Adjoint Method and Multiple-Frequency Reconstruction

1. 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

2. 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

3. 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

4. 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:

5. Dualmesh-Examples 2D FDTD forward mesh 2D order-2 recon. mesh 2D FEM forward mesh 2D order-1 recon. mesh

6. 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

7. Js Perturbation currents At Node n Source Receiver

8. Formulation Denoted as perturbation source J1• E2= J2 • E1 J2 J1 E1 E2 Reciprocal Media

9. 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

10. 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

11. 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

12. 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

13. Modeling of Dispersive Medium 1-1 mapping

14. Reconstruction Demo. Background (Init. Guess) Real Curve Key Frequencies Recon. Frequencies

15. 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

16. Single Frequency Real Form Pre-scaled Real Form of Gauss-Newton Formula: Need to supply extra information to make unknowns same for both frequencies

17. Combined System Solve Then replace into To get the change at each Key Frequencies

18. General Form for MFRA

19. 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

20. Single Freq. Recon at 900M Error plot

21. Lower Contrast Example • A low contrast Example 1:2

22. Dispersive Medium Simulation Lower end Permittivity Permittivity background larger object Conductivity Conductivity 1G 900M 100M 600M

23. Phantom Data Recon. Saline Background/Agar Phantom with inclusion Single Frequency Recon at 900M Using 500/700/900 Non-dispersive version

24. Time/Memory Issues -- Forward: 124X124 2D forward mesh -- Reconstruction: 281 2D parameter nodes

25. 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

26. 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?

27. Questions?