Variable Change Technique Applied in Constrained Inverse ...

Variable Change Technique Applied in Constrained Inverse

Transport Applications

Zeyun Wu & Marvin Adams

Department of Nuclear EngineeringTexas A&M University

ANS Meeting, San Diego, CA, June 15th, 2010

Nuclear Engineering, Texas A&M

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Introduction


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• Forward Transport Problem1

Note: Symbols in red are unknown variables in the problem.

Forward vs. Inverse

' ,'

,1 0

'( )( , ) ( , ( ) ( ) )( ) ( ,)G l

ltg lm sg g ext g

g l m lg g lm gr Y rr r Sr rψ ψ φ

∞

→= = =−

Ω ΩΩ ∇ +Σ = Ω Σ + Ω∑∑∑g

, ' ,'

'1 0

( )( , ) ( , ) ( ) ( ) ( , )( )G l

g g ll

tg m lm g ext gg m

sl l

g gr rr r Y r S rψ ψ φ∞

= = =−→Ω ∇ Ω + Ω = ΩΣ Σ + Ω∑∑∑g

• Inverse Transport Problem

1J.J. Duderstadt & L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, 1976


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Solve the Inverse Problem

The purpose of our project is to infer material distribution inside an object based upon detection and analysis of radiation emerging from the object. (Neutron tomography)

We are particularly interested in problems in which radiation can undergo significant scattering within the object.

?

Beam window

Investigated object

Radiation detectors


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Model Problem

Table: Physics properties of the materials in the model problem

sΣ

tΣ

g

trΣ

mfp

c

Material Water Iron

(1/cm) 0.736 0.967

(1/cm) 0.744 1.19

(2/3A) 3.70E-2 1.20E-2

(1/cm) 0.716 1.167

(cm) 1.35 0.848

0.990 0.820

Fig: Schematic diagram of the one iron inclusion model problem (X-Y view).

1, , str t s

t t

g mfp c ΣΣ = Σ − Σ = =

Σ Σ


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Gradient-based IterativeOptimization Method


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Minimization Based Optimization Approaches2,3

• Objective function

• Here ‘M’ is experimental measurements provided by accurate mathematical model

• ‘P’ is predicted measurementsprovided by forward model calculation, which is treated as a function of properties of the unknown object

( )( )

1

1

, , ,

, , ,t s s

t s s

P P σ σ σ

σ σ σ

=

⇒Φ =Φ

L

LA schematic diagram of the beam-object-detector system

( )2

1

12

N

i ii

P M=

Φ = −∑

2V. Scipolo, Scattered neutron tomography based on a neutron transport problem, Master Thesis, Texas A&M, 2004 3A. D. Klose et al., Optical tomography using the time-Independent equation of radiative transfer - part 1: forward modelJ. Quant. Spectrosc. Radiat. Transf. , 72, 691-713, 2002


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• Forward model: One-group neutron transport equation for a non-multiplying system with linearly anisotropic scattering

• Inverse model: Nonlinear conjugated gradient (CG) based iterative updating schemeInitialize variables , search direction , where Define termination tolerance and set iteration counter Start of iteration loopPerform line search to find that minimizes

Exit ifEnd of iteration loop

Forward and Inverse Model

[ ]( ) ( )1( , , ) ( , , ) ( , ) 3 ( , ) ( , , )4 xts etr t r t r t J r t Sr tg rrψ ψ φπ

Ω ∇ Ω + Ω = ΩΣ + + ΩΣg g

(1)x (1) (1)=d r ( )(1) (1) , { , , }t s g≡ −∇Φ = Σ Σx xrε 0k =

minα ( ) ( )( )k kαΦ +x d( 1) ( ) ( )

mink k kα+ = +x x d

( )( 1) ( 1)k k+ +≡ −∇Φr x

( 1) ( 1) ( )k k kkβ

+ += +d r d

( )( )k ε∇Φ <x

1k k= +


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Variable Change Technique


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Variable Change Technique

Purpose: Convert constrained optimization to non-constrained optimization( 1) ( ) ( )

, o ( r , )k k ki i x step ix d x gx α+ + Σ= =

( ) ( ) ( )1, ,

k k ki i y step y jy y dα+ = +

log( ) j j

jj

j j j j

y

dy dy

= Σ ⇒

Σ∂Φ ∂Φ ∂Φ= = Σ

∂ ∂Σ ∂Σ

2

tan 2

2 1 1 tan

2

j j

j

j j j

jj

y g

dgy g dy

g g

π

ππ

⎛ ⎞= ⇒⎜ ⎟⎝ ⎠

∂Φ ∂Φ=

∂ ∂

∂Φ=∂ ⎛ ⎞+ ⎜ ⎟

⎝ ⎠

(a)

(b)


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Initialize variables , search direction , where Define termination tolerance and set iteration counter Start of iteration loopChange variable x to yPerform line search to find that minimizes

Change variable y back to xExit ifEnd of iteration loop

(1)x (1) (1)=d r

Nonlinear CG Updating Scheme with Variable Change Technique Incorporated

( )(1) (1) , { , , }t s g≡ −∇Φ = Σ Σr x xε 0k =

minα ( ) ( )( )k kαΦ +y d( 1) ( ) ( )

mink k kα+ = +y y d

( )( 1) ( 1)k k+ +≡ −∇Φr y

( 1) ( 1) ( )k k kkβ

+ += +d r d

( )( )k ε∇Φ <x

( ) ( )( 1) ( 1), where is calculated based on k k+ +∇Φ ∇Φy x

1k k= +


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Results


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Deterministic Optimization Results for Model Problem

Fig: Transport cross section ( ) distribution obtained from deterministic CG based iterative search scheme for the one iron

problem. (a) The real (background is water and square inclusion is iron). (b) Initial guess for . (c) and (d) are results

after 100 and 1000 iterations, respectively.

0.7 0.75 0.8 0.85 0.9

X [cm](a) Real distribution

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

X [cm](b) Initial homogeneous distribution

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

X [cm](c) Results after 100 iterations

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

X [cm](d) Results after 1000 iterations

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

trΣtrΣ

trΣ

0 100 200 300 400 500 600 700 800 900 100010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

# of iterations

Φ

Objective function drastically reduced along iterative process

The objective function changes after each iteration for one iron problem


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Animation Show of the Iterative Reconstruction Procedure

Transport cross section ( ) distribution within the objectchanges after each updating iteration.

trΣ


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Another Model Problem


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Model Problem II

Fig: Schematic diagram of the two irons inclusion model problem (X-Y view).

10 cm

10 cm

2 cm2 cm

2 cm

1.5 cm

Water

Iron

2 cm

1.5 cm

2 cm1.5 cm

Beam Window


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Results for Model Problem II (Deterministic Stage)

0.7 0.75 0.8 0.85 0.9

X [cm](a) Real distribution

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

X [cm](b) Initial homogeneous distribution

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

X [cm](c) Results after 100 iterations

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

X [cm](d) Results after 1000 iterations

Y [c

m]

2.5 5 7.5 10

10

7.5

5

2.5

Fig. 1: Transport cross section ( ) distribution obtained from deterministic CG based iterative search scheme for

the two irons problem. (a) The real (background is water and square inclusions are irons). (b) Initial guess

for . (c) and (d) are results after 100 and 1000 iterations, respectively. trΣ

trΣ

trΣ

Fig. 2: Animation show of the reconstruction procedure for transport cross section ( ) distribution within the object changes with

each iteration.trΣ


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Conclusions


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Summary of This Talk

• We have introduced a variable change technique to convert constrained optimization into unconstrained one in inverse transport applications

• Results from our neutron tomography model problems demonstrate that the method works well and is robust

• The method is very easy to apply• We believe this technique may be beneficial to many gradient-

based iterative optimization problems.


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Questions?

Thank You!

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