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?