Scattering Problem Inverse Scattering Problem Numerical examples Inverse scattering problem from an impedance obstacle Lee, Kuo-Ming Department of Mathematics, NCKU 5 th Workshop on Boundary Element Methods, Integral Equations and Related Topics in Taiwan NSYSU, October 4, 2014 Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Numerical examples Outline Scattering Problem 1 Direct Scattering Problem Inverse Scattering Problem 2 Regularization Reconstruction Numerical examples 3 Ellipse Peanut Bean Lee, Kuo-Ming Impedance Problem
✄ � ✁ ✟ ✂ ✁ ✞ ✄ ✁ � ✁ ✂ ✁ Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Scattering problem Object : time harmonic acoustic scattering Modelling : Exterior boundary value problem for the Helmholtz equation ✄✆☎ ✁ ✂✁ ✄✆✝ Lee, Kuo-Ming Impedance Problem
✄ � ✁ ✟ ✂ ✁ ✞ ✄ ✁ � ✁ ✂ ✁ Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Scattering problem Object : time harmonic acoustic scattering Modelling : Exterior boundary value problem for the Helmholtz equation ✄✆☎ ✁ ✂✁ ✄✆✝ Lee, Kuo-Ming Impedance Problem
✄ � ✁ ✟ ✂ ✁ ✞ ✄ ✁ � ✁ ✂ ✁ Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Scattering problem Object : time harmonic acoustic scattering Modelling : Exterior boundary value problem for the Helmholtz equation ✄✆☎ ✁ ✂✁ ✄✆✝ Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Direct problem Definition 1 Find: u s ∈ C 2 ( R 2 \ ¯ D ) ∩ C ( R 2 \ D ) satisfies the Helmholtz equation ∆ u s + k 2 u s = 0 , in R 2 \ ¯ D 1 the impedance boundary condition 2 ∂ u ∂ν + λ u = 0 on ∂ D (1) for the total field u := u i + u s the Sommerfeld radiation condition(SRC) 3 √ r � ∂ν − iku s � ∂ u s x r := | x | , ˆ lim r →∞ = 0 , x := | x | Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Green’s representation formula − ∂ u s ( y ) � u s ( y ) ∂ Φ( x , y ) R 2 \ D u s ( x ) = ∂ν ( y ) Φ( x , y ) ds ( y ) , x ∈ I ∂ν ( y ) ∂ D (2) Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Solution ansatz � u ( y ) ∂ Φ( x , y ) u s ( x ) = R 2 \ D ∂ν ( y ) + λ ( y ) u ( y )Φ( x , y ) ds ( y ) , x ∈ I ∂ D (3) Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Integral operators � S ϕ ( x ) := 2 Φ( x , y ) ϕ ( y ) ds ( y ) (4) ∂ D � ∂ Φ( x , y ) K ϕ ( x ) := 2 ∂ν ( y ) ϕ ( y ) ds ( y ) (5) ∂ D Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Well-posedness of DP Theorem 1 The direct problem has a unique solution given by � u ( y ) ∂ Φ( x , y ) R 2 \ ¯ u s ( x ) = + λ ( y ) u ( y )Φ( x , y ) ds ( y ) , x ∈ I D ∂ν ( y ) ∂ D (6) where (the total field) u is the (unique) solution to the following boundary integral equation u − Ku − S ( λ u ) = 2 u i , on ∂ D (7) Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Far field pattern u ∞ The far field pattern or the scattering amplitude is given by � 1 u s ( x ) = e i k | x | � �� u ∞ (ˆ x ) + O | x | → ∞ � | x | | x | x ∈ Ω := { x ∈ R 2 || x | = 1 } . uniformly for all directions ˆ In our case � x > + c 2 λ ( y )) e − i k < ˆ x , y > u ( y ) ds ( y ) u ∞ (ˆ x ) = ( c 1 < ν ( y ) , ˆ (8) ∂ D � where c 1 = 1 − i 1 + i k π , c 2 = √ 4 4 k π Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Far field pattern u ∞ The far field pattern or the scattering amplitude is given by � 1 u s ( x ) = e i k | x | � �� u ∞ (ˆ x ) + O | x | → ∞ � | x | | x | x ∈ Ω := { x ∈ R 2 || x | = 1 } . uniformly for all directions ˆ In our case � x > + c 2 λ ( y )) e − i k < ˆ x , y > u ( y ) ds ( y ) u ∞ (ˆ x ) = ( c 1 < ν ( y ) , ˆ (8) ∂ D � where c 1 = 1 − i 1 + i k π , c 2 = √ 4 4 k π Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Summary : Direct Problem The direct problem can be understood as the process of calculating the far-field pattern from an impedance obstacle. Mathematically, it is equivalent to the solving of the system: � u − Ku − S ( λ u ) = 2 u i , on ∂ D (9) u ∞ (ˆ ˆ x ) = F ( ∂ D , λ, u ) , x ∈ Ω Lee, Kuo-Ming Impedance Problem
Scattering Problem Inverse Scattering Problem Direct Scattering Problem Numerical examples Summary : Direct Problem The direct problem can be understood as the process of calculating the far-field pattern from an impedance obstacle. Mathematically, it is equivalent to the solving of the system: � u − Ku − S ( λ u ) = 2 u i , on ∂ D (9) u ∞ (ˆ ˆ x ) = F ( ∂ D , λ, u ) , x ∈ Ω Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Inverse Problem Definition 2 (IP) Determine both the scatterer D and the impedance λ if the far field pattern u ∞ ( · , d ) is known for one incident direction d and one wave number k > 0. Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Unique solvability Uniqueness Not available Existence Not available ? Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Unique solvability Uniqueness Not available Existence Not available ? Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Unique solvability Uniqueness Not available Existence Not available ? Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Unique solvability Uniqueness Not available Existence Not available ? Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Unique solvability Uniqueness Not available Existence Not available ? Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Comments on Existence Solving the inverse problem means to solve the far field equation F ( ∂ D , λ, u ) = u ∞ (10) However (10) is an equation of the first kind The operator F is compact F has no bounded inverse in general This means that equation (10) cannot be resonably solved ! Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Comments on Existence Solving the inverse problem means to solve the far field equation F ( ∂ D , λ, u ) = u ∞ (10) However (10) is an equation of the first kind The operator F is compact F has no bounded inverse in general This means that equation (10) cannot be resonably solved ! Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Comments on Existence Solving the inverse problem means to solve the far field equation F ( ∂ D , λ, u ) = u ∞ (10) However (10) is an equation of the first kind The operator F is compact F has no bounded inverse in general This means that equation (10) cannot be resonably solved ! Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Fredholm integral Equations 2. Kind � 1 ϕ ( x ) − 1 ( x + 1 ) e − xy ϕ ( y ) dy = e − x − 1 2 + 1 2 e − ( x + 1 ) , 0 ≤ x ≤ 1 2 0 Trapzoidal rule n x = 0 x = 0 . 5 x = 1 4 -0.007146 -0.010816 -0.015479 8 -0.001788 -0.002711 -0.003882 16 -0.000447 -0.000678 -0.000971 32 -0.000112 -0.000170 -0.000243 Simpson’s rule n x = 0 x = 0 . 5 x = 1 4 -0.00006652 -0.00010905 -0.00021416 8 -0.00000422 -0.00000692 -0.00001366 16 -0.00000026 -0.00000043 -0.00000086 32 -0.00000002 -0.00000003 -0.00000005 Lee, Kuo-Ming Impedance Problem
Scattering Problem Regularization Inverse Scattering Problem Reconstruction Numerical examples Fredholm integral Equations 1. Kind � 1 ( x + 1 ) e − xy ϕ ( y ) dy = 1 − e − ( x + 1 ) , 0 ≤ x ≤ 1 0 Trapzoidal rule n x = 0 x = 0 . 5 x = 1 4 0.4057 0.3705 0.1704 8 -4.5989 14.6094 -4.4770 16 -8.5957 2.2626 -153.4805 32 3.8965 -32.2907 22.5570 64 -88.6474 -6.4484 -182.6745 Simpson’s rule n x = 0 x = 0 . 5 x = 1 4 0.0997 0.2176 0.0566 8 -0.5463 6.0868 -1.7274 16 -15.4796 50.5015 -53.8837 32 24.5929 -24.1767 67.9655 64 23.7868 -17.5992 419.4284 Lee, Kuo-Ming Impedance Problem
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