Neumann-Poincar e type operators, super-resolution, and - - PowerPoint PPT Presentation

. . . . . .

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility

Habib Ammari Department of Mathematics and Applications Ecole Normale Sup´ erieure, Paris

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Neumann-Poincar´ e operator

• Γ(x) =

1 2π log |x|: fundamental solution of ∆ in R2;

• D: bounded, smooth (C1,α, 0 < α < 1) domain in R2; ν: outward normal

at ∂D;

• Neumann-Poincar´

e operator (convolution with ∂Γ(x − y)/∂ν(x)) : K∗

D[ϕ](x) := 1

2π ∫

∂D

(x − y) · ν(x) |x − y|2 ϕ(y) dσ(y),

• Adjoint:

KD[ϕ](x) := 1 2π ∫

∂D

(y − x) · ν(y) |x − y|2 ϕ(y) dσ(y).

• KD, K∗

D : L2(∂D) → L2(∂D) compact. Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Neumann-Poincar´ e operator

• Kellog’s result (spectrum of K∗

D lies in ] − 1 2, 1 2]):

(λI − K∗

D) : L2(∂D) → L2(∂D) invertible, ∀ λ ∈] − ∞, −1

2]∪]1 2, +∞[.

• ( 1

2I − K∗ D) : L2 0(∂D) → L2 0(∂D) invertible; L2 0(∂D) : L2 with mean value 0.

• Single layer potential SD[ϕ](x) =

∫

∂D Γ(x − y)ϕ(y)dσ(y); Trace formula:

(±1 2I − K∗

D)[ϕ] = ∂

∂ν SD[ϕ]

• ±
• n ∂D.

+: limit from outside D, −: limit from inside D.

• Calder´
• n identity (SDK∗

D is selfadjoint): SDK∗ D = KDSD. Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Neumann-Poincar´ e operator

• Lim’s result: KD = K∗

D (KD is selfadjoint) iff D is a disk.

• Symmetrization technique: based on a Calder´
• n identity + a general

theorem on symmetrization of non-selfadjoint operators.

• Khavinson -Putinar-Shapiro: If M is a Hilbert-Schmidt operator

(∑ ||Mϕn||2 < ∞, ∀(ϕn) orthonormal basis) and there exists a strictly positive bounded selfadjoint operator R such that R2M is selfadjoint, then there is a bounded selfadjoint operator A such that AR = RM.

• M = K∗

D and R = √−SD; R2M: selfadjoint (Calder´

• n identity):

A = √ −SDK∗

D(

√ −SD)−1 (on Range(SD)) is selfadjoint (Ker(SD) ̸= {0} iff the logaritmic capacity (∂D) ̸= 1).

• Spectral representation theorem: multiply by √−SD and make a change
• f function: √−SD(λI − K∗

D)[ϕ] = (λI − A)[ψ], ψ = √−SD[ϕ]. Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Applications of Neumann-Poincar´ e type operators

• First application: super-resolution in conductivity imaging (quasi-static

regime).

• Super-resolution: reconstruction of small shape details from imaging data

with good signal-to-noise ratio (SNR).

• Concept of generalized polarization tensors (GPTs).
• GPTs for target identification: GPTs can capture topology and

high-frequency shape oscillations (with H. Kang, M. Lim, H. Zribi; with

• J. Garnier, H. Kang, M. Lim, S. Yu).
• Identification using dictionary matching of GPTs (with T. Boulier, J.

Garnier, W. Jing, H. Kang, H. Wang).

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Applications of Neumann-Poincar´ e type operators

• Second application: electromagnetic cloaking.
• Electromagnetic cloaking is to make a target invisible with respect to

probing by electromagnetic waves.

• Two schemes: change of variables (interior cloaking) and anomalous

resonance (exterior cloaking).

• New cancellation technique to achieve enhanced near-cloaking using the

change of variables scheme (with H. Kang, H. Lee, M. Lim): GPT-vanishing structures.

• Mathematical justification of cloaking due to anomalous localized

resonance (CALR) (with G. Ciraolo, H. Kang, H. Lee, G. Milton): spectral analysis of a Neumann-Poincar´ e type operator.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Generalized polarization tensors

Definition of GPTs:

• B: Lipschitz bounded domain.
• Multi-indices α, β ∈ N2 and |λ| > 1/2:

Mαβ(λ, B) := ∫

∂B

xβ(λI − K∗

D)−1[∂y α

∂ν ](x) dσ(x).

• Polya-Sze¨

go polarization tensor: |α| = |β| = 1 and λ = 1/2.

• Virtual mass (Schiffer-Sze¨

go): |α| = |β| = 1 and λ = −1/2.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Generalized polarization tensors

Properties of the GPTs:

• Symmetry: If {aα} and {bβ} are such that ∑ aαxα and ∑ bβxβ are

harmonic polynomials, then ∑ aαbβMαβ = ∑ aαbβMβα

• Positivity: If λ > 1/2, then

∑ aαaβMαβ > 0 (< 0, λ < −1/2).

• Unique determination of B and λ by GPTs: If

∑ aαbβMαβ(λ1, B1) = ∑ aαbβMαβ(λ2, B2) ∀aα, bβ, then λ1 = λ2 and B1 = B2.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Generalized polarization tensors

• Monotonicity: B′ B, λ > 1/2,

∑ aαaβMαβ(λ, B) > ∑ aαaβMαβ(λ, B′).

• Wiener-type bounds (harmonic moments of B can be estimated from the

GPTs): Let λ > 1/2 and f = ∑

α aαxα be a harmonic polynomial. Then

4λ 1 + 2λ ∫

B

|∇f |2 ≤ ∑ aαaβMαβ(λ, B) ≤ 4λ 2λ − 1 ∫

B

|∇f |2.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Generalized polarization tensors

• Contracted GPTs (CGPTs): particularly suitable linear combinations of

GPTs: Mmn := ∫

∂B

Pn(x)(λI − K∗

D)−1[∂Pm

∂ν ](x) dσ(x), Mc

mn :=

∫

∂B

Pn(x)(λI − K∗

D)−1[∂Pm

∂ν ](x) dσ(x), with Pn(x) := (x1 + ix2)n.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Generalized polarization tensors

• High-frequency oscillations of the boundary are only contained in its

high-order CGPTs.

• Rotation:

Mmn(RθB) = ei(m+n)θMmn(B), Mc

mn(RθB) = ei(n−m)θMc mn(B).

• Scaling: Mmn(sB) = s(m+n)Mmn(B),

Mc

mn(sB) = s(m+n)Mc mn(B)

• Translation:

Mmn(TzB) =

m

∑

l=1 n

∑

k=1

C z

mlMlk(B)C z nk,

Mc

mn(TzB) = m

∑

l=1 n

∑

k=1

C z

mlMlk(B)C z nk,

with C z

mn =

(m

n

) zm−n.

• CGPTs → Basis for shape representation.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Super-resolved shape imaging using GPTs

For a given entire harmonic function H, consider    ∇· ( χ(R2 \ D) + σ1χ(D) ) ∇u = 0 in R2, u(x) − H(x) = O(1/|x|) as |x| → ∞.

• Multipolar expansions:

u(x) = H(x) + ∑

α

∑

β

(−1)|β| α!β! ∂αH(z)∂βΓ(x − z)Mαβ, |x| → ∞.

• First-order (dipolar approximation): Friedman-Vogelius (89).
• {Mαβ} : GPTs associated with B and λ = (σ1 − 1)/(2(σ1 + 1)).
• |α| = |β| = 1, M : polarization tensor (PT).
• M: can not separate the size from the conductivity.
• M: canonical representation by ellipses.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Super-resolved shape imaging using GPTs

−1.5 −1 −0.5 0.5 1 1.5 2 −1.5 −1 −0.5 0.5 1 1.5

Figure: Multistatic configuration: (xi) array of point emitters = array of point receivers.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Super-resolved shape imaging using GPTs

• xi = point receiver; xj = point emitter.
• Imaging data = Multistatic response matrix (MSR): matrix of entry ij

given by u(xi) with H(x) = Γ(x − xj).

• Detection and localization algorithms in the presence of (measurement or

medium) noise: SVD-based algorithms; (weighted) subspace projection algorithms; optimal detection tests; stability and resolution analysis of localization algorithms; (with J. Garnier, H. Kang, W. Park, K. Sølna; with J. Garnier and K. Sølna; with J. Garnier, V. Jugnon).

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Super-resolved shape imaging using GPTs

• Once the location of the target is reconstructed, the GPTs can be
• btained from the MSR by a least squares method.
• Number of computed GPTs: depends only on the signal-to-noise ratio

(SNR) in the data.

• ϵ = characteristic size of the target/ the distance to the array of

transmitters/receivers.

• SNR = ϵ2/standard deviation of the measurement noise (Gaussian).
• Formula for the resolving power m as function of the SNR:

(mϵ1−m)2 = SNR.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Super-resolved shape imaging using GPTs

• Make use of

∑

|α|+|β|≤K

aαbβMαβ for a fixed K ≥ 2 to image fine details of the shape of the inclusion and separate material parameter/geometry.

• Higher is the order of the GPT, more resolved is the encoded shape

information.

• Recursive procedure (Hopping algorithm): minimize the discrepancy

between the first l GPTs. Use the result of step l as an initial guess for step l + 1. Stop at l = K.

• Initial guess: the equivalent ellipse.
• Recursive procedure for reconstructing a shape from its GPTs:robust.
• GPTs can capture the topology of the domain (level set version of the

recursive procedure).

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Super-resolved shape imaging using GPTs

Reconsructions from GPTs of order up to N = 5 and N = 6 (higher SNR), respectively:

−1 1 −1 1 −1 1 −1 1

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Dictionary matching of GPTs

• Suppose that we have a dictionary which is a collection of standard

shapes (for example Roman letters).

• Target obtained from one element of the dictionary after some rotation,

translation, and scaling (could also be a perturbed copy).

• Real-time target identification and tracking.
• New invariants for the CGPTs: rotation (Rθ), translation (Tz), and

scaling (s) formulas are not enough.

• New invariants obtained from simple relations:

Inv[GPTs](TzsRθB) = z + eiθInv[GPTs](B).

• A fast, cheap, and robust procedure for real-time target identification in

imaging based on matching on a dictionary of precomputed GPTs.

• Infinite number of invariants associated with the CGPTs.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Dictionary matching of GPTs

Dictionary letter Unknown shape 5 10 15 20 25 5 10 15 20 25 −7 −6 −5 −4 −3 −2 −1

Figure: Matching algorithm applied on the all 26 letters using the standard dictionary. The color indicates the relative error in logarithmic

• scale. All letters are correctly identified.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Dictionary matching of GPTs

5 10 15 20 25 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Letter Relative error of CGPT 5 10 15 20 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Letter Relative error of CGPT

Figure: Identification of the letter “P” using the first 2 (left), and 5

• rders CGPTs (right).

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Near-cloaking

• Cloak a region inside the cloaking device.
• Conductivity problem (quasi-static regime): the Dirichlet-to-Neumann

map is nearly the same as the one associated to the constant conductivity distribution.

• Change of variable scheme + structures with vanishing generalized

polarization tensors.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Transformation of PDE

Let Λ[σ] be the Dirichlet-to-Neumann map, i.e., Λ[σ](ϕ) = σ ∂u ∂ν where u is the solution to { ∇ · σ∇u = 0, in Ω, u = ϕ,

• n ∂Ω.

Let F be a diffeomorphism of Ω which is identity on ∂Ω. Push-forward of σ by F to obtain the anisotropic conductivity: F∗σ(y) = DF(x)σ(x)DF(x)t det(DF(x)) , x = F −1(y). Then Λ[σ] = Λ[F∗σ].

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Singular transformation

• Greenleaf-Lassas-Uhlmann (2003):

F : {x : 0 < |x| < 2} → {x : 1 < |x| < 2} is given by F(x) := ( 1 + |x| 2 ) x |x|. Then, anything inside the hole {|x| < 1} surround by a suitable anisotropic conductivity is invisible by the DtN map (perfect cloaking).

• Pendry et al (2006) used exactly the same transformation optics for

electromagnetic cloaking.

• Physically: selective bending of light rays, i.e., a ray is diverted in the

direction of the high conductivity, routed tangentially around |x| = 1, and then ejected out the other side to continue on its way.

• And then, many works on cloaking have been produced. (A review in

SIREV by Greeleaf-Lassas-Kurylev-Uhlmann (2009))

• Drawback: F∗1 is singular on |x| = 1 (0 in the normal direction, ∞ in

tangential direction, 2D)

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Near cloaking (Regularization)

Blowing-up a small ball (Kohn-Shen-Vogelius-Weinstein (2008))

• For a small number ρ, let

σ = { σ1 if |x| < ρ, 1 if ρ ≤ |x| ≤ 2. (σ1 can be 0 (the core is insulated) or ∞ (perfect conductor))

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Near cloaking (Regularization)

• Let

F(x) = {(

2−2ρ 2−ρ + 1 2−ρ|x|

)

x |x|

if ρ ≤ |x| ≤ 2,

x ρ

if 0 ≤ |x| ≤ ρ. Then F maps B2 onto B2 and blows up Bρ onto B1.

• Then, approximate cloaking

∥Λ[F∗σ] − Λ[1]∥ ≤ Cρ2.

• Conductivity in the inner cloaking region: O(ρ) in the normal direction,

0(1/ρ) in tangential direction, 2D (product = 1).

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Small volume expansions

• Λ[F∗σ] = Λ[σ] and

Λ[σ](ϕ)(x) = Λ[1](ϕ)(x) + ∇U(0) · M ∂ ∂νx ∇yG(x, 0) + h.o.t, x ∈ ∂Ω, where G is the Dirichlet Green function and U the solution to { ∆U = 0 in Ω, U = ϕ

• n ∂Ω,

M is the polarization tensor of Bρ, and G(x, y) is the Green function for Ω. (The expansion holds uniformly for σ1).

• PT for Bρ with conductivity σ1 (proportional to the volume):

M = 2(σ1 − 1) σ1 + 1 |Bρ|I.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Polarization tensor of a two-phase structure

• Make PT vanish enhances the cloaking.
• Not possible to make PT vanish with two phases.
• Multi-phase structures.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Hashin’s neutral inclusion

M = 0 (GPTs vanishing structure of order 1; a disc with a single coating)

x y

u

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Enhanced near cloaking

σN (GPTs vanishing structure of order N)

• Estimate:

∥Λ[F∗σN] − Λ[1]∥ = ∥Λ[σN] − Λ[1]∥ ≤ Cρ2N+2 for some C independent of ρ and N.

• Keep the conductivity in the inner cloaking O(ρ) in the normal direction,

0(1/ρ) in tangential direction, 2D.

• Make the h.o.t. vanish in the asymptotic expansion of the

Dirichlet-to-Neumann map.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Multiply layered structure

• Let u be the solution to

{ ∇ · ( σχ(D) + χ(R2 \ D) ) ∇u = 0 in R2, u(x) − H(x) = O(|x|−1) as |x| → ∞.

• Theorem: The far-field expansion holds as |x| → ∞:

(u − H)(x) = −

∞

∑

m,n=1

[ cos mθ 2πmr m (Mcc

mnac n + Mcs mnas n) + sin mθ

2πmr m (Msc

mnac n + Mss mnas n)

] where H(x) = H(0) + ∑∞

n=1 r n(ac n cos nθ + as n sin nθ).

• Mcc

mn, Mcs mn, Msc mn, Mss mn: contracted GPTs. Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Multiply layered structure

Disc with multiple coatings:

• For a positive integer N, let 1 = rN+1 < rN < . . . < r1 = 2 and define

Aj := {rj+1 < r ≤ rj}, j = 1, 2, . . . , N.

• A0 = R2 \ B2, AN+1 = B1.
• Set σj to be the conductivity of Aj for j = 1, 2, . . . , N + 1, and σ0 = 1.

Let σ =

N+1

∑

j=0

σjχ(Aj). (σN+1 may (or may not) be fixed: σN+1 is fixed to be 0 if the core is insulated.)

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

GPTs vanishing structure

• Let Mcc

mn[σ], etc, denote the GPTs associated with σ. Because of the

symmetry of the disc, Mcs

mn[σ] = Msc mn[σ] = 0

for all m, n, Mcc

mn[σ] = Mss mn[σ] = 0

if m ̸= n, and Mcc

nn[σ] = Mss nn[σ]

for all n.

• Let Mn = Mcc

nn, n = 1, 2, . . .. Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

GPTs vanishing structure

• To compute Mk, we look for solutions uk to

∇ · σ∇u = 0 in R2

• f the form

uk(x) = a(k)

j

r k cos kθ + b(k)

j

r k cos kθ in Aj, j = 0, 1, . . . , N + 1, with a(k) = 1 and b(k)

N+1 = 0.

• Then uk satisfies

(uk − H)(x) = b(k) r k cos kθ as |x| → ∞. with H(x) = r k cos kθ.

• Hence, Mk = −2πkb(k)

0 . Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

GPTs vanishing structure

• The transmission conditions on the interface {r = rj}:

[ a(k)

j

b(k)

j

] = 1 2σj [ σj + σj−1 (σj − σj−1)r −2k

j

(σj − σj−1)r 2k

j

σj + σj−1 ] [ a(k)

j−1

b(k)

j−1

] , and hence [ a(k)

N+1

] =

N+1

∏

j=1

1 2σj [ σj + σj−1 (σj − σj−1)r −2k

j

(σj − σj−1)r 2k

j

σj + σj−1 ] [ 1 b(k) ] .

• Let

P(k) = [ p(k)

11

p(k)

12

p(k)

21

p(k)

22

] :=

N+1

∏

j=1

1 2σj [ σj + σj−1 (σj − σj−1)r −2k

j

(σj − σj−1)r 2k

j

σj + σj−1 ] . Then, b(k) = −p(k)

21

p(k)

22

.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

GPTs vanishing structure

• GPTs vanishing structure of order N: Mk = 0 for k = 1, . . . , N, or

p(k)

21 = 0,

k = 1, . . . , N.

• Solve the equations for rN < . . . < r2 and σN, . . . , σ1.
• If N = 1, Hashin’s neutral inclusion.
• For N = 2, 3, ..., can be solved by hand.
• For arbitrary N, the equation is non-linear algebraic equation: numerical
• ptimization.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

GPTs vanishing structure

1 1.5 2 1 5 10 15

r σ

1 3 15 10

−15

10

−10

10

−5

10 10

5

k Mk

Figure: The conductivity of the core is fixed to be 0. N = 3

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Enhancement of near cloaking

• Theorem: Let σ be a multi-layered structure with r1 = 2 and rN+1 = 1. If

f = ∑∞

k=−∞ fkeikθ, then

( Λ[σ(1 ρx)] − Λ[1] ) (f ) =

∞

∑

k=−∞

2|k|ρ2|k|M|k|[σ] 2π|k| − ρ2|k|M|k|[σ]fkeikθ.

• Corollary: σ: a GPTs vanishing structure of order N, σN(x) = σ( 1

ρx).

Then ( Λ[σN] − Λ[1] ) (f ) = ∑

|k|>N

2|k|ρ2|k|M|k|[σ] 2π|k| − ρ2|k|M|k|[σ]fkeikθ.

• Lemma: |Mk[σ]| ≤ 2πk22k for all k.
• Theorem: Using the transformation blowing up a small ball, we can get a

near-cloaking structure such that ∥ Λ[σN] − Λ[1] ∥ = ∥ Λ[F∗σN] − Λ[1] ∥ ≤ Cρ2N+2.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Enhancement of near cloaking

Change of variables (sends the annulus [ρ, 2ρ] onto a fixed annulus): Fρ(x) :=            ( 3 − 4ρ 2(1 − ρ) + 1 4(1 − ρ)|x| ) x |x| for 2ρ ≤ |x| ≤ 2, (1 2 + 1 2ρ|x| ) x |x| for ρ ≤ |x| ≤ 2ρ, x ρ for |x| ≤ ρ. Anisotropic conductivity distributions:

log10(σ11), alternative blow−up of a 3 layer structure

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

log10(σ11), alternative blow−up of a 6 layer structure

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −1 −0.5 0.5 1

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Enhancement of near cloaking

log10 of the eigenvalues of Λ[σ( 1

ρx)] − Λ[1] for different values of N:

1 2 3 4 5 6 7 8 −20 −15 −10 −5 5

log10(|λk[σ]−λk[1]|) k Perturbation of the eigenvalues of the DtN map

hole of radius 1 hole of radius ρ=0.25 hole of radius ρ=0.25+1 layer hole of radius ρ=0.25+2 layers hole of radius ρ=0.25+3 layers hole of radius ρ=0.25+4 layers hole of radius ρ=0.25+5 layers hole of radius ρ=0.25+6 layers

N-layer vanishing GPTs structure: same first N DtN eigenvalues as Λ[1].

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Enhancement of near cloaking

• Measure of the invisibility of a cloak:

β(σ) = sup

k

|λk

B2,B1[σ] − λk B2[1]|

λk

B2,B1[σ], λk B2[1]: eigenvalues of Λ[σ( 1 ρx)], Λ[1].

• To obtain the same invisibility as a 6-layer structure with ρ = 0.25, one

has to use a change of variable cloak with size parameter ρeq ≈ 1.5.10−6 which will result in a much more singular conductivity distribution.

• Perform a statistical sensitivity analysis of the invisibility measure using

Sobol indices to explain the fluctuations of the invisibility measure of the multi-layer cloak in terms of the conductivities of the layers.

• Strong interaction between the conductivity values.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• Ω: bounded domain in R2; D ⊂⊂ Ω. Ω and D of class C1,µ, 0 < µ < 1.

For a given loss parameter δ > 0, the permittivity distribution in R2 is given by ϵδ =      1 in R2 \ Ω, −1 + iδ in Ω \ D, 1 in D.

• Configuration (plasmonic structure): core with permittivity 1 coated by

the shell Ω \ D with permittivity −1 + iδ.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• For a given function f compactly supported in R2 satisfying

∫

R2 fdx = 0

(conservation of charge), consider the following dielectric problem: ∇ · ϵδ∇Vδ = αf in R2, with the decay condition Vδ(x) → 0 as |x| → ∞.

• Dielectric problems: models the quasi-static (zero-frequency) transverse

magnetic regime.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• Fundamental problem: identify f such that when α = 1

Eδ := ∫

Ω\D

δ|∇Vδ|2dx → ∞ as δ → 0. |Vδ(x)| < C, when |x| > a for some constants C and a independent of δ.

• Eδ: proportional to the electromagnetic power dissipated into heat by the

time harmonic electrical field averaged over time.

• Infinite amount of energy dissipated per unit time in the limit δ → 0:

unphysical.

• Choose α = 1/√Eδ: αf produces the same power independent of δ and

the new associated solution Vδ approaches zero outside the radius a.

• Necessary and sufficient condition for CALR (with α = 1) Vδ/√Eδ goes

to zero outside some radius as δ → 0.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• Using layer potential techniques: we reduce the problem to a singularly

perturbed system of integral equations.

• The system is non-self-adjoint ⇒ we introduce a symmetrization

technique in order to express the solution in terms of the eigenfunctions

• f a self-adjoint compact operator.
• Symmetrization technique: based on a generalization of a Calder´
• n

identity to the system of integral equations.

• We provide a necessary and sufficient condition on the source term under

which the blowup of the power dissipation takes place. The condition is given in terms of the Newtonian potential of the source 1 2π ∫

R2 ln |x − y|f (y)dy,

x ∈ R2, which is the solution for the potential in the absence of the plasmonic structure.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• In the case of an annulus (D is the disk of radius ri and Ω =: Be is the

concentric disk of radius re), it is known (Milton et al.) that there exists a critical radius (the cloaking radius) r∗ = √ r 3

e ri −1.

such that any finite collection of dipole sources located at fixed positions within the annulus Br∗ \ Be is cloaked.

• We sufficient conditions for a source αf supported in E to be cloaked.

(In particular, quadrupole source inside the annulus Br∗ \ Be: cloaked).

• Conversely we show that if the source function f is supported outside Br∗

then no cloaking occurs.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• Notation: Γi := ∂D Γe := ∂Ω, F Newtonian potential of f ;

H = L2(Γi) × L2(Γe); zδ =

iδ 2(2−iδ).

• Representation formula:

Vδ(x) = F(x) + SΓi [φi](x) + SΓe[φe](x).

• Introduce:

Φ := [φi φe ] , g :=     ∂F ∂νi − ∂F ∂νe     .

• Singularly perturbed equation:

(zδI2 + K∗)Φ = g.

• K∗ : H → H Neumann-Poincar´

e-type operator (compact non-self-adjoint in general): K∗ :=     −K∗

Γi

− ∂ ∂νi SΓe ∂ ∂νe SΓi K∗

Γe

    .

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• The spectrum of K∗ lies in the interval [−1/2, 1/2].
• The operator

S = [SΓi SΓe SΓi SΓe ] is self-adjoint and −S ≥ 0 on H.

• Calder´
• n’s-type identity: SK∗ = KS.
• K∗ is Hilbert-Schmidt (in 2D; Schatten-von Neumann in 3D).
• K∗ is symmetrizable: there is a bounded self-adjoint operator A on

Range(S) such that A √ −S = √ −SK∗.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

• A self-adjoint ⇒ an orthogonal decomposition: H = KerA ⊕ (KerA)⊥,

and (KerA)⊥ = RangeA.

• P and Q = I − P: the orthogonal projections from H onto KerA and

(KerA)⊥, respectively. Let λ1, λ2, . . . with |λ1| ≥ |λ2| ≥ . . . be the nonzero eigenvalues of A and Ψn be the corresponding (normalized)

• eigenfunctions. A ∈ C2(H) ⇒

∞

∑

n=1

λ2

n < ∞,

and AΦ =

∞

∑

n=1

λn⟨Φ, Ψn⟩Ψn, Φ ∈ H.

• Theorem: If P

√ −Sg ̸= 0, then CALR takes place. If Ker(K∗) = {0}, then CALR takes place if and only if δ ∑

n

|⟨ √ −Sg, Ψn⟩|2 λ2

n + δ2

→ ∞ as δ → 0.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

Anomalous resonance in an annulus:

• Eigenvalues λ of A= {±ρ|n|}, ρ = ri

re .

• Theorem: (blow-up of power dissipation criterion) For a given source f

supported outside Be (with α = 1). If the Fourier coefficients of − ∂F

∂νe on

Γe, where F is the Newton potential of f satisfies a Gap condition (mild condition), then ∫

Be\Bi

δ|∇Vδ|2 → ∞ as δ → 0, and CALR occurs.

• Quadrupole satisfies the Gap condition.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Anomalous resonance

Anomalous resonance in an annulus:

• Any source supported outside Br∗ cannot make the blow-up of the power

dissipation happen and is not cloaked. Indeed, in the limit δ → 0 the annulus itself becomes invisible to sources that are sufficiently far away.

• Theorem: If f is supported in R2 \ Br∗, then

∫

Be\Bi

δ|∇Vδ|2 < C holds for some constant C independent of δ (with α = 1). Moreover, we have sup

|x|≥r∗

|Vδ(x) − F(x)| → 0 as δ → 0.

• Annulus itself becomes invisible to sources that are sufficiently far away.

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari

. . . . . .

Conclusion

• GPTs: key role in imaging.
• GPTs can be obtained from MSR matrix.
• GPTs yield super-resolved shape imaging.
• GPTs can be used as a shape descriptor: dictionary matching of GPTs

yields a fast and robust algorithm for target identification.

• GPTs vanishing structures: improve near cloaking effect.
• Justification of cloaking by anomalous localized resonance.
• Concept of GPTs: generalized to Helmholtz equation, elasticity

equations, Maxwell’s equations.

• Concept of GPTs: generalized to inhomogeneous conductivities (with Y.

Deng, H. Kang, H. Lee).

Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari