Adjoint method and multiple frequency 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. Outline. Generalized Dual-mesh Scheme

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Adjoint method and multiple frequency reconstruction

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


Outline
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
    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


    Dual mesh cond
    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:


    Dualmesh examples
    Dualmesh-Examples

    2D FDTD forward mesh

    2D order-2 recon. mesh

    2D FEM forward mesh

    2D order-1 recon. mesh


    Jacobian matrix

    Source=1, diff receivers

    Source=2, diff receivers

    Source=ns, diff receivers

    Source ID

    receiver ID

    parameter node ID

    Jacobian Matrix

    Provide the first order

    derivative information

    Sensitive Coefficient


    Js

    Perturbation

    currents

    At Node n

    Source

    Receiver


    Formulation
    Formulation

    Denoted as perturbation source

    J1• E2= J2 • E1

    J2

    J1

    E1

    E2

    Reciprocal Media


    Comparison
    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


    Computational cost
    ComputationalCost

    • Computational cost for Sensitive Equ. Method:

      For each iteration:

      Solving the AX=b for (Ns+Ns*Nc) times, where

      Ns= Source number

      Nc= Parameter node number

    • Computational cost for Adjoin method

      For each iteration:

      Solving the AX=b for (Ns+Nr) times, where

      Ns= Source number

      Nr= 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
    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
    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



    Reconstruction demo
    Reconstruction Demo.

    Background

    (Init. Guess)

    Real Curve

    Key Frequencies

    Recon. Frequencies


    Key questions
    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
    Single Frequency Real Form

    Pre-scaled Real Form of Gauss-Newton Formula:

    Need to supply extra information to

    make unknowns same for both frequencies


    Combined system
    Combined System

    Solve

    Then replace into

    To get the change at each Key Frequencies



    Results i
    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



    Lower contrast example
    Lower Contrast Example

    • A low contrast Example 1:2


    Dispersive medium simulation
    Dispersive Medium Simulation

    Lower end

    Permittivity

    Permittivity

    background

    larger object

    Conductivity

    Conductivity

    1G

    900M

    100M

    600M


    Phantom data recon
    Phantom Data Recon.

    Saline Background/Agar Phantom with inclusion

    Single Frequency

    Recon at 900M

    Using

    500/700/900

    Non-dispersive

    version


    Time memory issues
    Time/Memory Issues

    -- Forward: 124X124 2D forward mesh

    -- Reconstruction: 281 2D parameter nodes


    Conclusions
    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
    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?



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