PICOF 2012 Regularity Estimates in High Conductivity Homogenization Yves Capdeboscq Oxford Centre for Nonlinear PDE, University of Oxford With Marc Briane (Rennes) & Luc Nguyen (Princeton)
Outline Enhanced Resolution for finite contrast (Ammari-Bonnetier-C) The High Contrast case (Briane-C-Nguyen)
Time Reversal Experiment (Physics) C. Prada & M. Fink (Wave Motion ’94) . . . For acoustic waves, with random scatterers in a resonant chamber. Step 1 A pulse is emitted at a source location Step 2 The signal is recorded on an array of microphones, behind the scatterers (for a long time T ) Step 3 The signal is amplified and played back, backwards, from the array. At time 2 T , a peak appears at the location where the pulse was emitted.
Time Reversal Mirror (Maths) Extracted from Guillaume Bal’s webpage, http://www.columbia.edu/˜gb2030/ J. P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave propagation and time reversal in randomly layered media , Springer, 2007. G. Bal and L. Ryzhik Time reversal and refocusing in random media , SIMA, 2003. C. Bardos & M. Fink, Mathematical foundations of the time reversal mirror, Asymptotic Anal. 2002 http://www.claudebardos.com/pdf/retemp.pdf
Focusing Beyond the Diffraction Limit G. Lerosey, J. de Rosny, A. Tourin, A. Derode, M. Fink, Focusing Beyond the Diffraction Limit with Far-Field Time Reversal Science, 23 February 2007, p.1120-1122 A TRM made of eight commercial dipolar antennas operating at λ = 12 cm. is placed in a 1 m 3 reverberating chamber. Ten wavelength away from the TRM is placed a sub-wavelength receiving array consisting of 8 micro-structured antennas λ/ 30 apart from each other.
Focusing Beyond the Diffraction Limit
Focusing Beyond the Diffraction Limit Interpretation via homogenization and small volume asymptotics, for passive imaging.
Passive imaging : no source in the medium Model problem: Helmholtz equation (time harmonic, in TM (transverse magnetic) Polarization). The x 3 component of the magnetic field satisfies div ( a ( x ) ∇ u ( x )) + ω 2 µ ( x ) u ( x ) = 0 in B R , u = ϕ on S R 1 with a ( x ) = ε ( x ) + i σ ( x ) /ω, and ε ( x ): electric permittivity, σ ( x ) : conductivity, µ ( x ) : magnetic permeability, ω : 2 π λ .
Difference Imaging We compare the traces of div ( a ( x ) ∇ u ( x )) + ω 2 µ ( x ) u ( x ) = 0 in B R , and div ( a d ( x ) ∇ u d ( x )) + ω 2 µ d ( x ) u d ( x ) = 0 in B R , u , u d = ϕ on S R where u d corresponds to ( a d , µ d ) := ( a ( x ) , µ ( x )) + 1 D ( x )( a D ( x ) − a ( x ) , µ D ( x ) − µ ( x )) . We want to find inclusions from boundary measurements (or the far field). We consider differential measurements, that is, the map Λ : H 1 / 2 ( S R ) H − 1 / 2 ( S R ) → ϕ → a ∇ ( u d − u ) · n | S R
Resolution for passive systems Example: a homogeneous background ( ε 0 , µ 0 ) and a constant inclusion ( ε 1 , µ 1 ). Suppose D = B (0 , r ) and r → 0.
Resolution for passive systems Example: a homogeneous background ( ε 0 , µ 0 ) and a constant inclusion ( ε 1 , µ 1 ). Suppose D = B (0 , r ) and r → 0. � r d � (Λ( φ ) , φ ) = r d R ( φ, φ ) + o The response operator R is the bilinear form given by R ( φ, ψ ) = 1 � M B ∇ u φ · ∇ u ψ dy + 1 � ω 2 m B u φ u ψ dy r d r d B r B r ε 1 − ε 0 2 The term M B is the polarisation tensor, here M B = I d , ε 0 + ε 1 ε 0 and m B = µ 1 − µ 0 . The function u φ (resp. u ψ ) satisfies div( ǫ − 1 0 ∇ u φ ) + ω 2 µ 0 u φ = 0 in B R u φ = φ resp. ψ on S R .
Resolution for passive systems For constant coefficients, R is explicit yielding a representation formula for Λ. The permeability response is, up to o ( r d ), C 1 , B J 1 ( kr ) e i θ � � J 1 ( kr ) e i θ , · Λ = C 0 , B J 0 ( kr ) e i θ � � J 0 ( kr ) e i θ , · + when d = 2 , or C 1 , B j 1 ( kr ) e i θ � � j 1 ( kr ) e i θ , · Λ = C 0 , B j 0 ( kr ) e i θ � � j 0 ( kr ) e i θ , · + when d = 3 , where k 2 = µ 0 ε 0 ω 2
Resolution for passive system 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 K 10 K 5 0 5 10 K 10 K 5 0 5 10 x x The size of the focal spot, is given by the size first lobe(s) of the (spherical) Bessel Function J 0 ( √ µ 0 ε 0 ω · ) ( J 1 ( √ µ 0 ε 0 ω · ) )
Resolution for passive system 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 K 10 K 5 0 5 10 K 10 K 5 0 5 10 x x The size of the focal spot, is given by the size first lobe(s) of the (spherical) Bessel Function J 0 ( √ µ 0 ε 0 ω · ) ( J 1 ( √ µ 0 ε 0 ω · ) ) This size does not depend on the inclusion.
Enhanced resolution in structured media ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ | D |→ 0 ε = 0 . lim ( | D | 2 , − 1 / ln | D | . . . .. )
Enhanced resolution Ammari-Bonnetier-C ’09 Introduce scatterers in the medium, on a periodic grid (for example). B ε, j = j ε + ε B � ( a s , µ s ) in ∪ j ∈S ε B ε, j ( a ε ( x ) , µ ε ( x )) = (1) ( a 0 , µ 0 ) otherwise . The true (defective) medium, material parameters are ( a D , µ D ) in D ( a ε, d ( x ) , µ ε, d ( x )) = ( a s , µ s ) in ∪ j B ε, j \ D . ( a 0 , µ 0 ) otherwise .
Enhanced Resolution in Structured Media We want to image inclusions (imperfections) from boundary (far field) measurements. We consider difference measurements, namely the map Λ : H 1 / 2 ( S R ) H − 1 / 2 ( S R ) → ϕ → a ∇ ( u ε, D − u ε ) · n | S R
Enhanced Resolution in Structured Media We want to image inclusions (imperfections) from boundary (far field) measurements. We consider difference measurements, namely the map Λ : H 1 / 2 ( S R ) H − 1 / 2 ( S R ) → ϕ → a ∇ ( u ε, D − u ε ) · n | S R Result: if ε → 0 as | D | → 0, the response operator is given by � � M ∗ ∇ u ∗ ( x ) ·∇ u ∗ ( x ) dx + ω 2 m ∗ u 2 < Λ( ϕ ) , ϕ > = ∗ ( x ) dx + o ( | D | ) , D D where M ∗ and m ∗ are constant polarization terms that depend on the contrast in material constants, where o ( | D | ) / | D | → 0 uniformly for � ϕ � H 1 / 2 ≤ 1
Enhanced Resolution in Structured Media We want to image inclusions (imperfections) from boundary (far field) measurements. We consider difference measurements, namely the map Λ : H 1 / 2 ( S R ) H − 1 / 2 ( S R ) → ϕ → a ∇ ( u ε, D − u ε ) · n | S R Result: if ε → 0 as | D | → 0, the response operator is given by � � M ∗ ∇ u ∗ ( x ) ·∇ u ∗ ( x ) dx + ω 2 m ∗ u 2 < Λ( ϕ ) , ϕ > = ∗ ( x ) dx + o ( | D | ) , D D where M ∗ and m ∗ are constant polarization terms that depend on the contrast in material constants, where o ( | D | ) / | D | → 0 uniformly for � ϕ � H 1 / 2 ≤ 1 To be compared with the unscattered response operator, � M ∇ u 0 · ∇ u 0 + ω 2 µ u 2 � � R ( ϕ ) = dx + o ( | D | ) . 0 D
different asymptotic limits 1 If ε → 0 first , then | D | → 0: correct focal spot, but incorrect asymptotic formulae.
different asymptotic limits 1 If ε → 0 first , then | D | → 0: correct focal spot, but incorrect asymptotic formulae. 2 If | D | → 0 first, then ε → 0: the “reference” resolution calculation.
different asymptotic limits 1 If ε → 0 first , then | D | → 0: correct focal spot, but incorrect asymptotic formulae. 2 If | D | → 0 first, then ε → 0: the “reference” resolution calculation. 3 If ε d ≪ | D | , then (1) is correct.
Elements of proof We use the following tools: H-convergence (strong with correctors) � u ε, d − u ∗ � L 2 (Ω) , � u ε − u ∗ � L 2 (Ω) → 0 . (Murat 70s). Local regularity, and smoothness of u ∗ The scatterers have a smooth C 1 ,α boundary (important). Then Li & Vogelius (’01), Li & Nirenberg (’03), and Avellaneda & Lin (’97) give W 1 , ∞ convergence estimates || u ε − u ∗ || L ∞ ( D ) , ||∇ u ε − P ε ∇ u ∗ || L ∞ ( D ) → 0 , where P ε ∈ L ∞ (Ω) is the corrector matrix. Similar to Ben Hassen & Bonnetier ’06. Small volume fraction asymptotics This gives an asymptotic formula for 1 D ∇ u ε, D / | D | in terms of a polarization tensor, involving P ε , and ∇ u ∗ , not unlike C-Vogelius (’03, ’06).
Elements of proof We use the following tools: H-convergence (strong with correctors) � u ε, d − u ∗ � L 2 (Ω) , � u ε − u ∗ � L 2 (Ω) → 0 . (Murat 70s). Local regularity, and smoothness of u ∗ The scatterers have a smooth C 1 ,α boundary (important). Then Li & Vogelius (’01), Li & Nirenberg (’03), and Avellaneda & Lin (’97) give W 1 , ∞ convergence estimates || u ε − u ∗ || L ∞ ( D ) , ||∇ u ε − P ε ∇ u ∗ || L ∞ ( D ) → 0 , where P ε ∈ L ∞ (Ω) is the corrector matrix. Similar to Ben Hassen & Bonnetier ’06. Small volume fraction asymptotics This gives an asymptotic formula for 1 D ∇ u ε, D / | D | in terms of a polarization tensor, involving P ε , and ∇ u ∗ , not unlike C-Vogelius (’03, ’06). That’s it.
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