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# Adjoint Method and Multiple-Frequency Reconstruction - PowerPoint PPT Presentation

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

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

• 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

• 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

• 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:

2D FDTD forward mesh

2D order-2 recon. mesh

2D FEM forward mesh

2D order-1 recon. mesh

Source ID

parameter node ID

Jacobian Matrix

Provide the first order

derivative information

Sensitive Coefficient

Js

Perturbation

currents

At Node n

Source

Denoted as perturbation source

J1• E2= J2 • E1

J2

J1

E1

E2

Reciprocal Media

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

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

When using Tranceiver module, only Ns times forward solving is needed.

Which is 1/(Nc+1) of the time using by sensitive equation method

• 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

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

1-1 mapping

Background

(Init. Guess)

Real Curve

Key Frequencies

Recon. Frequencies

• 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?

Pre-scaled Real Form of Gauss-Newton Formula:

Need to supply extra information to

make unknowns same for both frequencies

Solve

Then replace into

To get the change at each Key Frequencies

• 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

Error plot

• A low contrast Example 1:2

Lower end

Permittivity

Permittivity

background

larger object

Conductivity

Conductivity

1G

900M

100M

600M

Saline Background/Agar Phantom with inclusion

Single Frequency

Recon at 900M

Using

500/700/900

Non-dispersive

version

-- Forward: 124X124 2D forward mesh

-- Reconstruction: 281 2D parameter nodes

• 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

• 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?