Identification of generalized impedance boundary conditions in - PowerPoint PPT Presentation

Page 1 Identification of generalized impedance boundary conditions in inverse scattering problems L. Bourgeois, N. Chaulet and H. Haddar INRIA/DEFI Project (Palaiseau, France) AIP, Vienna, 23/07/2009

Page 2 About the Generalized Impedance Boundary Conditions • Context : scattering problems in the harmonic regime • GIBCs : correspond to models involving small parameters → For example, perfect conductor coated with a layer for T E polarization (order 1 ), Z = δ ( ∂ ss + k 2 n ) , ∂ ν u + Zu = 0 on Γ , with δ : width of the layer, s : curvilinear abscissa, k : wave number, n : mean value of the thin coating index along ν We consider the following model of GIBC : ∂ ν u + µ ∆ Γ u + λu = 0 on Γ , with µ : complex constant, λ : complex function.

Page 3 Outline of the talk Typical inverse problem : the obstacle being known, determine λ and µ from the far field u ∞ associated to one incident wave at fixed frequency Nonlinear operator of interest : T : ( λ, µ ) − → u ∞ • The forward problem • Uniqueness for the inverse problem • Stability for the inverse problem • Numerical experiments • Perspectives

Page 4 The forward problem Obstacle D ⊂ R 3 , Ω := R 3 \ D Incident wave u i ( x ) = e ik d.x Governing equations for u s = u − u i :  ∆ u s + k 2 u s = 0 in Ω ,       ∂u s  ∂ν + µ ∆ Γ u s + λu s = f on Γ ,   � | ∂u s /∂r − iku s | 2 ds ( x ) = 0 ,   lim    R → + ∞ ∂B R with � � ∂u i ∂ν + µ ∆ Γ u i + λu i f := − | Γ

Page 5 The forward problem • Classical impedance problem µ = 0 : uniquely solvable in V 0 R = { H 1 (Ω ∩ B (0 , R )) } provided λ ∈ L ∞ (Γ) with Im( λ ) ≥ 0 • Generalized impedance problem µ � = 0 : uniquely solvable in V R = { v ∈ V 0 R , v | Γ ∈ H 1 (Γ) } provided λ ∈ L ∞ (Γ) with Im( λ ) ≥ 0 , Re( µ ) > 0 and Im( µ ) ≤ 0 . Remark : ∆ Γ v is defined in H − 1 (Γ) by � ∀ w ∈ H 1 (Γ) � ∆ Γ v, w � H − 1 (Γ) ,H 1 (Γ) = − ∇ Γ v. ∇ Γ w ds, Γ

Page 6 Uniqueness for inverse problem (the obstacle is known) • Classical impedance problem µ = 0 (Colton and Kirsch 81): uniqueness for piecewise continuous λ Proof : assume T ( λ 1 ) = T ( λ 2 ) = u ∞ . Rellich Lemma + unique continuation ⇒ u 1 = u 2 in Ω , then ( u 1 − u 2 ) | Γ = 0 and ∂ ν ( u 1 − u 2 ) | Γ = 0 . ∂ ν u 1 + λ 1 u 1 = ∂ ν u 1 + λ 2 u 1 = 0 on Γ Then ( λ 1 − λ 2 ) u 1 = 0 on Γ . For x 0 ∈ Γ not on a curve of discontinuity s.t. ( λ 1 − λ 2 )( x 0 ) � = 0 , then | ( λ 1 − λ 2 )( x ) | > 0 on B ( x 0 , η ) ∩ Γ . As a result u 1 = 0 , ∂ ν u 1 = 0 on B ( x 0 , η ) ∩ Γ , and unique continuation ⇒ u 1 = 0 in Ω . This contradicts the fact that u i is a plane wave. Hence λ 1 ( x ) = λ 2 ( x ) a.e. on Γ . �

Page 7 Uniqueness for the inverse problem • Generalized impedance problem µ � = 0 : non uniqueness A counterexample in 2D : D = B (0 , 1) , d = (1 , 0) , k = 1 , u 0 : solution of the classical impedance problem with λ 0 = i α := ∆ Γ u 0 /u 0 is a smooth function on Γ • µ 1 � = µ 2 s.t. | µ i | max Γ | α | ≤ 1 , Re( µ i ) > 0 , Im( µ i ) ≤ 0 • λ 1 � = λ 2 s.t. λ i := λ 0 − αµ i on Γ → We have on Γ : Im( λ i ) = Im( λ 0 ) − Im( αµ i ) ≥ Im( λ 0 ) − | µ i | max Γ | α | ≥ 0 ∂ ν u 0 + µ i ∆ Γ u 0 + λ i u 0 = ( − λ 0 + αµ i + λ i ) u 0 = 0 As a result, u ∞ = T ( i, 0) is the far field associated to the 0 generalized impedance problem with both ( λ 1 , µ 1 ) and ( λ 2 , µ 2 )

Page 8 Uniqueness for the inverse problem We can restore uniqueness with restrictions : two examples • λ and µ two complex constants + Geometric assumption : there exists x 0 ∈ Γ , η > 0 such that Γ 0 := Γ ∩ B ( x 0 , η ) is portion of a plane, cylinder or sphere and { x + γν ( x ) , x ∈ Γ 0 , γ > 0 } ⊂ Ω • λ piecewise continuous, and µ complex constant : Re( λ ) and Im( µ ) are fixed and known, the unknown being Im( λ ) and Re( µ ) + Geometric assumption : both D , λ are invariant by reflection against a plane which does not contain d or by a rotation around an axis which is not directed by d • More general conditions in Bourgeois & Haddar (2009, submitted)

Page 9 Uniqueness for the inverse problem Second case : sketch of the proof ∂ ν u + µ 1 ∆ Γ u + λ 1 u = ∂ ν u + µ 2 ∆ Γ u + λ 2 u = 0 on Γ If µ 1 � = µ 2 , then 1 � � |∇ Γ u | 2 ds = ( λ 2 − λ 1 ) | u | 2 ds µ 2 − µ 1 Γ Γ Hyp. : Re( λ ) and Im( µ ) are fixed and known Then ( λ 2 − λ 1 ) / ( µ 2 − µ 1 ) ∈ i R ⇒ u = C on Γ , and λ 1 = λ 2 = λ . u s + u i = C ∂ ν u s + ∂u i and ν = − Cλ on Γ

Page 10 Uniqueness for the inverse problem Second case : sketch of the proof (cont.) : Representation formulas for u s and u i on Γ :  u s ( x ) / 2 = T ( u s )( x ) − S ( ∂ ν u s ( x ))  u i ( x ) / 2 = −T ( u i )( x ) + S ( ∂ ν u i ( x ))  with S := γ − SL = γ + SL , T = ( γ + DL + γ − DL) / 2 (SL : single layer potential, DL : double layer potential) We obtain u i ( x ) = C 2 (1 − 2 T (1)( x ) − 2 S ( λ )( x )) on Γ This is forbidden by the geometric assumption. �

Page 11 Stability for the inverse problem The classical impedance problem : many results in the litterature (Labreuche 99, Sincich 06, ...) + (Γ) → u ∞ ∈ L 2 ( S 2 ) : Some proprieties of operator T : λ ∈ L ∞ • Injective (piecewise continuous λ ) • Differentiable in the sense of Fr´ echet dT λ : h → v ∞ h is defined by � x ∈ S 2 v ∞ h (ˆ x ) = p ( y, ˆ x ) u ( y, d ) h ( y ) ds ( y ) ∀ ˆ Γ where p ( ., ˆ x ) is the solution associated to Φ ∞ ( ., ˆ x ) . • dT λ injective (piecewise continuous λ ) ⇒ Some simple Lipschitz stability results can be derived in compact subsets of finite dimensional spaces

Page 12 Stability for the inverse problem The generalized impedance problem : Some proprieties of operator T : ( λ, µ ) ∈ V (Γ) → u ∞ ∈ L 2 ( S 2 ) : • Injective • Differentiable in the sense of Fr´ echet dT λ,µ : ( h, l ) → v ∞ h,l is defined by x ∈ S 2 v ∞ h,l (ˆ x ) = � p ( ., ˆ x ) , l ∆ Γ u ( ., d ) + u ( ., d ) h � H 1 ,H − 1 ∀ ˆ where p ( ., ˆ x ) is the solution associated to Φ ∞ ( ., ˆ x ) . • dT λ,µ injective ⇒ Some simple Lipschitz stability results can be derived in compact subsets of finite dimensional spaces

Page 13 Numerical experiments in 2 D • Minimize the cost function (classical impedance) F ( λ ) = 1 obs || 2 2 || T ( λ ) − u ∞ L 2 ( S 1 ) • Artificial data u ∞ obs obtained with a Finite Element Method • Projection of λ along the trace on Γ of the FE basis • Computation of gradient (classical impedance): h 1 = Re( h ) , h 2 = Im( h ) � ( dF ( λ ) , h ) = Re { ( h 1 ( y ) + ih 2 ( y )) u ( y ) Γ � S 1 p ( y, ˆ x )( T ( λ ) − u ∞ obs )(ˆ x ) d ˆ x } ds ( y ) • H 1 (Γ) regularization of gradient • Obstacle : B (0 , 1) , incident wave d = ( − 1 , 0) , k = 9

Page 14 Numerical experiments : classical impedance problem Initial guess, exact solution, retrieved solutions with 0 and 2% noise 1.2 1 0.8 0.6 Re( λ ) = 0 Im( λ ) = sin 2 ( θ ) 0.4 0.2 0 −0.2 −3 −2 −1 0 1 2 3

Page 15 Numerical experiments : classical impedance problem 4 directions of incident wave, measurements limited to 1 / 4 -th of S 1 1 0.9 0.8 0.7 0.6 Re( λ ) = 0 0.5 Im( λ ) = sin 2 ( θ ) 0.4 0.3 0.2 0.1 0 −3 −2 −1 0 1 2 3

Page 16 Numerical experiments : classical impedance problem 4 directions of incident wave, measurements limited to 1 / 4 -th of S 1 1.2 1 0.8 Re( λ ) = 0 0.6 Im( λ ) = sin 2 ( θ − π/ 4) 0.4 0.2 0 −0.2 −3 −2 −1 0 1 2 3

Page 17 Numerical experiments : generalized impedance problem 1.2 1 Second example : 0.8 Re( λ ) = 0 0.6 Im( µ ) = 0 0.4 Im( λ ) = sin 2 ( θ ) Re( µ ) = 0 . 5 0.2 0 −0.2 −3 −2 −1 0 1 2 3

Page 18 Perspectives • Improve uniqueness results for our GIBC • Obtain logarithmic stability results for our GIBC without restriction on the set of parameters • Other GIBCs, for example involving div Γ ( µ ( x ) ∇ Γ u ) • Uniqueness from backscattering data : an open problem