Global Optimization Techniques in Computational Electromagnetics Zbyněk Raida Dept. of Radio...

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

45.026.0

38.3 16.9

Global optimization techniques …

Optimizationobjective function (1)

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

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]

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)

More objectivesdirectivity patterns (1)

More objectivesdirectivity patterns (2)

More objectivesmulti-objective formulation

nhormaxD fEEF xx

nnS fsF xx

mnvertP fEF xx

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

importance

vector

P[w , w , w ]

single-object.optimization

F = w F + w F + w F

Searching for a minimumglobal versus local methods

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

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

absorbing reflecting invisible

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

0 10 20 30 400

[ 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

startingpoint

global local

Searching for a minimumglobal first, local later

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

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

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

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)

iterations

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.

GPS wire antennaGA v. PSO (2) b)

f[MHz]

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

Global Optimization Techniques in Computational Electromagnetics Zbyněk Raida Dept. of Radio...

Documents

Al- Raida Issue #30 - LAU

Metropolis Brno 481

428 magazin Brno

420 Brno infoposter

419 Brno infoposter

Brno Talk March27 2009

Brno Metropolitan Area - European Commission · Brno Metropolitan Area Number of inhabitants (2018) Total number (167 municipalities) 621 108 Municipalities without Brno 241 581 Brno

ICIS DA 2015 CZECH REPUBLIC BRNO. When? Where? SUNDAY 7th June 2015 - WEDNESDAY 10th June 2015 BRNO Barceló Brno Palace Hotel Address: Šilingrovo námestí

Zbyněk Roubalík - JBoss...7 Advanced Java EE Lab 2017 @ MUNI | Zbyněk Roubalík JDK tools – stack trace and JVM stats Java stack traces of threads jstack stack traces of Java

Gr underwater alain zbyněk

Jaromír Skorkovský KPH-ESF-MU Brno, Czech Republic Brno, 2015

Brno October 2008

Introduction to Linear Antennas Zbynek Raida Dept. of Radio Electronics Brno University of Technology Brno, Czechia

Pitbox 2014 brno

MASARYK UNIVERSITY BRNO

David Lehký , Zbyněk Keršner , Drahomír Novák

AR.03-Raida Chakroun-2012, Mandatory vs Voluntary Disclosures

424 Brno infoposter

Brno preview

TIC BRNO ↓ Adolf Loos in Brno · 3 Adolf Loos Sr Fountain with sculptures of three fishing boys in Lužánky 4 Besední dům, today the Brno Philharmonic (Filharmonie Brno) 5 Second