f g (3)Physical Background (Forward) Conductivity Problem Ω

Inverse Conductivity Problem

[ A motivation to study the Cauchy problem ]

이화여대 수학과 이 준 엽

The Cauchy Problem

(Forward) Conductivity Problem

Ω

∂

∂ =

= ∂

Ω

Ω

∈

on n g

OR u f

u

t s in

x u find

x x

given For

. . )

(

, ),

σ (

=0

∇

⋅

∇ σ u The Cauchy Problem

. .

. )

(

, )

( ₀

Ω

∂

⊂ Γ

=

=

Ω

Ω

=

∂∂ g on AND

f u

t s in

x u find

in x

given For

n u

σ σ

) Γ

, ( f g

Physical Background

(Forward) Conductivity Problem

Ω

∂

∂ =

= ∂

Ω

Ω

∈

on n g

OR u f

u

t s in

x u find

x x

given For

. . )

(

, ),

σ (

=0

∇

⋅

∇ σ u Inverse Conductivity Problem

Ω

∂

∂ =

= ∂ g on

n AND u

f u

given from

x Find σ ( )

Ω

) ∂

, ( f g

Impedance

~Voltage/Current

Tomography

Electrical Impedance Tomography (EIT)

A model problem with heart and lung

Methods of EIT/ICP

n g u u

x u

f_{k k}

Min∑ ^{− ( ) where∇⋅ ∇ = 0, ∂}_∂ ⁼

Objective _σ ² _σ

σ

σ

voltage f_i =

current g_i =

) σ (x

• Single Measurement

⇒ minimum information

• Many Measurement

⇒ full information (?) on σ

• Infinite Measurement

⇒ mathematical theory

Numerical Obstacles for EIT/ICP

σ σ

σ

σ σ

σ σ σ

better ~ Find

, 0 Solve

Guss

, 0 where

) ( )

( Objective

)

2(

→

∂ =

= ∂

∇

⋅

∇

→

∂ =

= ∂

∇

⋅

∇

∑ − ^Ω n g u u

σ(x)

n g u u

x u x

f L

Min

• Strongly nonlinear:

• Ill-posed problem:

) (

)

( ₂

2 Ω → ∈ ∂Ω

∈ L u_σ L σ

σ

σ σ

σ ≈/ ^~ → ^u ≈ ^u~

f u

n g

u u = → −

∂

= ∂

∇

⋅

∇ σ _σ 0, ^σ Find better σ~ using _σ Solve

σ 0 Regularity restrictiuon on σ condition

Stop u − f ≅ →

A case study

Finite Element Mesh High Conductivity Inc.

Finite Element Solution for σσσσ

• Minimize

• Newton’s method to solve the non-linear eq.

∑ ⁻

=

k

k

k u x

f

E(σ ) _σ ( ) ²

k i

k k

i

x u f x

E u

i σ

σ

σ _σ

σ ∂

⋅ ∂

−

∂ =

= ∂ ^{( ) 2}∑^{( ( ) ( ))}

0

i i

T

i

k k

T

J u u

H u

x f x

u J

H i

σ σ

σ δσ

σ σ

σ σ

∂

≈ ∂



 





∂

 ∂



 





∂

≈ ∂

−

−

≈ ⁻

,

where

)) (

) (

1 (

Ill-posed Nonlinear

• 50 iterations

Fundamental Difficulty of ICP

• Ill-poseness of Cauchy Problem

Cauchy Problem

Well-posed vs. Ill-posed

Well-posed Mixed boundary

value problem

Ill-posed

Cauchy (harmonic extention) problem

Singular Values of Cauchy Operator

Restrictions on conductivity Profile

• Mesh Grouping Algorithm

Requirements for Cauchy solver

• Numerically Stable

(small condition number)

• Numerically Accurate

(small computational error)

(1) Dotted : Error= 10^-5 (2) Dash-Dotted :

Error= 10^-10 (3) Solid :

Error=10^-15

(A) Small condition number (B) Large condition number