Optimization Using Broyden-Update Self-Adjoint Sensitivities Dongying Li, N. K. Nikolova, and M. H....
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Transcript of Optimization Using Broyden-Update Self-Adjoint Sensitivities Dongying Li, N. K. Nikolova, and M. H....
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: [email protected])
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.]
14
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
15
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