Some Old and Some New Results in Inverse Obstacle Scattering Rainer - PowerPoint PPT Presentation

Some Old and Some New Results in Inverse Obstacle Scattering Rainer Kress, Göttingen Analysis and Numerics of Acoustic and Electromagnetic Problems Linz, October 2016 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Inverse obstacle scattering ν � ✒ � ❳❳ ✄✄ ③ ❳ ✄ ✲ ✄ D u i = e i k x · d u = u i + u s u s , in R 3 \ ¯ ∆ u + k 2 u = 0 D u = 0 on ∂ D � 1 � ∂ u s ∂ r − i ku s = o , r = | x | → ∞ r � x � 1 � � �� u s ( x ) = e i k | x | u ∞ + O | x | → ∞ , | x | | x | | x | Given: Far field u ∞ for one (or several) incident plane wave Find: Shape and location of scatterer D 1. Uniqueness, i.e., identifiability 2. Reconstruction algorithms Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Outline Two old uniqueness proofs 1 Two new uniqueness results for the generalized impedance 2 boundary condition A recent iterative reconstruction algorithm for the 3 generalized impedance boundary condition Factorization method and transmission eigenvalues 4 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness, i.e., identifiability Rellich’s Lemma: u s = 0 in R 3 \ D on S 2 u ∞ = 0 ⇔ Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness, i.e., identifiability Rellich’s Lemma: u s = 0 in R 3 \ D on S 2 u ∞ = 0 ⇔ Far field u ∞ uniquely determines total field u = u i + u s u = 0 ❳❳ ❳ ③ ✄✄ ✄ ✄ ✲ D u i u s Question of uniqueness: ⇔ Existence of additional closed surfaces with u = 0 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness, i.e., identifiability Question of uniqueness: ⇔ Existence of additional closed surfaces on which u = 0 u = 0 ❳❳ ✄✄ ③ ❳ ✄ ✄ ✲ D u i u s u = 0 No!! Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness, i.e., identifiability Question of uniqueness: ⇔ Existence of additional closed surfaces on which u = 0 u = 0 ❳❳ ✄✄ ③ ❳ ✄ ✲ ✄ D u i u s u = 0 Do not know??? Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Schiffer’s theorem Theorem (Schiffer ≈ 1960) For a sound-soft scatterer, assume that u ∞ , 1 (ˆ x , d ) = u ∞ , 2 (ˆ x , d ) for all observation directions ˆ x and all incident directions d. Then D 1 = D 2 . D 1 D 2 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Schiffer’s proof D ∗ = unbounded component of R 3 \ ( D 1 ∪ D 2 ) D 1 D 2 u s 1 ( · , d ) = u s in D ∗ 2 ( · , d ) Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Schiffer’s proof D ∗ = unbounded component of R 3 \ ( D 1 ∪ D 2 ) D 1 D 2 u s 1 ( · , d ) = u s in D ∗ 2 ( · , d ) u 1 ( x , d ) = e i k x · d + u s 1 ( x , d ) △ u 1 + k 2 u 1 = 0 In shaded domain: On boundary: u 1 = 0 { u 1 ( · , d ) : d ∈ S 2 } linearly independent Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Schiffer’s proof D ∗ = unbounded component of R 3 \ ( D 1 ∪ D 2 ) D 1 D 2 u s 1 ( · , d ) = u s in D ∗ 2 ( · , d ) u 1 ( x , d ) = e i k x · d + u s 1 ( x , d ) △ u 1 + k 2 u 1 = 0 In shaded domain: On boundary: u 1 = 0 { u 1 ( · , d ) : d ∈ S 2 } linearly independent Lax and Philipps 1967 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Schiffer’s proof Correct shaded domain: ( R 3 \ D ∗ ) \ D 1 D ∗ D 1 D 2 Incorrect shaded domain: D 2 \ ( D 1 ∩ D 2 ) D 1 D 2 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Schiffer’s proof Schiffer’s proof does not work for other boundary conditions! D 1 D 2 Finite multiplicity of eigenvalues is based on the compact embedding of H 1 0 (Ω) into L 2 (Ω) for the shaded domain Ω . However, in general, the embedding H 1 (Ω) into L 2 (Ω) is not compact. Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Other boundary conditions = u 0 on ∂ D , sound-soft ∂ u = 0 on ∂ D , sound-hard ∂ν ∂ u ∂ν + i k λ = 0 on ∂ D , impedance , ℜ λ ≥ 0 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness of obstacle plus boundary condition Theorem (Kirsch, K. 1992) Assume that u ∞ , 1 (ˆ x , d ) = u ∞ , 2 (ˆ x , d ) for all observation directions ˆ x and all incident directions d. Then D 1 = D 2 and B 1 = B 2 . B 1 D 1 D 2 B 2 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof x ∗ z ∗ D w i ( x , z ) = e i k | x − z | | x − z | = incident field, point source w s ( x , z ) = scattered field, w ∞ (ˆ x , z ) = far field Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof x ∗ z ∗ D w i ( x , z ) = e i k | x − z | | x − z | = incident field, point source w s ( x , z ) = scattered field, w ∞ (ˆ x , z ) = far field w s ( x , z ) = w s ( z , x ) reciprocity : u ∞ (ˆ x , d ) = u ∞ ( − d , − ˆ x ) , u s ( z , d ) = w ∞ ( − d , z ) mixed reciprocity : Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof mixed reciprocity : u s ( z , d ) = w ∞ ( − d , z ) D ∗ B 1 x ⋆ ⋆ z D 2 D 1 B 2 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof mixed reciprocity : u s ( z , d ) = w ∞ ( − d , z ) D ∗ B 1 x ⋆ ⋆ z D 2 D 1 B 2 u ∞ , 1 (ˆ u ∞ , 2 (ˆ for | ˆ x , d ) = x , d ) x | = | d | = 1 u s u s for z ∈ D ∗ , | d | = 1 1 ( z , d ) = 2 ( z , d ) for z ∈ D ∗ , | d | = 1 w ∞ , 1 ( d , z ) = w ∞ , 2 ( d , z ) w s w s for x , z ∈ D ∗ 1 ( x , z ) = 2 ( x , z ) Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof D ∗ B 1 ⋆ x ∗ ⋆ z D 2 D 1 B 2 w s 1 ( x , z ) = w s for x , z ∈ D ∗ 2 ( x , z ) Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof D ∗ B 1 ⋆ x ∗ ⋆ z D 2 D 1 B 2 w s 1 ( x , z ) = w s for x , z ∈ D ∗ 2 ( x , z ) z → x ∗ B 1 w s 1 ( x ∗ , z ) = ∞ , z → x ∗ B 1 w s 2 ( x ∗ , z ) = finite lim lim ⇒ D 1 = D 2 Holmgren’s theorem yields B 1 = B 2 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Idea of proof D ∗ B 1 ⋆ x ∗ ⋆ z D 2 D 1 B 2 w s 1 ( x , z ) = w s for x , z ∈ D ∗ 2 ( x , z ) z → x ∗ B 1 w s 1 ( x ∗ , z ) = ∞ , z → x ∗ B 1 w s 2 ( x ∗ , z ) = finite lim lim ⇒ D 1 = D 2 Holmgren’s theorem yields B 1 = B 2 Use of mixed reciprocity: Potthast 1999 Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Generalized impedance condition ∂ u ∂ν + i k ( λ u − div ∂ D µ grad ∂ D u ) = 0 ℜ λ, ℜ µ ≥ 0 on ∂ D , Theorem (Bourgeois, Chaulet, Haddar 2012) Both the shape and the impedance functions of a scattering obstacle with generalized impedance condition are uniquely determined by the far field patterns for an infinite number of incident waves with distinct incident directions and one fixed wave number. Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness for impedance functions in R 2 � � ∂ u λ u − d ds µ du ∂ν + i k = 0 on ∂ D ds Theorem (Cakoni, K. 2013) In two dimensions, for a given shape ∂ D, three far field patterns corresponding to the scattering of three plane waves with different incident directions uniquely determine the impedance functions µ and λ . Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Uniqueness for impedance functions in R 2 � � ∂ u λ u − d ds µ du ∂ν + i k = 0 on ∂ D ds Theorem (Cakoni, K. 2013) In two dimensions, for a given shape ∂ D, three far field patterns corresponding to the scattering of three plane waves with different incident directions uniquely determine the impedance functions µ and λ . � � i k d du 2 du 1 ∂ u 2 ∂ u 1 ds µ u 1 ds − u 2 = u 1 ∂ν − u 2 on ∂ D ds ∂ν � �� � W ( u 1 , u 2 ) α 3 α 1 α 2 ⇒ W ( u 1 , u 2 ) = W ( u 2 , u 3 ) = W ( u 3 , u 1 ) Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Iterative methods versus qualitative methods Iterative methods: Reformulate inverse problem as nonlinear ill-posed operator equation. Solve by iteration methods such as regularized Newton methods, Landweber iterations or conjugate gradient methods Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Iterative methods versus qualitative methods Iterative methods: Reformulate inverse problem as nonlinear ill-posed operator equation. Solve by iteration methods such as regularized Newton methods, Landweber iterations or conjugate gradient methods Qualitative methods: Develop criterium in terms of behaviour of certain ill-posed linear integral equations that decide on whether a point lies inside or outside the scatterer. Linear sampling, factorization, probe methods, etc Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering

Single-layer approach Assume that k 2 is not a Dirichlet eigenvalue for − ∆ in D � e ik | x − y | u s ( x ) = 1 x ∈ R 3 \ D | x − y | ϕ ( y ) ds ( y ) , 4 π ∂ D Rainer Kress Some Old and Some New Results in Inverse Obstacle Scattering