1. Optimization Using Broyden-Update Self-Adjoint Sensitivities Dongying Li, N. K. Nikolova, and M. H. Bakr McMaster University, 1280 Main Street West, Hamilton, ON L8S 4K1, CANADA Department of Electrical and Computer Engineering Computational Electromagnetics Laboratory (e-mail: lid6@mcmaster.ca) IEEE AP-S International Symposium Albuquerque NM, June 26, 2006

2. 2 Outline objective & motivation sensitivity analysis –design sensitivity analysis (DSA) –finite difference approximation (FD) –self-adjoint sensitivity analysis (SASA) SASA-based gradient optimization –theory: FD-SASA, B-SASA, B/FD-SASA –numerical results & comparison conclusion and future work

3. 3 Objective & Motivation applications of DSA gradient based optimization yield and tolerance analysis design of experiments and models Gradient Based Optimizer Numerical EM Solver Design Sensitivity Analysis p (0) Specs p(i)p(i) F(p(i))F(p(i)) F(p(i))F(p(i)) p*p*

4. 4 Design Sensitivity Analysis Given FEM system equation design variables objective function find subject to

5. 5 Design Sensitivity Analysis via Finite Differences easy and simple method overhead: at least N additional system analyses

6. 6 Design Sensitivity Analysis via SASA SASA for S-parameters only original system solution needed [N. K. Nikolova, J. Zhu, D. Li, M. Bakr, and J. Bandler, IEEE T-MTT. vol. 54, pp. 670-681, Feb, 2006.]

7. 7 Design Sensitivity Analysis via SASA methodmatrix fillssystem solutions sensitivity formula computation FDNN0 SASAN0N computational overhead

8. 8 SASA-Based Gradient Optimization gradient-based algorithms quasi-Newton sequential quadratic programming (SQP) trust-region fast convergence vs. non-gradient based algorithms pattern search neural network-based algorithms genetic algorithms particle swarm guaranteed global minimum

9. 9 SASA-Based Gradient Optimization factors affecting efficiency 1. required number of iterations nature of the algorithm 2. number of simulation calls per iteration nature of the algorithm the Jacobian computation

10. 10 SASA-Based Gradient Optimization finite-difference SASA (FD-SASA) overhead: N matrix fill Broyden SASA (B-SASA) overhead: practically zero

11. 11 SASA-Based Gradient Optimization B/FD-SASA guarantees robust derivative computation with minimum time switch between B-SASA and FD-SASA switching criteria from B-SASA to FD-SASA

12. 12 Example of B/FD-SASA: H-Plane Filter design parameter p T =[L 1 L 2 L 3 W 1 W 2 W 3 W 4 ] initial design p (0)T = [12 14 18 14 11 11 11] (mm) design requirement optimization algorithm TR-minimax [G. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance– Matching Networks, and Coupling Structures. 1980, pp. 545-547.]

13. 13 Example of B/FD-SASA: H-Plane Filter Initial design FD optimal B/FD-SASA optimal

14. 14 Example of B/FD-SASA: H-Plane Filter parameter step size with respect to iterations function value with respect to iterations

15. 15 Example of B/FD-SASA: H-Plane Filter finite difference optimal design p T = [12.226 14.042 17.483 14 11 10.922 11.341] (mm) Iterations: 11 time: 3825 s B/FD-SASA optimal design p T = [12.131 13.855 17.809 14.01 11.1 11.098 11.191] (mm) Iterations: 7 time: 949 s [switching criterion I triggered at 5th iteration]

16. 16 Conclusion summary efficient SASA method for sensitivity analysis implementation of B/FD-SASA on gradient-based optimization: improving efficiency future work further verification of the switching criteria in B/FD- SASA

17. 17 Thank you