Adjoint Method and MultipleFrequency Reconstruction

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Adjoint Method and Multiple-Frequency
Reconstruction

• Qianqian Fang
• Thayer School of Engineering
• Dartmouth College
• Hanover, NH 03755
• Thanks to Paul Meaney, Keith Paulsen, Margaret
  Fanning, Dun Li, Sarah Pendergrass, Timothy
  Raynolds

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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

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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

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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

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Dualmesh-Examples
2D FDTD forward mesh 2D order-2 recon. mesh
2D FEM forward mesh 2D order-1 recon. mesh
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Jacobian Matrix
Provide the first order derivative information
Sensitive Coefficient
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Formulation
Denoted as perturbation source
J1 E2 J2 E1
J2
J1
E1
E2
Reciprocal Media
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Comparison
Old
New
Field generated by Js
Strength of auxiliary source, can be 1
Field generated by Jr
Very sparse matrix Geometry related only
Replace matrix inversion with matrix
multiplication
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Computational Cost

• Computational cost for Sensitive Equ. Method
• For each iteration
• Solving the AXb for (NsNsNc) times, where
• Ns Source number
• Nc Parameter node number
• Computational cost for Adjoin method
• For each iteration
• Solving the AXb for (NsNr) times, where
• Ns Source number
• Nr Receiver number
• When using Tranceiver module, only Ns times
  forward solving is needed.
• Which is 1/(Nc1) of the time using by sensitive
  equation method

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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

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Advantages of MF vs. SFMS
Potential

• 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

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Modeling of Dispersive Medium
1-1 mapping
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Reconstruction Demo.
Background (Init. Guess)
Real Curve
Key Frequencies
Recon. Frequencies
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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

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Single Frequency Real Form
Pre-scaled Real Form of Gauss-Newton Formula
Need to supply extra information to make
unknowns same for both frequencies
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Combined System
Solve
Then replace into
To get the change at each Key Frequencies
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General Form for MFRA
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Results-I

• Non-dispersive medium simulation large cylinder
  with inclusion
• D7.5cm, contrast 16/15 for real/imag
• Use 300M/600M/900M
• Non of the previous single frequency(900M) recon
  works

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Single Freq. Recon at 900M
Error plot
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Lower Contrast Example

• A low contrast Example 12

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Dispersive Medium Simulation
Lower end
Permittivity
Permittivity
background
larger object
Conductivity
Conductivity
1G
900M
100M
600M
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Phantom Data Recon.
Saline Background/Agar Phantom with inclusion
Single Frequency Recon at 900M
Using 500/700/900 Non-dispersive version
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Time/Memory Issues
-- Forward 124X124 2D forward mesh --
Reconstruction 281 2D parameter nodes
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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

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Still need works

• How to qualify the improvement of the
  ill-posedness of inversion (cond. number is not
  always good)
• Whats 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?

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Questions?