D ,
implies u = w = 0 in D.The Factorization Method for Inverse Scattering Problems 14/30
Karlsruhe Institute of TechnologyThe Factorization MethodFirst: Born approximation with far field operator FB; that is,
(FBg)(x) =
∫Sd−1
u∞B (x , θ) g(θ) ds(θ)
= k2∫
Sd−1
∫∫D
q(y) g(θ) eik(θ−x)·ydy ds(θ)
= k2∫∫
Dq(y) e−ik x ·y
∫Sd−1
g(θ) eik θ·y ds(θ︸ ︷︷ ︸=: (Hg)(y)
) dy
= H∗ TB Hg with TBf = k2q f .
Therefore, FB = H∗ TB H .
This is factorization of FB. Note: FB is self-adjoint for real q while Fis only normal!
The Factorization Method for Inverse Scattering Problems 15/30
Karlsruhe Institute of TechnologyThe Factorization MethodNow general nonlinear case:
Recall: u∞(·, θ) ←→ uinc(·, θ)
superposition: Fg ←→∫
Sd−1uinc(·, θ) g(θ) ds(θ) =: vg
Theorem: F has factorization F = H∗T H with
(Hg)(x) =
∫Sd−1
g(θ) eik θ·x ds(θ) , x ∈ Rd ,
(H∗ϕ)(x) =
∫∫Dϕ(y) e−ik x ·y dy , x ∈ Sd−1 ,
and Tf = k2q (f + k2v) where v solves
∆v + k2(1 + q)v = −q f , v satisfies SRC.
Compare to FB = H∗TB H with TBf = k2q f .
This is factorization, what’s the method?The Factorization Method for Inverse Scattering Problems 16/30
Karlsruhe Institute of TechnologyThe Factorization MethodTheorem: Let Rd \ D be connected. For any z ∈ Rd defineφz ∈ L2(Sd−1) by
φz(x) = e−ik x ·z , x ∈ Sd−1 .
Then z ∈ D if, and only if, φz ∈ R(H∗).
Proof: (H∗ϕ)(x) =
∫∫Dϕ(y) e−ik x ·y dy ?
= e−ik x ·z , x ∈ Sd−1 .
This is equivalent to (because complement of D is connected)
(∗)∫∫
Dϕ(y) Φ(x , y) dy = Φ(x , z) , x /∈ (D ∪ {z}) .
z ∈ D: Choose Φ ∈ C∞(Rd) with Φ(x) = Φ(x , z) outside of D anddefine ϕ by
∫∫D ϕ(y) Φ(·, y) dy = Φ in Rd ; that is, −ϕ = ∆Φ + k2Φ.
z /∈ D: (∗) can not have a solution! (Left hand side bounded, righthand side unbounded for x → z.)
The Factorization Method for Inverse Scattering Problems 17/30
Karlsruhe Institute of TechnologyRange IdentityRecall: F = H∗ T H and z ∈ D ⇔ φz ∈ R(H∗) .
Goal: Express range R(H∗) by known operator F !General situation:
X X
Y Y-
-
?
6
H H∗
T
F
Theorem: If T : X → X is selfadjoint and coercive; that is,
〈ψ,Tϕ〉 = 〈Tψ,Tϕ〉 , 〈ϕ,Tϕ〉 ≥ c‖ϕ‖2 for all ψ,ϕ ∈ X ,
then R(H∗) = R(F 1/2) .
The Factorization Method for Inverse Scattering Problems 18/30
Karlsruhe Institute of TechnologyRange IdentityTheorem: Let F = H∗T H : Y → Y be one-to-one and such thatI + ir F is unitary for some r > 0. Furthermore, let T : X ∗ → X becomp. perturb. of s.a. and coercive operator and Im〈ϕ,Tϕ〉 6= 0 for
all ϕ ∈ closureR(H) with ϕ 6= 0. Then R(H∗) = R((F ∗F )1/4
).
Idea of proof: I + irF unitary implies F normal and thusFψj = λjψj . Then F = H∗T H = |F |1/2S|F |1/2 where
|F |1/2ψ =∑
j
√|λj | 〈ψ,ψj〉Y ψj ,
Sψ =∑
j
λj
|λj |〈ψ,ψj〉Y ψj .
|〈Sψ,ψ〉| =∣∣∣∣∑j
λj
|λj |∣∣〈ψ,ψj〉Y
∣∣2∣∣∣∣≥ c‖ψ‖2Y
The Factorization Method for Inverse Scattering Problems 19/30
Karlsruhe Institute of TechnologyApplication to Scattering ProblemLet k2 be no int. transm. eigenvalue, q real, q(x) ≥ q0 on D.Recall: F = H∗T H and F is one-to-one and I + ik
2πF is unitaryand T : L2(D)→ L2(D), f 7→ k2q(f + v) is compact perturbation ofcoercive operator and and Im〈ϕ,Tϕ〉 > 0 for all
ϕ ∈ closureR(H), ϕ 6= 0. Thus R(H∗) = R((F ∗F )1/4
).
Combination of previous theorems:Theorem: Let again φz(x) = exp(−ik x · z), x ∈ Sd−1.Under above assumptions:
z ∈ D ⇐⇒ φz ∈ R((F ∗F )1/4)
Let {λj : j ∈ N} ⊂ C be eigenvalues of (normal!) operator F withnormalized eigenfunctions ψj ∈ L2(Sd−1) for j ∈ N. Then:
z ∈ D ⇐⇒∑j∈N
|〈φz , ψj〉L2 |2
|λj |<∞ ⇐⇒
∑j∈N
|〈φz , ψj〉L2 |2
|λj |
−1
> 0 .
The Factorization Method for Inverse Scattering Problems 20/30
Karlsruhe Institute of TechnologyMedia with AbsorptionNow n = 1 + q ∈ L∞(Rd) complex valued, Im n ≥ 0. StillF = H∗TF but not normal anymore. Define
Re F =12(F + F ∗) = H∗(Re T ) H
Im F =12i
(F − F ∗) = H∗(Im T ) H
F# =def |Re F | + Im F = H∗T H
with coercive T .
Theorem: z ∈ D ⇐⇒ φz ∈ R(F 1/2# )
Let {λj : j ∈ N} ⊂ R be eigenvalues of (selfadjoint!) operator F# withnormalized eigenfunctions ψj ∈ L2(Sd−1) for j ∈ N. Then:
z ∈ D ⇐⇒∑j∈N
|〈φz , ψj〉L2 |2
λj<∞ ⇐⇒
∑j∈N
|〈φz , ψj〉L2 |2
λj
−1
> 0 .
The Factorization Method for Inverse Scattering Problems 21/30
Karlsruhe Institute of TechnologyConnection to Linear Sampling MethodRecall Factorization Method:
z ∈ D ⇐⇒ (F ∗F )1/4h = φz solvable in L2(Sd−1)
Linear Sampling Method:
Solve (approximately): Fg = φz in L2(Sd−1)
If Fg = φz solvable then (F ∗F )1/4S(F ∗F )1/4g = φz , thus z ∈ D.For example, apply Tikhonov regularization to Fg = φz :
(εI + F ∗F ) gz,ε = F ∗φz
for z ∈ Rd and ε > 0. Let vg be Herglotz function with kernel g.Then (Arens, Lechleiter 2004, 2007):
(a) x ∈ D: c ‖hz‖2L2(S2) ≤ limε→0
∣∣vgz,ε(z)∣∣ ≤ ‖hz‖2L2(S2)
(b) x /∈ D: limε→0|vgz,ε(z)| =∞
The Factorization Method for Inverse Scattering Problems 22/30
Karlsruhe Institute of TechnologyConnection to Time Reversal
Recall: Fg ←→ vg(x) =
∫Sd−1
g(θ) eikx ·θds(θ)
Define (Rg)(x) = g(−x). Then (Hazard, Ramdani 2004):
g F7→ Fg R7→ RFg F7→ FRFg R7→ RFR︸ ︷︷ ︸= F∗
Fg = F ∗Fg
Focusing: Choose (largest) eigenvalue λ of F and correspondingeigenfunction ψ and plot |vψ(z)|2 =
∣∣(ψ, φz)L2
∣∣2.
Note: vψ(z) =4πk2
λ
∫∫D
q(y) uψ(y) j0(k |z − y |) dy , z ∈ R3 .
Factorization Method: Plot w(z) =
∑j∈N
∣∣(ψj , φz)L2
∣∣2|λj |
−1
The Factorization Method for Inverse Scattering Problems 23/30
Karlsruhe Institute of TechnologyNumerical SimulationsRecall:
z ∈ D ⇐⇒∑j∈N
|〈φz , ψj〉L2 |2
|λj |< ∞
⇐⇒ w(z) =
∑j∈N
|〈φz , ψj〉L2 |2
|λj |
−1
> 0 .
Therefore, sign(w) is the characteristic function of D!
The following examples show plots of
wN(z) =
N∑j=1
|〈φz , ψj〉|2
|λj |
−1
, z ∈ R2 :
for N = 32 or N = 36, respectively.The Factorization Method for Inverse Scattering Problems 24/30
Karlsruhe Institute of TechnologyNumerical Simulations
Dirichlet boundary conditions:
The Factorization Method for Inverse Scattering Problems 25/30
Karlsruhe Institute of TechnologyNumerical Simulations
Dirichlet boundary conditions:
The Factorization Method for Inverse Scattering Problems 26/30
Karlsruhe Institute of TechnologyNumerical Simulations
Scattering by an open arc:
The Factorization Method for Inverse Scattering Problems 27/30
Karlsruhe Institute of TechnologyNumerical Simulations
Real data:
The Factorization Method for Inverse Scattering Problems 28/30
Karlsruhe Institute of TechnologyNumerical Simulations3D-Example (joint work with A, Kleefeld): Scattering underconductive transmission conitions
∆u + k2u = 0 in R3 \ ∂D ,
u+ = u− ,∂u+
∂ν− ∂u−
∂ν= λ u on ∂D .
The Factorization Method for Inverse Scattering Problems 29/30
Karlsruhe Institute of TechnologyFinal Remarks
Rigorous justification of Factorization Method for:
Scattering by anisotropic media: ∇ · (A∇u) + k2u = 0
Scattering of electromagnetic and elastic waves
Scattering by arcs, periodic structures, in wave guides
Nonlinear Helmholtz equation (Lechleiter, Minisymposium M04)
Tomography (Bruhl, Hanke, Hyvonen, Lechleiter, von Harrach,Griesmaier)
References:
A. Kirsch, N. Grinberg: The Factorization Method for InverseProblems. Oxford University Press, 2008.
A. Kirsch: Introduction to the Mathematical Theory of InverseProblems. Springer, 2nd edition, 2011.
Thank you for your attention!
The Factorization Method for Inverse Scattering Problems 30/30