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(1)

Topological derivatives for shape and

parameter reconstruction

Ana Carpio

1

María–Luisa Rapún

2

1

Matemática Aplicada, Universidad Complutense de Madrid, Spain

2

Fundamentos Matemáticos, Universidad Politécnica de Madrid, Spain

(2)

1

Inverse scattering problems

2

Topological derivative methods

TD for shape reconstruction

TD for shapes and parameters

3

Conclusions

(3)

Description of the problem

Medium

R

with obstacles

:

How many? how big? where?

physical properties in

?

Some applications

Medicine (tumors, fracture)

Geophysics (oil, gas)

(4)

Scattering problem

An incident wave

u

inc

interacts with a medium

R

containing objects

.

Forward (direct) problem

The shape, size, location and physical properties of the

objects are known

Compute the response of the system at the detectors "

×

"

(5)

Scattering problem

An incident wave

u

inc

interacts with a medium

R

containing objects

.

Forward (direct) problem

The shape, size, location and physical properties of the

objects are known

Compute the response of the system at the detectors "

×

"

(6)

Scattering problem

An incident wave

u

inc

interacts with a medium

R

containing objects

.

Inverse problem

Measurements

umeas

are taken at the receptors

Find the scatters

and the interior parameters s.t.

u

=

u

meas

on

Γ

meas

,

u= sol. forward problem

(7)

Model problem

We assume that

We generate

acustic waves

Incident waves are

time–harmonic

U

inc

(x

,

t

) =

Re

[e

i

ω

t

u

inc

(x)]

,

u

inc

is a

planar wave

in the direction

d

,

u

inc

(x) =

e

ik

x

·

d

The solution to the direct problem is time–harmonic

(8)

We assume that

We generate

acustic waves

Incident waves are

time–harmonic

U

inc

(

x

,

t

) =

Re

[

e

i

ωt

u

inc

(

x

)]

,

u

inc

is a

planar wave

in the direction

d

,

u

inc

(x) =

e

ik

x

·

d

The solution to the direct problem is time–harmonic

(9)

Model problem

We assume that

We generate

acustic waves

Incident waves are

time–harmonic

U

inc

(

x

,

t

) =

Re

[

e

i

ωt

u

inc

(

x

)]

,

u

inc

is a

planar wave

in the direction

d

,

u

inc

(

x

) =

e

ik

x

·

d

The solution to the direct problem is time–harmonic

(10)

We assume that

We generate

acustic waves

Incident waves are

time–harmonic

U

inc

(

x

,

t

) =

Re

[

e

i

ωt

u

inc

(

x

)]

,

u

inc

is a

planar wave

in the direction

d

,

u

inc

(

x

) =

e

ik

x

·

d

The solution to the direct problem is time–harmonic

(11)

A simple forward problem

is a penetrable known obstacle. The incident field generates

a scattered wave

usc

in

R

n

\

and a transmitted wave

utr

in

.

The total field

u

=

u

inc

+

u

sc

in

R

n

\

and

u

=

u

tr

in

solves

∆u

+

k

e

2

u

=

0

in

R

n

\

∆u

+

k

i

2

u

=

0

in

u

=

u

+

, ∂

n

u

=

n

u

+

on

lim

r

→∞

r

(n

1

)/

2

(

r

(u

u

inc

)

ik

e

(u

u

inc

)) =

0

(12)

1

Inverse scattering problems

2

Topological derivative methods

TD for shape reconstruction

TD for shapes and parameters

3

Conclusions

(13)

Constrained optimization

Original problem

(we assume that

k

i

is known)

Find

such that

u

=

u

meas

on

Γ

meas

A weaker formulation

Find

minimizing

J

(Ω) =

1

2

Γ

meas

|u

u

meas

|

2

for

u

solving the forward problem with objects

The domain

is the variable

(14)

Original problem

(we assume that

k

i

is known)

Find

such that

u

=

u

meas

on

Γ

meas

A weaker formulation

Find

minimizing

J

(Ω) =

1

2

Γ

meas

|u

u

meas

|

2

for

u

solving the forward problem with objects

The domain

is the variable

(15)

Constrained optimization

Original problem

(we assume that

k

i

is known)

Find

such that

u

=

u

meas

on

Γ

meas

A weaker formulation

Find

minimizing

J

(Ω) =

1

2

Γ

meas

|u

u

meas

|

2

for

u

solving the forward problem with objects

The domain

is the variable

(16)

Some alternatives

Modified gradient methods:

differ on how an initial guess is

deformed from one iteration to the next in such a way that the

cost functional decreases

Classical deformations

following a vector field

Problem: The number of scatterers has to be known from

the beginning

Kirsch 1993, Hettlich 1995, Potthast 1996

Level set based deformations

allow changes in topology

Problem: Slow evolution. Initial guess?

Santosa 1996, Dorn 2005

Topological derivatives

Provide good initial guesses

(17)

Some alternatives

Modified gradient methods:

differ on how an initial guess is

deformed from one iteration to the next in such a way that the

cost functional decreases

Classical deformations

following a vector field

Problem: The number of scatterers has to be known from

the beginning

Kirsch 1993, Hettlich 1995, Potthast 1996

Level set based deformations

allow changes in topology

Problem: Slow evolution. Initial guess?

Santosa 1996, Dorn 2005

Topological derivatives

Provide good initial guesses

(18)

Some alternatives

Modified gradient methods:

differ on how an initial guess is

deformed from one iteration to the next in such a way that the

cost functional decreases

Classical deformations

following a vector field

Problem: The number of scatterers has to be known from

the beginning

Kirsch 1993, Hettlich 1995, Potthast 1996

Level set based deformations

allow changes in topology

Problem: Slow evolution. Initial guess?

Santosa 1996, Dorn 2005

Topological derivatives

Provide good initial guesses

(19)

Some alternatives

Modified gradient methods:

differ on how an initial guess is

deformed from one iteration to the next in such a way that the

cost functional decreases

Classical deformations

following a vector field

Problem: The number of scatterers has to be known from

the beginning

Kirsch 1993, Hettlich 1995, Potthast 1996

Level set based deformations

allow changes in topology

Problem: Slow evolution. Initial guess?

Santosa 1996, Dorn 2005

Topological derivatives

Provide good initial guesses

(20)

Definition of Topological Derivative (Sokowloski–Zochowski ’99)

The TD of a shape functional

J

(

R

)

at a point

x

R

is

D

T

(

x,

R

) =

lim

ε

0

J

(

R

\

B

ε(

x

)

)

J

(

R

)

Vol(

B

ε

(

x

)

)

It is a scalar function of

x

It measures sensitivity to removing balls around

x

D

T

(

x

,

R

)

<<

0

=

high probability of finding an object

Equivalently, for

x

∈ R

and

h(

ε

) =

Vol

(B

ε

(x))

(21)

Definition of Topological Derivative (Sokowloski–Zochowski ’99)

The TD of a shape functional

J

(

R

)

at a point

x

R

is

D

T

(

x,

R

) =

lim

ε

0

J

(

R

\

B

ε(

x

)

)

J

(

R

)

Vol(

B

ε

(

x

)

)

It is a scalar function of

x

It measures sensitivity to removing balls around

x

D

T

(

x

,

R

)

<<

0

=

high probability of finding an object

Equivalently, for

x

∈ R

and

h(

ε

) =

Vol

(B

ε

(x))

(22)

Definition of Topological Derivative (Sokowloski–Zochowski ’99)

The TD of a shape functional

J

(

R

)

at a point

x

R

is

D

T

(

x,

R

) =

lim

ε

0

J

(

R

\

B

ε(

x

)

)

J

(

R

)

Vol(

B

ε

(

x

)

)

It is a scalar function of

x

It measures sensitivity to removing balls around

x

D

T

(

x

,

R

)

<<

0

=

high probability of finding an object

Equivalently, for

x

∈ R

and

h

(

ε

) =

Vol

(

B

ε

(

x

))

(23)

Transmission problem:

u

=

u

+

,

n

u

=

n

u

+

Case I: No a priori information on the obstacles,

R

=

R

n

,

Ω =

Theorem.

For any

x

R

n

the topological derivative of

J

(

R

n

) =

1

2

Γ

meas

|

u

u

meas

|

2

is

D

T

(x

,

R

n

) =

Re

(k

i

2

k

e

2

)

u

(x)w

(x)

(24)

Case I: No a priori information on the obstacles,

R

=

R

n

,

Ω =

Theorem.

For any

x

R

n

the topological derivative of

J

(

R

n

) =

1

2

Γ

meas

|u

u

meas

|

2

is

D

T

(

x

,

R

n

) =

Re

(

k

i

2

k

e

2

)

u

(

x

)

w

(

x

)

(25)

Forward problem with

Ω =

:

u

+

k

e

2

u

=

0

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

(

u

u

inc

)

ik

e

(

u

u

inc

)) =

0

Therefore,

u

=

u

inc

(

x

) =

e

ike

x

·

d

Adjoint problem with

Ω =

:

w

+

k

e

2

w

= (

u

meas

u

)

δ

Γ

meas

in

R

n

lim

r

→∞

r

(

n

1

)/

2

(

r

w

ik

e

w

) =

0

Therefore,

w

=

Γ

meas

G

k

e

(x

y)(

u

meas

u

)(y)

dl

y

The true obstacles enter in the TD through the measured

data at the adjoint field

(26)

Forward problem with

Ω =

:

u

+

k

e

2

u

=

0

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

(

u

u

inc

)

ik

e

(

u

u

inc

)) =

0

Therefore,

u

=

u

inc

(

x

) =

e

ike

x

·

d

Adjoint problem with

Ω =

:

w

+

k

e

2

w

= (

u

meas

u

)

δ

Γ

meas

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

r

w

ik

e

w

) =

0

Therefore,

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

The true obstacles enter in the TD through the measured

data at the adjoint field

(27)

Forward problem with

Ω =

:

u

+

k

e

2

u

=

0

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

(

u

u

inc

)

ik

e

(

u

u

inc

)) =

0

Therefore,

u

=

u

inc

(

x

) =

e

ike

x

·

d

Adjoint problem with

Ω =

:

w

+

k

e

2

w

= (

u

meas

u

)

δ

Γ

meas

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

r

w

ik

e

w

) =

0

Therefore,

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

The true obstacles enter in the TD through the measured

data at the adjoint field

(28)

Forward problem with

Ω =

:

u

+

k

e

2

u

=

0

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

(

u

u

inc

)

ik

e

(

u

u

inc

)) =

0

Therefore,

u

=

u

inc

(

x

) =

e

ike

x

·

d

Adjoint problem with

Ω =

:

w

+

k

e

2

w

= (

u

meas

u

)

δ

Γ

meas

in

R

n

lim

r

→∞

r

(

n

1

)

/

2

(

r

w

ik

e

w

) =

0

Therefore,

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

The true obstacles enter in the TD through the measured

data at the adjoint field

(29)
(30)

"

×

"= observation points, 24 incident directions in

[

0

,

2

π

)

,

ke

=

2 and

k

i

=

1

/

2. Level of noise=1%

(31)

Some examples

"

×

"= observation points, 24 incident directions in

[

0

,

2

π

)

,

ke

=

2 and

k

i

=

1

/

2. Level of noise=1%

(32)

Results depend on the wave length (1 w.l.=2

π/k

):

1

st

row:

k

e

=

2 and

k

i

=

1

/

2

(33)

TD with an initial guess

Case II:

ap

first guess,

R

=

R

n

\

ap

,

Ω = Ω

ap

Theorem.

For any

x

R

n

\

ap

the topological derivative of

J

(

R

n

\

ap

) =

1

2

Γ

meas

|

u

u

meas

|

2

is

D

T

(x

,

R

n

\

ap

) =

Re

(k

2

i

k

e

2

)

u

(x)w

(x)

(34)

Case II:

ap

first guess,

R

=

R

n

\

ap

,

Ω = Ω

ap

Theorem.

For any

x

R

n

\

ap

the topological derivative of

J

(

R

n

\

ap

) =

1

2

Γ

meas

|u

umeas

|

2

is

D

T

(

x

,

R

n

\

ap

) =

Re

(

k

i

2

k

e

2

)

u

(

x

)

w

(

x

)

(35)

Forward problem with

Ω = Ω

ap

:

u

+

k

e

2

u

=

0

in

R

n

\

ap

u

+

k

i

2

u

=

0

in

ap

u

=

u

+

, ∂

n

u

=

n

u

+

on

ap

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

(

u

u

inc

)

ike

(

u

u

inc

)) =

0

Adjoint problem with

Ω = Ω

ap

:

w

+

k

e

2

w

= (

umeas

u

)

δ

Γ

meas

in

R

n

\

ap

w

+

k

i

2

w

=

0

in

ap

w

=

w

+

, ∂

nw

=

n

w

+

on

ap

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

w

ike

w

) =

0

(36)

Same examples as before with

Ω =

(37)

An iterative method

Algorithm

1

Compute the TD when

Ω =

2

Take

Ω1

=

{x

,

D

T

(

x

,

R

n

)

<

−C

1

},

C

1

>

0

3

For j=1:jmax

Compute the TD in

R

n

\

j

Select

j

+

1

j

(38)

C

j

First step

:

1

=

{

x

,

D

T

(

x

,

R

2

)

<

−C

1

}

C

1

=

3

5

|

min

D

T

|

Accept

C

1

if

J

1

<

J

0

Otherwise

C

1

<

C

1

Iterations:

j

+

1

= Ω

j

∪ {

x

,

D

T

(x

,

R

2

\

j

)

<

C

j

+

1

}

C

j

+

1

=

9

10

|

min

D

T

|

Stopping criteria?

(39)

How to choose

C

j

?

First step

:

1

=

{

x

,

D

T

(

x

,

R

2

)

<

−C

1

}

C

1

=

3

5

|

min

D

T

|

Accept

C

1

if

J

1

<

J

0

Otherwise

C

1

<

C

1

Iterations:

j

+

1

= Ω

j

∪ {

x

,

D

T

(

x

,

R

2

\

j

)

<

−C

j

+

1

}

C

j

+

1

=

9

10

|

min

D

T

|

Stopping criteria?

(40)

C

j

First step

:

1

=

{

x

,

D

T

(

x

,

R

2

)

<

−C

1

}

C

1

=

3

5

|

min

D

T

|

Accept

C

1

if

J

1

<

J

0

Otherwise

C

1

<

C

1

Iterations:

j

+

1

= Ω

j

∪ {

x

,

D

T

(

x

,

R

2

\

j

)

<

−C

j

+

1

}

C

j

+

1

=

9

10

|

min

D

T

|

Stopping criteria?

(41)
(42)
(43)
(44)

1

Inverse scattering problems

2

Topological derivative methods

TD for shape reconstruction

TD for shapes and parameters

3

Conclusions

(45)

Direct problem

u

+

k

e

2

u

=

0

in

R

n

\

u

+

k

i

2

u

=

0

in

u

=

u

+

, ∂n

u

=

∂n

u

+

on

lim

r

→∞

r

(

n

1

)

/

2

(

∂r

(

u

u

inc

)

ik

e

(

u

u

inc

)) =

0

(46)

Idea

In the first computation of the TD, i.e. when

Ω =

, we do

not need to know

k

i

:

D

T

(

x

,

R

2

) =

Re

(

k

i

2

k

e

2

)

u

(

x

)

w

(

x

)

where

u

=

u

inc

and

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

We compute the TD taking

k

i

0

k

e

to get an initial guess

Ω1

(47)

Idea

In the first computation of the TD, i.e. when

Ω =

, we do

not need to know

k

i

:

D

T

(

x

,

R

2

) =

Re

(

k

i

2

k

e

2

)

u

(

x

)

w

(

x

)

where

u

=

u

inc

and

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

We compute the TD taking

k

i

0

k

e

to get an initial guess

1

(48)

Idea

In the first computation of the TD, i.e. when

Ω =

, we do

not need to know

k

i

:

D

T

(

x

,

R

2

) =

Re

(

k

i

2

k

e

2

)

u

(

x

)

w

(

x

)

where

u

=

u

inc

and

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

We compute the TD taking

k

i

0

k

e

to get an initial guess

1

In the next step, we update

k

i

by a gradient method

(49)

Idea

In the first computation of the TD, i.e. when

Ω =

, we do

not need to know

k

i

:

D

T

(

x

,

R

2

) =

Re

(

k

i

2

k

e

2

)

u

(

x

)

w

(

x

)

where

u

=

u

inc

and

w

=

Γ

meas

G

ke

(

x

y

)(

umeas

u

)(

y

)

dly

We compute the TD taking

k

i

0

k

e

to get an initial guess

1

In the next step, we update

k

i

by a gradient method

(50)
(51)

Outline

1

Inverse scattering problems

2

Topological derivative methods

TD for shape reconstruction

TD for shapes and parameters

3

Conclusions

Other problems

(52)

A non–monotone method

Generalization of the concept of the topological derivative

Allows to detect annular defects and to remove spurious

regions

Other problems and boundary conditions

Dirichlet and Neumann problems:

u

|

Γ

=

0

or

n

u

|

Γ

=

0

General transmission problems:

u

+

=

u

, α

+

n

u

+

=

α

n

u

Heterogeneous materials:

(53)

Other problems and generalizations

A non–monotone method

Generalization of the concept of the topological derivative

Allows to detect annular defects and to remove spurious

regions

Other problems and boundary conditions

Dirichlet and Neumann problems:

u

|

Γ

=

0

or

n

u

|

Γ

=

0

General transmission problems:

u

+

=

u

, α

+

n

u

+

=

α

n

u

Heterogeneous materials:

(54)

Non–steady problems

We combine topological derivatives in space with Laplace

transforms in time

The observation of the system over an interval of time

(55)

Outline

1

Inverse scattering problems

2

Topological derivative methods

TD for shape reconstruction

TD for shapes and parameters

3

Conclusions

Other problems

(56)

The topological derivative is a powerful tool to solve

inverse problems dealing with

shape reconstruction in

different areas

: acoustics, phototermal problems, elasticity,

tomography,...

The TD gives a

good approximation

of the number, size

and location of the objects buried in a medium

Iterative procedures improve

their shape, and catch small

objects, if missed in the first trial

The algorithm for shape reconstruction can be combined

with a gradient method to recover both

shapes and

parameters

Work in progress

:

(57)

Conclusions

The topological derivative is a powerful tool to solve

inverse problems dealing with

shape reconstruction in

different areas

: acoustics, phototermal problems, elasticity,

tomography,...

The TD gives a

good approximation

of the number, size

and location of the objects buried in a medium

Iterative procedures improve

their shape, and catch small

objects, if missed in the first trial

The algorithm for shape reconstruction can be combined

with a gradient method to recover both

shapes and

parameters

Work in progress

:

(58)

The topological derivative is a powerful tool to solve

inverse problems dealing with

shape reconstruction in

different areas

: acoustics, phototermal problems, elasticity,

tomography,...

The TD gives a

good approximation

of the number, size

and location of the objects buried in a medium

Iterative procedures improve

their shape, and catch small

objects, if missed in the first trial

The algorithm for shape reconstruction can be combined

with a gradient method to recover both

shapes and

parameters

Work in progress

:

(59)

Conclusions

The topological derivative is a powerful tool to solve

inverse problems dealing with

shape reconstruction in

different areas

: acoustics, phototermal problems, elasticity,

tomography,...

The TD gives a

good approximation

of the number, size

and location of the objects buried in a medium

Iterative procedures improve

their shape, and catch small

objects, if missed in the first trial

The algorithm for shape reconstruction can be combined

with a gradient method to recover both

shapes and

parameters

Work in progress

:

(60)

The topological derivative is a powerful tool to solve

inverse problems dealing with

shape reconstruction in

different areas

: acoustics, phototermal problems, elasticity,

tomography,...

The TD gives a

good approximation

of the number, size

and location of the objects buried in a medium

Iterative procedures improve

their shape, and catch small

objects, if missed in the first trial

The algorithm for shape reconstruction can be combined

with a gradient method to recover both

shapes and

parameters

Work in progress

:

(61)

More information

A Carpio, ML Rapún

. Inverse Problems 24 (2008) Art.

045014

A Carpio, ML Rapún

. J Comput Physics 227 (2008)

8083–8106

A Carpio, ML Rapún

. Lecture Notes in Mathematics,

Springer 2008

A Carpio, ML Rapún

. Inv Probl Sci Eng 18 (2010) 35–50

A Carpio, T Johansson, ML Rapún

. J Math Imag Vision 36

(2010) 185–199

Referencias

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