Global Optimization Techniques in Computational Electromagnetics Zbyněk Raida Dept. of Radio Electronics Brno University of Technology Brno, Czechia

Global Optimization Techniquesin Computational Electromagnetics

Zbyněk Raida

Dept. of Radio ElectronicsBrno University of TechnologyBrno, Czechia

Outline

• What does the optimization mean

• Classification of optimization tasks- single-objective versus multi-objective- local versus global

• Genetic optimization vs. particle swarm one

• Local tuning of global solutions

• An example

Global optimization techniques …ITSS 2007, Pforzheim

Optimizationdefinition

• Searching for such values of state variables to meet desired parameters as close as possible

ITSS 2007, Pforzheim

32.3

45.026.0

21.1

38.3 16.9

22.3

24.8

Global optimization techniques …

Optimizationobjective function (1)

• Deviation of the actual parameters of the system from the desired ones

4

111 ,

nnfsF xx

ITSS 2007, Pforzheim Global optimization techniques …

Optimizationobjective function (2)

0.5 1.0 1.5 2.0 2.5 3.0f [GHz]

computedmeasured

S11[dB]

-10

-15

-20

-25

-30


More objectivespolarization purity (1)

ČÁP, A., RAIDA, Z., HERAS PALMERO, E., LAMADRID RUIZ, R. Multi-band planar antennas: a comparative study. Radioengineering, 2005, vol. 14, no. 4, p. 11–20.


More objectivespolarization purity (2)

RAIDA, Z., HERAS PALMERO, E., LAMADRID RUIZ, R. Four-band patch antenna with U-shaped notches. In Proc. of the16th international Conference on Microwaves, Radar and Wireless Communications MIKON 2006. Krakow (Poland), 2006, pp. 111–114.


a) b) c)

d)


More objectivesdirectivity patterns (1)


More objectivesdirectivity patterns (2)

More objectivesmulti-objective formulation

4

1

0,,n

nhormaxD fEEF xx

4

111 ,

nnS fsF xx

4

1

90

90

,,n m

mnvertP fEF xx


F

F

2

F

13

S

P

D


Multi-objective optimizationtwo approaches min F ( x)S

min F ( x)D

min F ( x)P

multi-objectiveoptimizer

trade-offsolutions

higher-levelinformation

choose onesolution

min F ( x)S

min F ( x)D

min F ( x)P

single-object.optimizer

higher-levelinformation

one optimumsolution

estim. relative

S

importance

D

vector

P[w , w , w ]

single-object.optimization

F = w F + w F + w F

S

D P

S

D P


Searching for a minimumglobal versus local methods


f(x)

x

startingpoint

global local


Global methodsgenetic algorithms (1)

mm00.9mm,00.1Amm050.0mm,001.0B

2.2,0.2,6.1,0.1r mm5.1mm,0.1h

GHz30f



initial populationquality evaluation

selection



crossover

mutation


function x = main( G, I, pc, pm)

% x(1)= A, x(2)= B, x(3)= h, x(4)= eps

load dip_616; % loading neural model

Rd = 200.0; % desired input resistanceXd = 0.0; % desired input reactancebit = [ 8 8 1 2]; % bits per A, B, h, epsgeb = norm( bit, 1) + 1; % bits in chromosome

gen = round( rand( I, geb-1)); % 1st generationfor g=1:G X = decode( I, bit, gen); % chromosome to A,B,h,eps Z = Tmax * sim( net, X'); % analysis gen(:,geb) = ((Rd-Z(1,:)).^2 + (Xd-Z(2,:)).^2; e(g) = min( gen( :,geb)); % minimum error [val,ind] = min( gen( :,geb)); x = X( ind, :); % best parameters gen = decim( gen, pc, pm, I, geb);end

plot( e);


0 5 10 15 20 25 30 35 40 iter.0

500

1000

1500

2000

2500

cost[ ]2


cost [2]

A [mm]

B [mm]

h [mm]

eps [ – ]

Rin []

Xin []

19 836 7.469 0.008 1.0 2.2 61.0 22.7

20 650 3.875 0.035 1.5 2.0 67.2 –54.9

402 5.156 0.026 1.5 1.0 183.3 –11.1

99 5.188 0.032 1.0 1.0 190.8 3.8

50 generations, 20 individuals, 90 % crossover, 10 % mutation, population decimation


Global methodsparticle swarm optimization (1)

ROBINSON, J., RAHMAT-SAMII, Y. Particle swarm optimization in electromagnetics. IEEE Transactions on Antennas and Propagation. 2004, vol. 52, no. 2, p. 397–407.


Global methodsPSO (2)

Tnnnnn hBA x

nnnnnn rcrcw xgxpvv 2211

nnn t vxx x

x

1

2

p2

g2

2

1

p11



absorbing reflecting invisible


x

x

2

1

x

x

2

1

x

x

2

1


function out = main( G, I)

% x(1)= A, x(2)= B, x(3)= h, x(4)= eps

load dip_616; % loading antenna model

Rd = 200; % required input resistanceXd = 0; % required input reactance

dt = 0.1; % time stepc1 = 1.49; % personal scaling factorc2 = 1.49; % global scaling factor

x = zeros( I, 5); % agents’ positionp = zeros( I, 5); % personal best

for n=1:I x(n,1) = 1.000 + 8.000*rand(); p(n,1) = x(n,1); x(n,2) = 0.001 + 0.049*rand(); p(n,2) = x(n,2); x(n,3) = 1.0 + 0.5 * rand(); p(n,3) = x(n,3); x(n,4) = 1.0 + 1.2 * rand(); p(n,4) = x(n,4); p(n,5) = 1e+6;end

v = rand( I, 4); % agent velocityg = zeros( 1, 4); % global beste = zeros( G+1, 1); e(1) = 1e+6;

for m=1:G % +++ MAIN ITERATION LOOP +++

w = 0.5*(G-m)/G + 0.4; % inertial weight Z = Tmax * sim( net, x(:,1:4)'); % impedance of agents x(:,5) = ((Rd-Z(1,:)).^2 + (Xd-Z(2,:)).^2 [e(m+1),ind] = min( x( :,5)); % the lowest error

if e(m+1)<e(m) g = x( ind, 1:4); % the global best end

for n=1:I if x(n,5)<p(n,5) % the personal best p(n,:) = x(n,:); end v(n,:) = w*v(n,:) + c1*rand()*( p(n,1:4)-x(n,1:4)); v(n,:) = v(n,:) + c2*rand()*( g(1,1:4)-x(n,1:4)); x(n,1:4) = x(n,1:4) + dt*v(n,:); if x(n,1) > 9.00, x(n,1)=9.00; end % absorbing walls if x(n,2) > 0.05, x(n,2)=0.05; end if x(n,3) > 1.5, x(n,3)=1.5; end if x(n,4) > 2.2, x(n,4)=2.2; end end

end


iter.

cost

0 10 20 30 400

1

2

3

4

5

6

7

[ 10 ]25

Global methodsPSO (6)

cost [2]

A [mm]

B [mm]

h [mm]

eps [ – ]

Rin []

Xin []

534 5.481 0.050 1.46 1.57 176.9 -0.1

2 288 5.794 0.050 1.46 1.69 152.2 1.7

154 5.333 0.050 1.44 1.50 187.6 -0.5

21 5.406 0.050 1.48 1.54 196.7 3.2

50 iterations, 20 agents, c1 = c2 = 1.49, w = 0.9 -> 0.4, absorbing walls


Searching for a minimumglobal first, local later


f(x)

x

startingpoint

global local


Searching for a minimumglobal first, local later


f(x)

x

startingpoint

globallocal methodmethod


Local minimizationgeneral algorithm (1)

1. Testing convergence. If the actual estimate of the optimum xk is accurate enough, then the algorithm is terminated. Otherwise, go to 2.

2. Computing search direction. Estimate the best direction pk moving the actual estimate of the optimum xk towards the optimum.


Local minimizationgeneral algorithm (2)

3. Computing step length. Estimate scalar k ensuring the significant decrease of the value of the objective function: F(xk + kpk) < F(xk)

4. Updating the estimate of the minimum. Setxk+1 xk + k pk, k k + 1. Go back to 1.


Testing algorithmsRosenbrock function

21

221221 1100, xxxxxF

function F = rosenbrock( x)

F = 100*( x(2,1) - x(1,1)^2)^2 +... ( 1 - x(1,1))^2;


Steepest descentanalytical approach

kk gp kkkk gxx 1

function sda( alpha)

M = 10000;x = [ -1; +1];

for m=1:M g(1,1) = -400*x(1,1)*( x(2,1)-x(1,1)^2)-2*(1-x(1,1)); g(2,1) = 200*( x(2,1) - x(1,1)^2); x = x - alpha*g; out(m,:) = x';end

1k


Steepest descentnumerical approach

function sdn( h)

M = 10000; alpha = 1e-3;x = [ -1; +1];

for m=1:M X1(1,1) = rosenbrock( [x(1,1) + h/2; x(2,1)]); X1(2,1) = rosenbrock( [x(1,1) - h/2; x(2,1)]); X2(1,1) = rosenbrock( [x(1,1); x(2,1) + h/2]); X2(2,1) = rosenbrock( [x(1,1); x(2,1) - h/2]); g(1,1) = (X1(1,1) - X1(2,1)) / h; g(2,1) = (X2(1,1) - X2(2,1)) / h; x = x - alpha*g; out(m,:) = x';end


Newton methoddirection, step

pGppgpx kkkk FF T21T

kkk gGp 1

kkkk gGxx

11

x

y

x x x012


Newton methodcode

function newton( x1, x2)

M = 10;x = [ x1; x2];

for m=1:M g(1,1) = -400*x(1,1)*(x(2,1)-x(1,1)^2)-2*(1-x(1,1)); g(2,1) = 200*( x(2,1) - x(1,1)^2); H(1,1) = 1200*x(1,1)^2 - 400*x(2,1) + 2; H(1,2) = -400*x(1,1); H(2,1) = -400*x(1,1); H(2,2) = 200; x = x - inv( H)*g; out(m,:) = x'end


Steepest descent vs. Newtoncomparison

Steepest descent Newton method

• Properly chosen step length k

• Step length k = 1 all the time

• Convergence for Rosenbrock: 7000 steps

• Convergence for Rosenbrock: 3 steps


ExampleGPS wire antenna

• Operation in frequency bands:– L1: central frequency fL1 = 1 575.4 MHz

– L2: central frequency fL2 = 1 227.6 MHz

• Omni-directional constant gain for the elevation from 5° to 90°

• Right-hand circular polarization


GPS wire antennaGA v. PSO (1)

a)

iterations

F

b)

iterations

F

LUKEŠ, Z., RAIDA, Z. Multi-objective optimiza-tion of wire antennas: genetic algorithms versus particle swarm optimization. Radioengineering, 2005, vol. 14, no. 4, p. 91–97.


a)

GPS wire antennaGA v. PSO (2) b)



a)



b)


Z

a)

f[MHz]



f[MHz]

Z

b)



a)

b)

a)

b)

Conclusions

• Multi-objective optimization:a complex view on the structure

• Global optimization:perspective designs of a structure

• Local optimization:tuning of a relatively good design


Documents

Global Optimization Techniques in Computational Electromagnetics Zbyněk Raida Dept. of Radio Electronics Brno University of Technology Brno, Czechia