Adjoint Method and

Multiple-Frequency Reconstruction

Qianqian FangThayer School of EngineeringDartmouth College Hanover, NH 03755

Thanks to Paul Meaney, Keith Paulsen, Margaret Fanning, Dun Li, Sarah Pendergrass, Timothy Raynolds

Outline Generalized Dual-mesh Scheme Adjoint formulation for dual-mesh

• Graphical interpretations• Formulations• Comparisons with old method

Multiple-Frequency Reconstruction Algorithm• Description of dispersive medium• How it works (animation)• General form for dispersive media• Time-Domain Reconstruction Algorithm• Results

Conclusions and prospects

Dual-mesh - Math Form

Definition: Independent discretization for state space and parameter space and the mapping rules between the two sets of base functions.

Rf is called forward space, discretized by basis

Rr is called reconstruction space, discretized by basis Mostly, we have

Single-mesh/Sub-mesh schemes are special cases of dual-mesh

1 2

1 2

( , ,..., )

( , ,..., )

f

r

f n

r n

span

span

r f i

i

Dual-mesh cond. Field values are defined on forward mesh Properties defined on reconstruction mesh So that

Field on recon. mesh need to interpolate from forward mesh

Properties on forward mesh need to interpolate from recon mesh

Mapping:

2 2

( ) ( )

( ) ( )

R Fii

r f

F Rjj

E r E r

rk r k r

Dualmesh-Examples

2D FDTD forward mesh2D order-2 recon. mesh

2D FEM forward mesh2D order-1 recon. mesh

1,1 1,1 1,1

2 2 21 2

1,2 1,2 1,2

2 2 21 2

1, 1,1,

2 2 21 2

2,1 2,1 2,1

2 2 21 2

2, 2, 2,

2 2 21 2

......

......

......

......

......

......

......

..

r r

nc

nc

n nnr

nc

nc

nr nr nr

nc

k k k

k k k

k k k

k k k

k k k

E E E

E E E

E EE

E E E

JE E E

,1 ,1 ,1

2 2 21 2

,2 ,2 ,2

2 2 21 1 1

, , ,

2 2 21 2

....

......

......

......

......

ns ns ns

nc

ns ns ns

ns nr ns nr ns nr

nc

k k k

k k k

k k k

E E E

E E E

E E E

Jacobian Matrix

Source=1, diff receivers

Source=2, diff receivers

Source=ns, diff receivers

,

2( , , ) { }s r

n

r rs r n

r

r r rk

EJ

Source ID

receiver ID

parameter node ID

Sensitive Coefficient

dfdx df

dx J k E

Provide the first order derivative information

Receiver

JsSource

Perturbation currentsAt Node n

npJ

12 2{ } ( ) { } ( )s

s

n n

rr r r

r r

r rk k

E AA E

1{ } { ( ), }sr s jA j r E J

Formulation

2 2

1

2

2

2

{ } { ( ), }

{ }

( , , ) , ( ) , ( )

{ } { ( ), }

,

(

{ }

{

( ),

}

,

)

s s s j

ss

n n

ss r n r r

n

r r r j

sr r

n

sr r i j

p

p

p

nn

j r

k k

J r r r r A rk

j r

k

j r rk

A E b J

E AA E

E

A E b J

E

b

b

bb E

EJ

,

1( , , ) ,

( )

s r

s r n i j n s rr

J r r rj r

E E

E EJ

J1• E2= J2 • E1

J1J2

E2E1

Reciprocal Media

Denoted as perturbation source

Comparison

1( , , ) ,

( )s r n i j n s rr

J r r rj r

E EJ

12

( , , ) [ ] { } , ( )s r n s rn

J r r r rk

AA E

Replace matrix inversion with matrix multiplication

Old:

New:

Field generated by Js

Field generated by JrVery sparse matrixGeometry related only

Strength of auxiliary source, can be 1

Computational Cost Computational cost for Sensitive Equ. Method:For each iteration:Solving the AX=b for (Ns+Ns*Nc) times, where

Ns= Source numberNc= Parameter node number

Computational cost for Adjoin methodFor each iteration:Solving the AX=b for (Ns+Nr) times, where

Ns= Source numberNr= Receiver number

When using Tranceiver module, only Ns times forward solving is needed.Which is 1/(Nc+1) of the time using by sensitive equation method

Multiple Frequency Reconstruction Algorithm Ill-posedness of the inversion problem due

to insufficient data input and linear dependence of the data.-> rank deficient matrix

Instability and Local minima Method: improve the condition of the

matrix: More antenna under single frequency(SFMS) Fixed antenna #, more frequencies

Advantages of MF vs. SFMS More sources & receiver will increase the expenses of

building DAQ system. Under single frequency illumination, the increasing

number of source will not always bring proportional increasing in stability.(???)

Single frequency reconstruction is hard to reconstruct large/high-contrast object due to the similarity of the info.(???)

In multi-frequency Recon.: lower frequency stabilize the convergence and provide information at different scales, supply more linearly independent measurements.

Need Eigen-analysis to prove

Computational Considerations: TD solver Hardware Considerations: TD system

Potential

1 2

1 2

'( ) ( , , ,..., ) ( , )

''( ) ( , , ,..., ) ( , )N

N

f f

g g

Modeling of Dispersive Medium

2 2( ) ( )

( ) '( ) ''( )

k

j

1 * * *1 2

1 * * *1 2

( '( ), '( ),..., '( ))

( ''( ), ''( ),..., ''( ))

N

N

f

g

1-1 mapping

*

*

'( )

''( )

*1 *

2 *3

Reconstruction Demo.2

1 3'( ) e

*1 *

2 *3

21

Key Frequencies Recon. Frequencies

Background (Init. Guess)

Real Curve

* * * *1 2{ , ,..., }N 1 2{ , ,..., }M

Key Questions? How to calculate the change with

multiple reconstruction frequencies for each step?

How to determine the Change at key frequencies from the Changes at reconstruction frequencies?

Answers see back

Single Frequency Real Form

1 12 2 2

11 1 112

11 1 112 2

1 1

( ) ( )1

'( )( ) ( ) ( )0

( )( ) ( ) 0''( )

( ) ( )

R R

R I R

II I

R I

E E

k k EQQ

EE E

k k

Pre-scaled Real Form of Gauss-Newton Formula:

2 22 2 2

22 2 222

22 2 222 2

2 2

( ) ( )1

'( )( ) ( ) ( )0

( )( ) ( ) 0''( )

( ) ( )

R R

R I R

II I

R I

E E

k k EQQ

EE E

k k

'

"

'( ) ( ) '

"( ) ( ) "

S

S

Need to supply extra information to make unknowns same for both frequencies

2 21 12 ' 1 2 " 12 2

1 1

2 21 12 ' 1 2 "

2 22 22 ' 2 2 " 22 2

2 2

2 22 ' 2 2

12 21 1

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )

( )(

( ) ( )( ) ( )

)

R R

R I

I

R

R R

R I

I I

R I

E EQ

E EQ S S

k k

E E

S Sk k

EQ S

Q S Sk k

k

1

2

2

2 22 " 2 2

2 2

1

( )

( )1

'

( )''

( )

( )( )

( ) ( )

R

I

I

R

I

IQ

E

E

E

ES

k

E

Combined System

* *'

* *"

'( ) ( ) '

"( ) ( ) "

i i

i i

S

S

Solve ' "and

Then replace into

To get the change at each Key Frequencies

General Form for MFRA2 21 12 ' 1 2 " 12 2

1 1

2 21 12 ' 1 2 " 12 2

1 1

2 22 22 ' 2 2 " 22 2

2 2

2 22 ' 2 2

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )

( )( )

R R

R I

I I

R I

R R

R I

I

R

E EQ S S

k k

E EQ S S

k k

E EQ S S

k k

EQ S

k

2 22 " 2 2

2 2

2 2' 2 "2 2

2 2' 2 "2 2

( )( )

( ) ( )

...... ......

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )

I

I

R M R MM M M

R M I M

I M I MM M M

R M I M

ES

k

E EQ S S

k k

E EQ S S

k k

1

1

2

2

( )

( )

( )1'

( )

'' ...

( )

( )

R

I

R

I

R M

I M

E

E

E

Q E

E

E

Results-I Non-dispersive medium simulation: large cylinder with inclusion D~7.5cm, contrast 1:6/1:5 for real/imag Use 300M/600M/900M Non of the previous single frequency(900M) recon works

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

Reconstructed Permitivity usingMulti-Frequency-Point Method

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Reconstructed Permitivity usingMulti-Frequency-Point Method

100.0092.8685.7178.5771.4364.2957.1450.0042.8635.7128.5721.4314.29

7.140.00

I2.52.321432.142861.964291.785711.607141.428571.251.071430.8928570.7142860.5357140.3571430.1785710

Single Freq. Recon at 900M

0 2 4 6 8 10 12 14 16 18 2010

-2

10-1

100

300/600/900900M only

100.0092.8685.7178.5771.4364.2957.1450.0042.8635.7128.5721.4314.29

7.140.00

I2.52.321432.142861.964291.785711.607141.428571.251.071430.8928570.7142860.5357140.3571430.1785710

Error plot

Lower Contrast Example A low contrast Example 1:2

0 2 4 6 8 10 12 14 16 18 2010

-2

10-1

100

600/900 R10,O2300/600/900 R10,O2900 R10,O2

Dispersive Medium Simulation

2

2

1

1

( )

( )

e

e

100M 1G

60.0056.4352.8649.2945.7142.1438.5735.0031.4327.8624.2920.7117.1413.5710.00

I2.52.321432.142861.964291.785711.607141.428571.251.071430.8928570.7142860.5357140.3571430.1785710

I1.81.671431.542861.414291.285711.157141.028570.90.7714290.6428570.5142860.3857140.2571430.1285710

100.0093.5787.1480.7174.2967.8661.4355.0048.5742.1435.7129.2922.8616.4310.00

600M 900M

Lower end

Permittivity

Conductivity

Permittivity

Conductivity

background

larger object

Phantom Data Recon.

Saline Background/Agar Phantom with inclusion

100.0092.8685.7178.5771.4364.2957.1450.0042.8635.7128.5721.4314.29

7.140.00

I2.52.321432.142861.964291.785711.607141.428571.251.071430.8928570.7142860.5357140.3571430.1785710

100.0092.8685.7178.5771.4364.2957.1450.0042.8635.7128.5721.4314.29

7.140.00

I2.52.321432.142861.964291.785711.607141.428571.251.071430.8928570.7142860.5357140.3571430.1785710

Single FrequencyRecon at 900M

Using 500/700/900Non-dispersiveversion

Time/Memory Issues

Methods Based on Time (alpha)

Memory

Multi-Frequency Recon. Algo.

Pre-scaled Real code

n_freq*12s/iterParallel: 3s

Jacobian get bigger, Inverse problem size double

TD-FFT Recon. Algo.

MFRA 36s/iter.For 600 freq componentsParallel: 10s

Jacobian bigger, big matrix needed for forward solver

Fast FDTDSingle freq.

Green function & FDTD

n_freq*4s/Iter.Parallel: 1s

One 8M matrix is need to hold all forward fields of previous step for 16 antennae

-- Forward: 124X124 2D forward mesh-- Reconstruction: 281 2D parameter nodes

Conclusions For simulations and recon. of phantom data,

MFRA shows stable, robust, and achieve better images.

Shows the abilities of reconstructing large-high contrast object.

Good for current wide-band measurement system

General form, fit for even complex dispersive medium

Still need works… How to qualify the improvement of the ill-

posedness of inversion (cond. number is not always good)

What’s the best number for transmitter/receiver under single frequency? and under multiple frequencies?

How to select frequencies? How they interact with each other?

How to weight a multi-freq equation? Is it possible to build TD measurement system?

(use microwave/electrical/optical signals). what are the difficulties need to accounted?

Questions?