Optimization Using Broyden-Update Self-Adjoint Optimization Using Broyden-Update Self-Adjoint SensitivitiesSensitivities

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 SymposiumAlbuquerque NM, June 26, 2006

omputationallectro-

agnetics

aboratory

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

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)

F(p(i))

F(p(i))

p*

4

Design Sensitivity Analysis

Given

FEM system equation design variablesobjective function

find subject to

Ax b1 2[ ... ]T

Np p pp( , ( ))f p x p

fp Ax b

1 2

N

f f ff

p p p

p

5

Design Sensitivity Analysis via Finite Differences

easy and simple method

overhead: at least N additional system analyses

( ) ( ) ( )i

i i

f f p f

p p

ip p e p

0

th element1

0

ie i

6

Design Sensitivity Analysis via SASA

( ) , , 1, ,Tkj kj k jS j k K p px Ax

SASA for S-parameters

0

-port

1

2 ( ) ( )kj

inck n n j jj

j

E ds

a E a e

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.]

( )kj Tkj k j

i i

S

p p

A

x x

7

Design Sensitivity Analysis via SASA

method matrix fills system solutionssensitivity formula

computation

FD N N 0

SASA N 0 N

computational overhead

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

SASA-Based Gradient Optimization

factors affecting efficiency

1. required number of iterationsnature of the algorithm

2. number of simulation calls per iterationnature of the algorithm the Jacobian computation

10

SASA-Based Gradient Optimization

finite-difference SASA (FD-SASA)

overhead: N matrix fill

Broyden SASA (B-SASA)

overhead: practically zero

( ) ( ) ( ), 1, ,i i

i i

pi N

p p

A p A p e A p

( )

( )( ) ( ) ( )( 1) ( )

( )

( ) ( )

( ) ( )

1, ,

k

kk k kjk k

jj ki

k T ki i

hp

hp p

i N

AA p h A p

A A

h h

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( ) ( 2)( ) ( )k kG G p p

k dh

12

Example of B/FD-SASA: H-Plane Filter

design parameterpT=[L1 L2 L3 W1 W2 W3 W4]

initial designp(0)T = [12 14 18 14 11 11 11]

(mm)

design requirement

optimization algorithmTR-minimax

2W1

2W1

2W2

2W2

2W3

2W3

2W4

2a L1

L1

L2

L2

L3

L3b

x

y

z

21

21

21

0.52 5.0 GHz

0.98 5.5 9.0 GHz

0.70 9.5 GHz

S f

S f

S f

[G. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance–Matching Networks, and Coupling Structures. 1980, pp. 545-547.]

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Example of B/FD-SASA: H-Plane Filter

Initial design

FD optimal

B/FD-SASA optimal

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Example of B/FD-SASA: H-Plane Filter

parameter step size with respect to iterations

function value with respect to iterations

1 2 3 4 5 6 7 8 9 100

2

4

6

8x 10

-4

Iterations

|xi +

1-xi| (

mm

)

FDFD-SASAB-SASAMixed B/FD-SASA

0 1 2 3 4 5 6 7 8 9 100.12

0.14

0.16

0.18

0.2

0.22

Iterations

f(x)

FDFD-SASAB-SASAMixed B/FD-SASA

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Example of B/FD-SASA: H-Plane Filter

finite difference

optimal designpT = [12.226 14.042 17.483 14 11 10.922 11.341] (mm)

Iterations: 11

time: 3825 s

B/FD-SASA

optimal designpT = [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

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

Thank you