chesnel/Documents/IHPChesnel.pdf · Thecaseθ inc = θ n I Inthepreviousapproach,weneededtoassume...

Séminaire “Problèmes spectraux en physique mathématique”

Construction of invisible defects for acousticproblems with a finite number of measurements

Lucas Chesnel1

Coll. with A. Bera2, A.-S. Bonnet-Ben Dhia2 and S.A. Nazarov3.

1Defi team, CMAP, École Polytechnique, France2Poems team, Ensta ParisTech, France3FMM, St. Petersburg State University, Russia

Institut Henri-Poincaré, Paris, 09/05/2016

General settingI We are interested in methods based on the propagation of waves todetermine the shape, the physical properties of objects, in an exact orqualitative manner, from given measurements.

I General principle of the methods:i) send waves in the medium;ii) measure the scattered field;iii) deduce information on the structure.

• Many techniques: Xray, ultrasound imaging, seismic tomography, ...• Many applications: biomedical imaging, non destructive testing ofmaterials, geophysics, ...

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Goal of the talk

I The goal of imaging techniques is to find features of the structure fromthe knowledge of measurements.

I In this talk, we are interested in questions of invisibility when one has afinite number of measurements .

- Less ambitious than usual cloaking and therefore, more accessible.- Also relevant for applications.

I We will consider two types of problems:

1 Scattering in free space

2 Scattering in waveguides

I At least two reasons to study invisibility questions:- We can wish to hide objects.- It allows to understand limits of imaging techniques.

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Goal of the talk

3 / 37

Goal of the talk

3 / 37

Goal of the talk

3 / 37

Outline of the talk

1 Invisibility in free space

The general scheme

The forbidden case

Numerical experiments

2 Invisibility for waveguide problems

Construction of invisible penetrable defects

Can one hide a small Dirichlet obstacle?

Can one hide a perturbation of the wall?

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The general scheme

The forbidden case

5 / 37

Model problemI Scattering in time-harmonic regime of an incident plane wave by abounded penetrable inclusion D (coefficients ρ) in R2.

ui := eikθinc·x (incident dir. θinc ∈ S1)

ρ = 1 Dρ 6= 1

Find u such that−∆u = k2ρ u in R2,

u = ui + us in R2,

limr→+∞

(∂us

∂r− ikus

Definition: ui = incident field (data)u = total field (uniquely defined by (1))us = scattered field (uniquely defined by (1)).

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ρ = 1 Dρ 6= 1

u = ui + us in R2,

limr→+∞

(∂us

∂r− ikus

6 / 37

ρ = 1 Dρ 6= 1

u = ui + us in R2,

limr→+∞

(∂us

∂r− ikus

6 / 37

Far field pattern

<e ui <e u <e us

I The scattered field of an incident plane wave of direction θinc behaves ineach direction like a cylindrical wave at infinity:

us(x,θinc) = eikr

(u∞s (θsca,θinc) +O(1/r)

Definition: The map u∞s (·, ·) : S1×S1 → C is called the far field pattern.

At infinity, one measures the far field pattern (other terms are too small).

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Far field pattern

<e ui <e u <e us

us(x,θinc) = eikr

7 / 37

Far field pattern

<e ui <e u <e us

us(x,θinc) = eikr

7 / 37

Far field pattern

<e ui <e u <e us

us(x,θinc) = eikr

At infinity, one measures the far field pattern (other terms are too small).7 / 37

SettingI The goal of imaging techniques is to find features of the inclusion fromthe knowledge of u∞s (·, ·) on a subset of S1 × S1.

– In literature, most of the techniques require a continuum of data.(Nachman, 1988, Sylvester & Uhlmann, 1987, Bukhgeim, 2008, Imanuvilov &Yamamoto, 2012)

– In practice, one has a finite number of emitters and receivers.

I To simplify the presentation, only one incident direction θinc and Nscattering directions θ1, . . . ,θN (given).

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

GOALWe explain how to construct inclusions such that

u∞s (θ1) = · · · = u∞s (θN ) = 0.

I These inclusions cannot be detected from far field measurements.I We assume that k and the support of the inclusion D are given.

Find a real valued function ρ 6≡ 1, with ρ− 1 supported in D, suchthat the solution to the problem

Find u = us + eikθinc·x such that−∆u = k2ρ u in R2,

us is outgoing

satisfies u∞s (θ1) = · · · = u∞s (θN ) = 0.

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SettingI The goal of imaging techniques is to find features of the inclusion fromthe knowledge of u∞s (·, ·) on a subset of S1 × S1.

– In literature, most of the techniques require a continuum of data.(Nachman, 1988, Sylvester & Uhlmann, 1987, Bukhgeim, 2008, Imanuvilov &Yamamoto, 2012)

– In practice, one has a finite number of emitters and receivers.

I To simplify the presentation, only one incident direction θinc and Nscattering directions θ1, . . . ,θN (given).

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

u∞s (θ1) = · · · = u∞s (θN ) = 0.

us is outgoing

8 / 37

SettingI To simplify the presentation, only one incident direction θinc and Nscattering directions θ1, . . . ,θN (given).

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

u∞s (θ1) = · · · = u∞s (θN ) = 0.

us is outgoing

8 / 37

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

→ We measure u∞s (θ1), . . . , u∞s (θN ).

u∞s (θ1) = · · · = u∞s (θN ) = 0.

us is outgoing

8 / 37

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

u∞s (θ1) = · · · = u∞s (θN ) = 0.

us is outgoing

8 / 37

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

u∞s (θ1) = · · · = u∞s (θN ) = 0.

I These inclusions cannot be detected from far field measurements.

I We assume that k and the support of the inclusion D are given.

us is outgoing

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−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

u∞s (θ1) = · · · = u∞s (θN ) = 0.

us is outgoing

8 / 37

−→θinc

−→θ2

−→θ3

−→θ1

−→θ4

u∞s (θ1) = · · · = u∞s (θN ) = 0.

us is outgoing

8 / 37

Sketch of the method

I We will work as in the proof of the implicit functions theorem.

The idea was used in Nazarov 11 to construct waveguides for whichthere are embedded eigenvalues in the continuous spectrum.

(N complex measurements ⇒ 2N real measurements)

Assume that dF (0) : L∞(D)→ R2N is onto.

∃µ0, µ1, . . . , µ2N ∈ L∞(D) s.t. dF (0)(µ0) = 0[dF (0)(µ1), . . . , dF (0)(µ2N )] = Id2N .

I Take µ = µ0 +2N∑

n=1τnµn where the τn are real parameters to set:

0 = F (εµ) ⇔ where ~τ = (τ1, . . . , τ2N )>

andGε(~τ) = −εF ε(µ).

If Gε is a contraction, the fixed-point equation has a unique solution ~τ sol.

Set σsol := εµsol. We have F (σsol) = 0 (invisible inclusion).

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Sketch of the methodI Define σ = ρ− 1 and gather the measurements in the vector

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

I We look for small perturbations of the reference medium: σ = εµ whereε > 0 is a small parameter and where µ has be to determined.Assume that dF (0) : L∞(D)→ R2N is onto.

0 = F (εµ) ⇔

0 = ε

2N∑n=1

τndF (0)(µn) + ε2F ε(µ)

where ~τ = (τ1, . . . , τ2N )>

and Gε(~τ) = −εF ε(µ).

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F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

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F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

I No obstacle leads to null measurements ⇒ F (0) = 0.

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

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F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

Our goal: to find σ ∈ L∞(D) such that F (σ) = 0 (with σ 6≡ 0).

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

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F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

I We look for small perturbations of the reference medium: σ = εµ whereε > 0 is a small parameter and where µ has be to determined.

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

I Taylor: F (εµ) = F (0) + εdF (0)(µ) + ε2F ε(µ)

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

I Taylor: F (εµ) = εdF (0)(µ) + ε2F ε(µ).

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔

0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔

0 = ε2N∑

n=1τndF (0)(µn) + ε2F ε(µ)

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ 0 = ε

2N∑n=1

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ 0 = ε~τ + ε2F ε(µ)

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ 0 = ε~τ + ε2F ε(µ)

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ ~τ = Gε(~τ)

where ~τ = (τ1, . . . , τ2N )>

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ ~τ = Gε(~τ)

where ~τ = (τ1, . . . , τ2N )> and Gε(~τ) = −εF ε(µ).

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ ~τ = Gε(~τ)

9 / 37

F (σ) = (F1(σ), . . . , F2N (σ))> ∈ R2N .

0 = F (εµ) ⇔ ~τ = Gε(~τ)

If Gε is a contraction, the fixed-point equation has a unique solution ~τ sol.Set σsol := εµsol. We have F (σsol) = 0 (invisible inclusion).

9 / 37

Calculus of dF (0) 1/2I For our problem, we have (σ = ρ− 1)

F (σ) = (<e u∞s (θ1), . . . ,<e u∞s (θN ),=mu∞s (θ1), . . . ,=mu∞s (θN )).

To compute dF (0)(µ), we take ρε = 1 + εµ with µ supported in D.

I We denote uε, uεs the functions satisfying

Find uε = uεs + eikθinc·x,with uε

s outgoing, such that−∆uε = k2ρε uε in R2.

uε∞s (θn) =

• We can prove that uεs = O(ε).

I We obtain the expansion (Born approx.), for small ε

uε∞s (θn) = 0 + ε c k2

∫Dµ eik(θinc−θn)·x dx +O(ε2).

dF (0)(µ) =(∫Dµ cos(k(θinc − θ1)·x) dx, . . . ,

∫Dµ cos(k(θinc − θN )·x) dx,∫

Dµ sin(k(θinc − θ1)·x) dx, . . . ,

∫Dµ sin(k(θinc − θN )·x) dx

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uε∞s (θn) =

uε∞s (θn) = 0 + ε c k2

10 / 37

uε∞s (θn) =

uε∞s (θn) = 0 + ε c k2

10 / 37

• uε∞s (θn) =c k2

(ρε − 1) (ui + uεs) e−ikθn·x dx

eiπ/4√

uε∞s (θn) = 0 + ε c k2

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• uε∞s (θn) =c k2

(ρε − 1) (ui + uεs) e−ikθn·x dx.

uε∞s (θn) = 0 + ε c k2

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• uε∞s (θn) = ε c k2

∫Dµ (ui + uε

s) e−ikθn·x dx.

uε∞s (θn) = 0 + ε c k2

10 / 37

∫Dµ (ui + uε

s) e−ikθn·x dx.

uε∞s (θn) = 0 + ε c k2

10 / 37

∫Dµui e

−ikθn·x dx+O(ε2).

uε∞s (θn) = 0 + ε c k2

10 / 37

∫Dµui e

uε∞s (θn) = 0 + ε c k2

10 / 37

F (σ) = (<e u∞s (θ1)ck2 , . . . ,<e u

∞s (θN )ck2 ,=m u∞s (θ1)

ck2 , . . . ,=m u∞s (θN )ck2 ).

∫Dµui e

uε∞s (θn) = 0 + ε c k2

10 / 37

F (σ) = (<e u∞s (θ1)ck2 , . . . ,<e u

∞s (θN )ck2 ,=m u∞s (θ1)

ck2 , . . . ,=m u∞s (θN )ck2 ).

∫Dµui e

uε∞s (θn) = 0 + ε c k2

)10 / 37

Calculus of dF (0) 2/2

Is dF (0) : L∞(D)→ R2N onto

I Clearly, we need to avoid the configuration θinc − θn = 0.

θinc θn = θinc

Emitter Receiver

I And one can prove that if θn 6= θinc , n = 1, . . . , N , the answer is yes.

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θinc θn = θinc

Emitter Receiver

11 / 37

θinc θn = θinc

Emitter Receiver

11 / 37

θinc θn = θinc

Emitter Receiver

I And one can prove that if θn 6= θinc , n = 1, . . . , N , the answer is yes.11 / 37

Main result

Proposition: Assume that θn 6= θinc for n = 1, . . . , N . For ε smallenough, define ρ sol = 1 + εµ sol with

µ sol = µ0 +2N∑

n=1τ sol

n µn.

Then the solution of the scattering problem

Find uε = uεs + eikθinc·x such that

−∆u = k2ρ sol u in R2,

us is outgoing

Comments:→ We need ε to be small enough to prove that Gε is a contraction.→ We have µ sol 6≡ 0 (non trivial inclusion). To see it, compute dF (0)(µ sol).

→ We have τ sol = O(ε) ⇒ µ sol ≈ µ0. We control the main form of the defect.

Existence of invisible inclusions may appear not so surprising since there are2N measurements and ρ ∈ L∞(D). Let us see the case θn = θinc ...

12 / 37

Main result

µ sol = µ0 +2N∑

n=1τ sol

n µn.

us is outgoing

12 / 37

Main result

µ sol = µ0 +2N∑

n=1τ sol

n µn.

us is outgoing

Existence of invisible inclusions may appear not so surprising since there are2N measurements and ρ ∈ L∞(D).

Let us see the case θn = θinc ...

12 / 37

Main result

µ sol = µ0 +2N∑

n=1τ sol

n µn.

us is outgoing

12 / 37

The general scheme

The forbidden case

13 / 37

The case θinc = θn

I In the previous approach, we needed to assume θn 6= θinc , n = 1, . . . , N .What if θn = θinc?

θinc θn = θinc

Emitter Receiver

I There holds

u∞s (θn) = c k2∫D

(ρ− 1) (ui + us) e−ikθn·x dx.

I This allows to prove the optical theorem (already obtained by Rayleigh)

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

Imposing invisibility in the direction θinc requires to impose invisi-bility in all directions θ ∈ S1! (not only for small but for all defects)

Rellich’s lemma ⇒ us ≡ 0 in R2 \ D. An incident plane wave whichproduces a scattered field null outside the defect... Is it possible?

• No solution if D has corners and under certain assumptions on ρ.

- Corners always scatter, E. Blåsten, L. Päivärinta, J. Sylvester, 2014

- Corners and edges always scatter, J. Elschner, G. Hu, 2015

• And if D is smooth? ⇒ The problem seems open.

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θinc θn = θinc

Emitter Receiver

I There holds

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

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θinc θn = θinc

Emitter Receiver

I There holds

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

14 / 37

θinc θn = θinc

Emitter Receiver

I There holds

u∞s (θinc) = c k2∫D

(ρ− 1) (ui + us)ui dx.

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

14 / 37

θinc θn = θinc

Emitter Receiver

I There holds

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

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θinc θn = θinc

Emitter Receiver

I There holds

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

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θinc θn = θinc

Emitter Receiver

I There holds

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

14 / 37

θinc θn = θinc

Emitter Receiver

I There holds

=m (c−1 u∞s (θinc)) = k

∫S1|u∞s (θ)|2 dθ.

14 / 37

The general scheme

The forbidden case

15 / 37

Data and algorithm

I We can solve the fixed point problem using an iterative procedure: weset ~τ 0 = (0, . . . , 0)> then define

~τ n+1 = Gε(~τ n).

I At each step, we solve a scattering problem. We use a P2 finite elementmethod set on the ball B8. On ∂B8, a truncated Dirichlet-to-Neumann mapwith 13 harmonics serves as a transparent boundary condition.

I For the numerical experiments, we take D = B1, M = 3 (3 directions ofobservation) and

θinc = (cos(ψinc), sin(ψinc)), ψinc = 0

θ1 = (cos(ψ1), sin(ψ1)), ψ1 = 90

θ2 = (cos(ψ2), sin(ψ2)), ψ2 = 180

θ3 = (cos(ψ3), sin(ψ3)), ψ3 = 225

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Results: coefficient ρ at the end of the process

17 / 37

Results: scattered field

Figure: |us| at the end of the fixed point procedure in logarithmic scale. Asdesired, we see it is very small far from D in the directions corresponding tothe angles 90, 180 and 225. The domain is equal to B8.

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Results: far field pattern

0 50 100 150 200 250 300 3500

Far field pattern at iteration 0Far field pattern at the end of the fixed point procedure

Figure: The dotted lines show the directions where we want u∞s to vanish.19 / 37

The general scheme

The forbidden case

20 / 37

Waveguide problemI Scattering in time-harmonic regime of an incident plane wave by abounded penetrable inclusion D (coefficients ρ) in Ω := (x, y) ∈ R× (0; 1).

ρ = 1D

ρ 6= 1

+`−`

Find u = ui + us s. t.−∆u = k2ρ u in Ω,∂nu = 0 on ∂Ω,

us is outgoing.

I Set k ∈ (0;π), w± = e±ikx/√

2k and ui = w−.

I us is outgoing means that there are some s± ∈ C such that

us = χ+s+w+ + χ−s−w− + us,

where us is exponentially decaying at ±∞.(χ± are smooth cut-off functions s.t. χ± = 1 for ±x ≥ 2`, χ± = 0 for ±x ≤ `)

I Conservation of energy implies |s−|2 + |1 + s+|2 = 1.

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ρ = 1D

ρ 6= 1

+`−`

us is outgoing.

I Set k ∈ (0;π), w± = e±ikx/√

2k and ui = w−.

us = χ+s+w+ + χ−s−w− + us,

21 / 37

ρ = 1D

ρ 6= 1

+`−`

us is outgoing.

I Set k ∈ (0;π), w± = e±ikx/√

2k and ui = w−.

us = χ+s+w+ + χ−s−w− + us,

21 / 37

ρ = 1D

ρ 6= 1

+`−`

us is outgoing.

I Set k ∈ (0;π), w± = e±ikx/√

2k and ui = w−.

us = χ+s+w+ + χ−s−w− + us,

I Conservation of energy implies |s−|2 + |1 + s+|2 = 1.21 / 37

Invisibility for waveguides

us = χ+s+w+ + χ−s−w− + us,

where us is exponentially decaying at ±∞.

At infinity, one measures the reflection coefficient s− and/or the transmissioncoefficient 1 + s+ (other terms are too small).

Definition: Inclusion is said non reflective if s− = 0completely invisible if s+ = 0.

I Due to conservation of energy |s−|2 + |1 + s+|2 = 1,

s+ = 0 ⇒ s− = 0 (and us is expo. decay. at ±∞).

The converse is wrong (s− = 0 6⇒ s+ = 0).

22 / 37

us = χ+s+w+ + χ−s−w− + us,

The converse is wrong (s− = 0 6⇒ s+ = 0).

22 / 37

us = χ+s+w+ + χ−s−w− + us,

The converse is wrong (s− = 0 6⇒ s+ = 0).22 / 37

Penetrable inclusionI For σ = ρ− 1 (contrast), gather the measurements in

F (σ) = (<e (s−/(ik)),=m (s−/(ik)),<e (s+/(ik)),=m (s+/(ik))).

Again, we wish to find σ 6≡ 0 such that F (σ) = 0.

I As in free space, we obtain

dF (0)(µ) =(∫Dµ cos(2kx) dx,

∫Dµ sin(2kx) dx,

∫Dµdx, 0

I We can get s− = 0 but dF (0) : L∞(D)→ R4 is not onto .

Can we get s+ = 0 or

ui s+ = 0

Waveguide˜ θinc u∞s (θinc) = 0

Free space

Impose s− = 0=ms+ = 0 . Then, |s

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

With this approach, we produce small contrast invisible perturbations ofthe reference medium.

Can we increase the perturbation to obtain larger defects

23 / 37

∫Dµ sin(2kx) dx,

∫Dµdx, 0

ui s+ = 0

Free space

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

23 / 37

∫Dµ sin(2kx) dx,

∫Dµdx, 0

ui s+ = 0

Free space

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

23 / 37

∫Dµ sin(2kx) dx,

∫Dµdx, 0

ui s+ = 0

Free space

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

23 / 37

∫Dµ sin(2kx) dx,

∫Dµdx, 0

ui s+ = 0

Free space

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

23 / 37

∫Dµ sin(2kx) dx,

∫Dµdx, 0

ui s+ = 0

Free space

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

23 / 37

∫Dµ sin(2kx) dx,

∫Dµdx, 0

ui s+ = 0

Free space

−|2 + |1 + s+|2 = 1s+ = O(ε) ⇒ s+ = 0 .

23 / 37

Can we increase the perturbation?

I Schematic view of what we did (F : R2 → R is the measurements map):

F (σ) = 0

F (εµ0)

F (εµsol)

Can we reiterate the process

24 / 37

F (σ) = 0

F (εµ0)

F (εµsol)

24 / 37

F (σ) = 0

F (εµ0)

F (εµsol)

24 / 37

F (σ) = 0

F (εµ0)

F (εµsol)

24 / 37

I Schematic view of the process to construct larger invisible defects:

F (σ) = 0

F (εµ0)

F (εµsol)

24 / 37

F (σ) = 0

F (εµ0)

F (εµsol)

24 / 37

F (σ) = 0

F (εµ0)

F (εµsol)

24 / 37

Numerical results to impose s− = 0I We set k = 3, D = (−π/k;π/k)× (1/4; 3/4), 3 steps of iterations.

σ = ρ− 1

→ First results are encouraging. Still some questions: at each step, how tochoose the new directions?→ We are not able to prove that ds−(σ) : L∞(D)→ C is onto for σ 6≡ 0.

25 / 37

σ = ρ− 1

25 / 37

σ = ρ− 1

25 / 37

The general scheme

The forbidden case

26 / 37

Small Dirichlet obstacleI Can one hide a small Dirichlet obstacle centered at M1?

Find u = ui + us s. t.−∆u = k2u in Ωε := Ω \ Oε

u = 0 on ∂Ωε,us is outgoing.

I Again, us is outgoing means that there are some s± ∈ C such that

us = χ+s+w+ + χ−s−w− + us, with us is expo. decaying at ±∞.

Due to Dirichlet B.C., w± are not the same as previously (but this notimportant).I In 3D, we obtain

s− = 0 + ε (4iπ cap(O)w+(M1)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)|w+(M1)|2) +O(ε2).

Non zero terms!(cap(O) > 0)

⇒ One single small obstacle cannot even be non reflective.

27 / 37

Due to Dirichlet B.C., w± are not the same as previously (but this notimportant).

I In 3D, we obtain

s− = 0 + ε (4iπ cap(O)w+(M1)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)|w+(M1)|2) +O(ε2).

27 / 37

I In 3D, we obtain

s− = 0 + ε (4iπ cap(O)w+(M1)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)|w+(M1)|2) +O(ε2).

27 / 37

I In 3D, we obtain

s− = 0 + ε (4iπ cap(O)w+(M1)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)|w+(M1)|2) +O(ε2).Non zero terms!

(cap(O) > 0)

27 / 37

I In 3D, we obtain

s− = 0 + ε (4iπ cap(O)w+(M1)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)|w+(M1)|2) +O(ε2).Non zero terms!

(cap(O) > 0)

⇒ One single small obstacle cannot even be non reflective.27 / 37

Small Dirichlet obstacles

I Let us try with two small Dirichletobstacles centered at M1, M2.

Oε1Oε

I We obtain s− = 0 + ε (4iπ cap(O)2∑

n=1w+(Mn)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)2∑

n=1|w+(Mn)|2) +O(ε2).

We can findM1, M2 such that s− = O(ε2). Then moving Oε1 fromM1 to

M1 + ετ , and choosing a good τ ∈ R3 (fixed point), we can get s− = 0 .

Comments:→ Hard part is to justify the asymptotics for the fixed point problem.→ We cannot impose s+ = 0 with this strategy.→ When there are more propagative waves, we need more obstacles.

Acting as a team, flies can become invisible!

28 / 37

Oε1Oε

n=1w+(Mn)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)2∑

n=1|w+(Mn)|2) +O(ε2).

28 / 37

Oε1Oε

n=1w+(Mn)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)2∑

n=1|w+(Mn)|2) +O(ε2).

We can findM1, M2 such that s− = O(ε2).

Then moving Oε1 fromM1 to

28 / 37

Oε1Oε

n=1w+(Mn)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)2∑

n=1|w+(Mn)|2) +O(ε2).

28 / 37

Oε1Oε

n=1w+(Mn)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)2∑

n=1|w+(Mn)|2) +O(ε2).

28 / 37

Oε1Oε

n=1w+(Mn)2) +O(ε2)

s+ = 0 + ε (4iπ cap(O)2∑

n=1|w+(Mn)|2) +O(ε2).

Acting as a team, flies can become invisible!28 / 37

The general scheme

The forbidden case

29 / 37

Can one hide a perturbation of the wall?I Pick h ∈ C∞0 (−`; `), ` > 0. Set k ∈ (0;π), w± = e±ikx/

√2k, ui = w−.

1 + εh(x)

−` ` Find u = ui + us s. t.−∆u = k2u in Ωε,∂nu = 0 on ∂Ωε,

us is outgoing.

I We obtain

s− = 0 + ε

∂xh(x)(w+(x))2 dx

)+O(ε2)

s+ = 0 + ε 0 +O(ε2).

⇒ With this approach, we can impose s− = 0 but not s+ = 0 .

30 / 37

√2k, ui = w−.

1 + εh(x)

us is outgoing.

I We obtain

s− = 0 + ε

∂xh(x)(w+(x))2 dx

)+O(ε2)

s+ = 0 + ε 0 +O(ε2).

⇒ With this approach, we can impose s− = 0 but not s+ = 0 .

30 / 37

√2k, ui = w−.

1 + εh(x)

us is outgoing.

I We obtain

s− = 0 + ε

∂xh(x)(w+(x))2 dx

)+O(ε2)

s+ = 0 + ε 0 +O(ε2).

⇒ With this approach, we can impose s− = 0 but not s+ = 0 .30 / 37

I More generally, for any Neumann wave-guide, one can show that s+ = 0 implies∫

Ω|∇us|2 − k2|us|2 dx = 0.

Ωc` Ω`

−` `

• Decomposing in Fourier series, one finds∫Ωc

|∇us|2 − k2|us|2 dx ≥ 0.

• Note that s+ = 0⇒ us ∈ Y := v ∈ H1(Ω`) |∫

x=±`v dσ = 0. Define

λ† := infv∈Y\0

(∫Ω`

|∇v|2 dx)/(∫

|v|2 dx)> 0.

Proposition: For a given shape, s+ = 0 cannot hold for k2 ∈ (0;λ†).

→ To impose invisibility at low frequency, we need to work with special shapes.

31 / 37

Ω|∇us|2 − k2|us|2 dx = 0.

ΩΩc` Ω`

−` `

|∇us|2 − k2|us|2 dx ≥ 0.

λ† := infv∈Y\0

(∫Ω`

|∇v|2 dx)/(∫

|v|2 dx)> 0.

31 / 37

Ω|∇us|2 − k2|us|2 dx = 0.

ΩΩc` Ω`

−` `

|∇us|2 − k2|us|2 dx ≥ 0.

λ† := infv∈Y\0

(∫Ω`

|∇v|2 dx)/(∫

|v|2 dx)> 0.

31 / 37

Ω|∇us|2 − k2|us|2 dx = 0.

ΩΩc` Ω`

−` `

|∇us|2 − k2|us|2 dx ≥ 0.

λ† := infv∈Y\0

(∫Ω`

|∇v|2 dx)/(∫

|v|2 dx)> 0.

→ To impose invisibility at low frequency, we need to work with special shapes.31 / 37

Non smooth perturbation of the wallI We study the same problem in the geometry Ωε

h1h2 h3

M1 M2 M3

I We obtain s− = 0 + ε(ik∑3

n=1(w+(Mn))2 tan(khn))

+O(ε2)

s+ = 0 + ε(i/2∑3

n=1 tan(khn))

+O(ε2)

1) We can find Mn, hn such that s− = O(ε2) and s+ = O(ε2) .2) Then changing hn into hn + τn, and choosing a good τ = (τ1, τ2, τ3) ∈ R3

(fixed point), we can get s− = 0 and =ms+ = 0 .3) Energy conservation + [s+ = O(ε)] ⇒ s+ = 0 .

32 / 37

h1h2 h3

M1 M2 M3

+O(ε2)

s+ = 0 + ε(i/2∑3

n=1 tan(khn))

+O(ε2)

1) We can find Mn, hn such that s− = O(ε2) and s+ = O(ε2) .

2) Then changing hn into hn + τn, and choosing a good τ = (τ1, τ2, τ3) ∈ R3

32 / 37

h1h2 h3

M1 M2 M3

+O(ε2)

s+ = 0 + ε(i/2∑3

n=1 tan(khn))

+O(ε2)

(fixed point), we can get s− = 0 and =ms+ = 0 .

3) Energy conservation + [s+ = O(ε)] ⇒ s+ = 0 .

32 / 37

h1h2 h3

M1 M2 M3

+O(ε2)

s+ = 0 + ε(i/2∑3

n=1 tan(khn))

+O(ε2)

32 / 37

Numerical results

33 / 37

RemarkI We could also have worked with gardens of flowers!

34 / 37

The general scheme

The forbidden case

35 / 37

Conclusion

What we did

♠ We explained how to construct invisible perturbations of a referencesituation in a setting with a finite number of measurements.

Future work

1) We want to continue the analysis of the reiteration process toconstruct large invisible defects of the reference medium.

2) It would be interesting to consider other models (Maxwell, elasticity,...) and to investigate cases where the differential is not onto.

3) For a given perturbation, can we study the frequencies (invisiblemodes) such that invisibility holds?

4) We wish to better understand the link between the invisible modesand the so-called trapped modes in waveguides.

36 / 37

Thank you for your attention!!!

37 / 37

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