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 2 π log | x | : fundamental solution of ∆ in R 2 ; 1 • Γ( x ) = • D : bounded, smooth ( C 1 ,α , 0 < α < 1) domain in R 2 ; ν : outward normal at ∂ D ; • Neumann-Poincar´ e operator (convolution with ∂ Γ( x − y ) /∂ν ( x )) : D [ ϕ ]( x ) := 1 ∫ ( x − y ) · ν ( x ) K ∗ ϕ ( y ) d σ ( y ) , | x − y | 2 2 π ∂ D • Adjoint: K D [ ϕ ]( x ) := 1 ∫ ( y − x ) · ν ( y ) ϕ ( y ) d σ ( y ) . 2 π | x − y | 2 ∂ D • K D , K ∗ D : L 2 ( ∂ D ) → L 2 ( ∂ 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 ]): D ) : L 2 ( ∂ D ) → L 2 ( ∂ D ) invertible , ∀ λ ∈ ] − ∞ , − 1 2] ∪ ]1 ( λ I − K ∗ 2 , + ∞ [ . 0 ( ∂ D ) : L 2 with mean value 0. • ( 1 2 I − K ∗ D ) : L 2 0 ( ∂ D ) → L 2 0 ( ∂ D ) invertible; L 2 ∫ • Single layer potential S D [ ϕ ]( x ) = ∂ D Γ( x − y ) ϕ ( y ) d σ ( y ); Trace formula: � ( ± 1 D )[ ϕ ] = ∂ 2 I − K ∗ � ∂ν S D [ ϕ ] on ∂ D . � � ± +: limit from outside D , − : limit from inside D . on identity ( S D K ∗ D is selfadjoint): S D K ∗ • Calder´ D = K D S D . . . . . . . Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari
Neumann-Poincar´ e operator • Lim’s result: K D = K ∗ D ( K D is selfadjoint) iff D is a disk. • Symmetrization technique: based on a Calder´ on identity + a general theorem on symmetrization of non-selfadjoint operators. • Khavinson -Putinar-Shapiro: If M is a Hilbert-Schmidt operator ( ∑ || M ϕ n || 2 < ∞ , ∀ ( ϕ n ) orthonormal basis) and there exists a strictly positive bounded selfadjoint operator R such that R 2 M is selfadjoint, then there is a bounded selfadjoint operator A such that AR = RM . D and R = √−S D ; R 2 M : selfadjoint (Calder´ • M = K ∗ on identity): √ √ −S D K ∗ −S D ) − 1 A = D ( (on Range ( S D )) is selfadjoint (Ker( S D ) ̸ = { 0 } iff the logaritmic capacity ( ∂ D ) ̸ = 1). • Spectral representation theorem: multiply by √−S D and make a change of function: √−S D ( λ I − K ∗ D )[ ϕ ] = ( λ I − A )[ ψ ] , ψ = √−S D [ 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 ∈ N 2 and | λ | > 1 / 2: D ) − 1 [ ∂ y α ∫ x β ( λ I − K ∗ M αβ ( λ, B ) := ∂ν ]( x ) d σ ( x ) . ∂ B • 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 { a α } and { b β } are such that ∑ a α x α and ∑ b β x β are harmonic polynomials, then ∑ ∑ a α b β M αβ = a α b β M βα • Positivity: If λ > 1 / 2, then ∑ a α a β M αβ > 0 ( < 0 , λ < − 1 / 2) . • Unique determination of B and λ by GPTs: If ∑ ∑ a α b β M αβ ( λ 1 , B 1 ) = a α b β M αβ ( λ 2 , B 2 ) ∀ a α , b β , then λ 1 = λ 2 and B 1 = B 2 . . . . . . . Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari
Generalized polarization tensors • Monotonicity: B ′ � B , λ > 1 / 2, ∑ ∑ a α a β M αβ ( λ, B ′ ) . a α a β M αβ ( λ, B ) > • Wiener-type bounds (harmonic moments of B can be estimated from the α a α x α be a harmonic polynomial. Then GPTs): Let λ > 1 / 2 and f = ∑ ∫ ∫ 4 λ 4 λ |∇ f | 2 ≤ ∑ |∇ f | 2 . a α a β M αβ ( λ, B ) ≤ 1 + 2 λ 2 λ − 1 B B . . . . . . Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari
Generalized polarization tensors • Contracted GPTs (CGPTs): particularly suitable linear combinations of GPTs: ∫ D ) − 1 [ ∂ P m P n ( x )( λ I − K ∗ M mn := ∂ν ]( x ) d σ ( x ) , ∂ B ∫ D ) − 1 [ ∂ P m M c P n ( x )( λ I − K ∗ mn := ∂ν ]( x ) d σ ( x ) , ∂ B with P n ( x ) := ( x 1 + ix 2 ) 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: M mn ( R θ B ) = e i ( m + n ) θ M mn ( B ) , M c mn ( R θ B ) = e i ( n − m ) θ M c mn ( B ) . • Scaling: M mn ( sB ) = s ( m + n ) M mn ( B ) , mn ( sB ) = s ( m + n ) M c M c mn ( B ) • Translation: m n ∑ ∑ C z ml M lk ( B ) C z M mn ( T z B ) = nk , l =1 k =1 m n ∑ ∑ M c ml M lk ( B ) C z mn ( T z B ) = C z nk , l =1 k =1 with C z ( m z m − n . mn = ) 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 ( R 2 \ D ) + σ 1 χ ( D ) ) in R 2 , ∇· ∇ u = 0 u ( x ) − H ( x ) = O (1 / | x | ) as | x | → ∞ . • Multipolar expansions: ( − 1) | β | ∑ ∑ α ! β ! ∂ α H ( z ) ∂ β Γ( x − z ) M αβ , u ( x ) = H ( x ) + | x | → ∞ . α β • First-order (dipolar approximation): Friedman-Vogelius (89). • { M αβ } : GPTs associated with B and λ = ( σ 1 − 1) / (2( σ 1 + 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 −0.5 −1 −1.5 −1.5 −1 −0.5 0 0.5 1 1.5 2 Figure: Multistatic configuration: ( x i ) 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 • x i = point receiver; x j = point emitter. • Imaging data = Multistatic response matrix (MSR): matrix of entry ij given by u ( x i ) with H ( x ) = Γ( x − x j ). • 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 obtained 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 = ϵ 2 /standard deviation of the measurement noise (Gaussian). • Formula for the resolving power m as function of the SNR: ( m ϵ 1 − m ) 2 = SNR . . . . . . . Neumann-Poincar´ e type operators, super-resolution, and electromagnetic invisibility Habib Ammari
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