Inverse scattering problem from an impedance obstacle Lee, Kuo-Ming - - PowerPoint PPT Presentation

Scattering Problem Inverse Scattering Problem Numerical examples

Inverse scattering problem from an impedance

• bstacle

Lee, Kuo-Ming

Department of Mathematics, NCKU

5th 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

1

Scattering Problem Direct Scattering Problem

2

Inverse Scattering Problem Regularization Reconstruction

3

Numerical examples Ellipse Peanut Bean

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

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 Numerical examples Direct Scattering Problem

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 Numerical examples Direct Scattering Problem

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 Numerical examples Direct Scattering Problem

Direct problem

Definition 1 Find: us ∈ C2(R2 \ ¯ D) ∩ C(R2 \ D) satisfies

1

the Helmholtz equation ∆us + k2us = 0, in R2 \ ¯ D

2

the impedance boundary condition ∂u ∂ν + λu = 0

• n ∂D

(1) for the total field u := ui + us

3

the Sommerfeld radiation condition(SRC) limr→∞ √r

• ∂us

∂ν − ikus

= 0, r := |x|, ˆ x :=

x |x|

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

Green’s representation formula

us(x) =

• ∂D

us(y)∂Φ(x, y) ∂ν(y) − ∂us(y) ∂ν(y) Φ(x, y)ds(y), x ∈ I R2 \ D (2)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

Solution ansatz

us(x) =

• ∂D

u(y)∂Φ(x, y) ∂ν(y) +λ(y)u(y)Φ(x, y)ds(y), x ∈ I R2\D (3)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

Integral operators

Sϕ(x) := 2

• ∂D

Φ(x, y)ϕ(y)ds(y) (4) Kϕ(x) := 2

• ∂D

∂Φ(x, y) ∂ν(y) ϕ(y)ds(y) (5)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

Well-posedness of DP

Theorem 1 The direct problem has a unique solution given by us(x) =

• ∂D

u(y)∂Φ(x, y) ∂ν(y) +λ(y)u(y)Φ(x, y)ds(y), x ∈ I R2 \ ¯ D (6) where (the total field) u is the (unique) solution to the following boundary integral equation u − Ku − S(λu) = 2ui,

• n ∂D

(7)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

Far field pattern u∞

The far field pattern or the scattering amplitude is given by us(x) = eik|x|

• |x|
• u∞(ˆ

x) + O 1 |x|

• |x| → ∞

uniformly for all directions ˆ x ∈ Ω := {x ∈ R2||x| = 1}. In our case

u∞(ˆ x) =

• ∂D

(c1 < ν(y), ˆ x > +c2λ(y)) e−ik<ˆ

x,y>u(y)ds(y)

(8)

where c1 = 1−i

4

• k

π,

c2 =

1+i 4 √ kπ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

Far field pattern u∞

The far field pattern or the scattering amplitude is given by us(x) = eik|x|

• |x|
• u∞(ˆ

x) + O 1 |x|

• |x| → ∞

uniformly for all directions ˆ x ∈ Ω := {x ∈ R2||x| = 1}. In our case

u∞(ˆ x) =

• ∂D

(c1 < ν(y), ˆ x > +c2λ(y)) e−ik<ˆ

x,y>u(y)ds(y)

(8)

where c1 = 1−i

4

• k

π,

c2 =

1+i 4 √ kπ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

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(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(∂D, λ, u), ˆ x ∈ Ω (9)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Direct Scattering Problem

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(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(∂D, λ, u), ˆ x ∈ Ω (9)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

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

• ne wave number k > 0.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Unique solvability

Uniqueness Not available Existence Not available

?

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Unique solvability

Uniqueness Not available Existence Not available

?

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Unique solvability

Uniqueness Not available Existence Not available

?

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Unique solvability

Uniqueness Not available Existence Not available

?

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Unique solvability

Uniqueness Not available Existence Not available

?

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

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 Inverse Scattering Problem Numerical examples Regularization Reconstruction

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 Inverse Scattering Problem Numerical examples Regularization Reconstruction

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 Inverse Scattering Problem Numerical examples Regularization Reconstruction

Fredholm integral Equations 2. Kind

ϕ(x)−1 2 1 (x+1)e−xyϕ(y)dy = e−x−1 2+1 2e−(x+1), 0 ≤ x ≤ 1

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 Inverse Scattering Problem Numerical examples Regularization Reconstruction

Fredholm integral Equations 1. Kind

1 (x + 1)e−xyϕ(y)dy = 1 − e−(x+1), 0 ≤ x ≤ 1

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

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Ill-Posed Problems : Regularization

Definition 3 (Regularization) Assume X, Y are normed spaces. Let the operator A : X → Y be linear, bounded and injective. A family of bounded linear operators Rα : Y → X, α > 0 is called a regularization scheme for Aϕ = f, if it satisfies the following pointwise convergence lim

α→0 RαAϕ = ϕ, for all ϕ ∈ X

In this case, the parameter α is called the regularization parameter.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Ill-Posed Problems : Regularization

Definition 3 (Regularization) Assume X, Y are normed spaces. Let the operator A : X → Y be linear, bounded and injective. A family of bounded linear operators Rα : Y → X, α > 0 is called a regularization scheme for Aϕ = f, if it satisfies the following pointwise convergence lim

α→0 RαAϕ = ϕ, for all ϕ ∈ X

In this case, the parameter α is called the regularization parameter.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Error

Find a stable approximation to the equation Aϕ = f The regularized approximation ϕδ

α := Rαf δ

The total approximation error ϕδ

α − ϕ = Rαf δ − Rαf + RαAϕ − ϕ

We have ϕδ

α − ϕ ≤ δRα + RαAϕ − ϕ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Error

Find a stable approximation to the equation Aϕ = f The regularized approximation ϕδ

α := Rαf δ

The total approximation error ϕδ

α − ϕ = Rαf δ − Rαf + RαAϕ − ϕ

We have ϕδ

α − ϕ ≤ δRα + RαAϕ − ϕ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Parameter

How to choose the regularization parameter α ?

1

a priori choice based on some information of the solution. In general not available

2

a posteriori choice based on the data error level δ Discrepancy Principle of Morozov : ARαf δ − f δ = γδ, γ ≥ 1

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Parameter

How to choose the regularization parameter α ?

1

a priori choice based on some information of the solution. In general not available

2

a posteriori choice based on the data error level δ Discrepancy Principle of Morozov : ARαf δ − f δ = γδ, γ ≥ 1

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Parameter

How to choose the regularization parameter α ?

1

a priori choice based on some information of the solution. In general not available

2

a posteriori choice based on the data error level δ Discrepancy Principle of Morozov : ARαf δ − f δ = γδ, γ ≥ 1

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Parameter

How to choose the regularization parameter α ?

1

a priori choice based on some information of the solution. In general not available

2

a posteriori choice based on the data error level δ Discrepancy Principle of Morozov : ARαf δ − f δ = γδ, γ ≥ 1

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Example

X, Y Hilbert spaces. Theorem 2 Assume A : X → Y compact and linear. Then for every α > 0, the operator αI + A∗A : X → X is bijective and has a bounded inverse. Furthermore, if the operator A is injective, then Rα := (αI + A∗A)−1 A∗, α > 0 describes a regularization scheme with Rα ≤

1 2√α.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Regularization : Example

X, Y Hilbert spaces. Theorem 2 Assume A : X → Y compact and linear. Then for every α > 0, the operator αI + A∗A : X → X is bijective and has a bounded inverse. Furthermore, if the operator A is injective, then Rα := (αI + A∗A)−1 A∗, α > 0 describes a regularization scheme with Rα ≤

1 2√α.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Tikhonov Regularization

Theorem 3 Let A : X → Y be a linear and bounded operator. Assme α > 0. Then for each f ∈ Y there exists a unique ϕα ∈ X such that Aϕα − f + αϕα = infϕ∈X

• Aϕ − f2 + αϕ2

The minimizer ϕα is given by the unique solution of the equation αϕα + A∗Aϕα = A∗f and depends continuously on f.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Reconstruction: Newton’s method

F(∂D, λ, u) = u∞ ∇3F · Θ = u∞ − F (11)     αI βI γI   + A∗A   Θ = A∗(u∞ − F) (12) where A =  

∂F ∂∂D ∂F ∂λ ∂F ∂u

 

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Reconstruction: Newton’s method

F(∂D, λ, u) = u∞ ∇3F · Θ = u∞ − F (11)     αI βI γI   + A∗A   Θ = A∗(u∞ − F) (12) where A =  

∂F ∂∂D ∂F ∂λ ∂F ∂u

 

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method

Recall the solution of direct problem (9)

• u − Ku − S(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(γ, λ, u), ˆ x ∈ Ω Split the inverse problem into two parts:

1

B(γ, λ)u = 2ui (13)

2

F(u)(γ, λ) = u∞ (14)

(13) is solved as a (well-posed) direct problem. (14) is solved as an ill-posed problem with two regularization parameters.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method

Recall the solution of direct problem (9)

• u − Ku − S(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(γ, λ, u), ˆ x ∈ Ω Split the inverse problem into two parts:

1

B(γ, λ)u = 2ui (13)

2

F(u)(γ, λ) = u∞ (14)

(13) is solved as a (well-posed) direct problem. (14) is solved as an ill-posed problem with two regularization parameters.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method

Recall the solution of direct problem (9)

• u − Ku − S(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(γ, λ, u), ˆ x ∈ Ω Split the inverse problem into two parts:

1

B(γ, λ)u = 2ui (13)

2

F(u)(γ, λ) = u∞ (14)

(13) is solved as a (well-posed) direct problem. (14) is solved as an ill-posed problem with two regularization parameters.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method

Recall the solution of direct problem (9)

• u − Ku − S(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(γ, λ, u), ˆ x ∈ Ω Split the inverse problem into two parts:

1

B(γ, λ)u = 2ui (13)

2

F(u)(γ, λ) = u∞ (14)

(13) is solved as a (well-posed) direct problem. (14) is solved as an ill-posed problem with two regularization parameters.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method

Recall the solution of direct problem (9)

• u − Ku − S(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(γ, λ, u), ˆ x ∈ Ω Split the inverse problem into two parts:

1

B(γ, λ)u = 2ui (13)

2

F(u)(γ, λ) = u∞ (14)

(13) is solved as a (well-posed) direct problem. (14) is solved as an ill-posed problem with two regularization parameters.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method

Recall the solution of direct problem (9)

• u − Ku − S(λu) = 2ui,
• n ∂D

u∞(ˆ x) = F(γ, λ, u), ˆ x ∈ Ω Split the inverse problem into two parts:

1

B(γ, λ)u = 2ui (13)

2

F(u)(γ, λ) = u∞ (14)

(13) is solved as a (well-posed) direct problem. (14) is solved as an ill-posed problem with two regularization parameters.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method : iterative scheme

1

Given a pair of initial guesses γ0, λ0.

2

Solve (13) for u.

3

Solve the regularized version of (14) for updates of γ, λ : αI βI

• + A∗A

χ q

• = A∗(u∞ − F)

(15) where A =

• ∂F

∂γ ∂F ∂λ

• 4

Set γ0 = γ0 + χ, λ0 = λ0 + q and repeat steps 2,3 until some criterion is fullfilled.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method : iterative scheme

1

Given a pair of initial guesses γ0, λ0.

2

Solve (13) for u.

3

Solve the regularized version of (14) for updates of γ, λ : αI βI

• + A∗A

χ q

• = A∗(u∞ − F)

(15) where A =

• ∂F

∂γ ∂F ∂λ

• 4

Set γ0 = γ0 + χ, λ0 = λ0 + q and repeat steps 2,3 until some criterion is fullfilled.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method : iterative scheme

1

Given a pair of initial guesses γ0, λ0.

2

Solve (13) for u.

3

Solve the regularized version of (14) for updates of γ, λ : αI βI

• + A∗A

χ q

• = A∗(u∞ − F)

(15) where A =

• ∂F

∂γ ∂F ∂λ

• 4

Set γ0 = γ0 + χ, λ0 = λ0 + q and repeat steps 2,3 until some criterion is fullfilled.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method : iterative scheme

1

Given a pair of initial guesses γ0, λ0.

2

Solve (13) for u.

3

Solve the regularized version of (14) for updates of γ, λ : αI βI

• + A∗A

χ q

• = A∗(u∞ − F)

(15) where A =

• ∂F

∂γ ∂F ∂λ

• 4

Set γ0 = γ0 + χ, λ0 = λ0 + q and repeat steps 2,3 until some criterion is fullfilled.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Modified Newton’s Method : iterative scheme

1

Given a pair of initial guesses γ0, λ0.

2

Solve (13) for u.

3

Solve the regularized version of (14) for updates of γ, λ : αI βI

• + A∗A

χ q

• = A∗(u∞ − F)

(15) where A =

• ∂F

∂γ ∂F ∂λ

• 4

Set γ0 = γ0 + χ, λ0 = λ0 + q and repeat steps 2,3 until some criterion is fullfilled.

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Major Advantages

No extra equations Two smaller systems Fr´ echet derivatives are easily obtained The unique solvability of the numerical scheme followed straight forward

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Numerical Settings: Solution spaces

Vm := span{1, cos t, cos 2t, . . . , cos mt; sin t, . . . , sin(m − 1)t} Γ := (γ1(t), γ2(t)) ∈ Vm × Vm λ ∈ Vk Stopping criterion for the Newton’s method: Discrepancy Principle: u∞,k − u∞ ≤ ǫ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Numerical Settings: Solution spaces

Vm := span{1, cos t, cos 2t, . . . , cos mt; sin t, . . . , sin(m − 1)t} Γ := (γ1(t), γ2(t)) ∈ Vm × Vm λ ∈ Vk Stopping criterion for the Newton’s method: Discrepancy Principle: u∞,k − u∞ ≤ ǫ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Regularization Reconstruction

Numerical Settings: Solution spaces

Vm := span{1, cos t, cos 2t, . . . , cos mt; sin t, . . . , sin(m − 1)t} Γ := (γ1(t), γ2(t)) ∈ Vm × Vm λ ∈ Vk Stopping criterion for the Newton’s method: Discrepancy Principle: u∞,k − u∞ ≤ ǫ

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Ellipse with exact data

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

Γ = (0.4 cos t, 0.3 sin t)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Ellipse with exact data

1 2 3 4 5 6 7 −0.2 0.2 0.4 0.6 0.8

λ = 0.5 + 0.2 cos t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Ellipse with 3% noise

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

Γ = (0.4 cos t, 0.3 sin t)

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Ellipse with 3% noise

1 2 3 4 5 6 7 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

λ = 0.5 + 0.2 cos t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Peanut A

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

Γ = γ(t)(cos t, sin t), γ(t) =

• cos2 t + 0.25 sin2 t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Peanut A

1 2 3 4 5 6 7 −0.2 0.2 0.4 0.6 0.8

λ = 0.3 + 0.1 sin t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Peanut B

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

Γ = γ(t)(cos t, sin t), γ(t) =

• cos2 t + 0.25 sin2 t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Peanut B

1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

λ = 0.1 sin 2t + 0.3 − 0.2 cos t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Peanut C

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

Γ = γ(t)(cos t, sin t), γ(t) =

• cos2 t + 0.25 sin2 t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Peanut C

1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

λ = 0.3e0.25 cos3 t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Bean with exact data

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

Γ = γ(t)(cos t, sin t), γ(t) = 1 + 0.9 cos t + 0.1 sin(2t) 1 + 0.75 cos t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Bean with exact data

1 2 3 4 5 6 7 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

λ = 0.3 + 0.1 sin t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Bean with 3% noise

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

Γ = γ(t)(cos t, sin t), γ(t) = 1 + 0.9 cos t + 0.1 sin(2t) 1 + 0.75 cos t

Lee, Kuo-Ming Impedance Problem

Scattering Problem Inverse Scattering Problem Numerical examples Ellipse Peanut Bean

Bean with 3% noise

1 2 3 4 5 6 7 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

λ = 0.3 + 0.1 sin t

Lee, Kuo-Ming Impedance Problem