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Global Optimization Techniques in Computational Electromagnetics

Global Optimization Techniques in Computational Electromagnetics. Zbyněk Raida. Dept. of Radio Electronics Brno University of Technology Brno, Czechia. Outline. What does the optimization mean

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Global Optimization Techniques in Computational Electromagnetics

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  1. Global Optimization Techniquesin Computational Electromagnetics Zbyněk Raida Dept. of Radio ElectronicsBrno University of TechnologyBrno, Czechia

  2. 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 ITSS 2007, Pforzheim Global optimization techniques …

  3. Optimizationdefinition • Searching for such values of state variables to meet desired parameters as close as possible ITSS 2007, Pforzheim Global optimization techniques …

  4. Optimizationobjective function (1) • Deviation of the actual parameters of the system from the desired ones ITSS 2007, Pforzheim Global optimization techniques …

  5. Optimizationobjective function (2) ITSS 2007, Pforzheim Global optimization techniques …

  6. 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. ITSS 2007, Pforzheim Global optimization techniques …

  7. 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. ITSS 2007, Pforzheim Global optimization techniques …

  8. More objectivesdirectivity patterns (1) ITSS 2007, Pforzheim Global optimization techniques …

  9. More objectivesdirectivity patterns (2)

  10. More objectivesmulti-objective formulation ITSS 2007, Pforzheim Global optimization techniques …

  11. Multi-objective optimizationtwo approaches ITSS 2007, Pforzheim Global optimization techniques …

  12. Searching for a minimumglobal versus local methods ITSS 2007, Pforzheim Global optimization techniques …

  13. Global methodsgenetic algorithms (1) ITSS 2007, Pforzheim Global optimization techniques …

  14. Global methodsgenetic algorithms (2) initial populationquality evaluation selection ITSS 2007, Pforzheim

  15. Global methodsgenetic algorithms (3) crossover mutation ITSS 2007, Pforzheim Global optimization techniques …

  16. 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 resistance Xd = 0.0; % desired input reactance bit = [ 8 8 1 2]; % bits per A, B, h, eps geb = norm( bit, 1) + 1; % bits in chromosome gen = round( rand( I, geb-1)); % 1st generation for 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);

  17. Global methodsgenetic algorithms (5)

  18. Global methodsgenetic algorithms (6) ITSS 2007, Pforzheim Global optimization techniques …

  19. 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. ITSS 2007, Pforzheim Global optimization techniques …

  20. Global methodsPSO (2) ITSS 2007, Pforzheim

  21. Global methodsparticle swarm optimization (3) absorbing reflecting invisible ITSS 2007, Pforzheim Global optimization techniques …

  22. 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 resistance Xd = 0;% required input reactance dt = 0.1;% time step c1 = 1.49;% personal scaling factor c2 = 1.49;% global scaling factor x = zeros( I, 5);% agents’ position p = 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 velocity g = zeros( 1, 4);% global best e = zeros( G+1, 1); e(1) = 1e+6;

  23. 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

  24. Global methodsparticle swarm optimization (5)

  25. Global methodsPSO (6) ITSS 2007, Pforzheim Global optimization techniques …

  26. Searching for a minimumglobal first, local later ITSS 2007, Pforzheim Global optimization techniques …

  27. Searching for a minimumglobal first, local later ITSS 2007, Pforzheim Global optimization techniques …

  28. Local minimizationgeneral algorithm (1) • Testing convergence. If the actual estimate of the optimum xk is accurate enough, then the algorithm is terminated. Otherwise, go to 2. • Computing search direction. Estimate the best direction pk moving the actual estimate of the optimum xk towards the optimum. ITSS 2007, Pforzheim Global optimization techniques …

  29. Local minimizationgeneral algorithm (2) • Computing step length. Estimate scalar kensuring the significant decrease of the value of the objective function: F(xk + kpk) < F(xk) • Updating the estimate of the minimum. Setxk+1xk + k pk, kk + 1. Go back to 1. ITSS 2007, Pforzheim Global optimization techniques …

  30. Testing algorithmsRosenbrock function function F = rosenbrock( x) F = 100*( x(2,1) - x(1,1)^2)^2 +... ( 1 - x(1,1))^2; ITSS 2007, Pforzheim Global optimization techniques …

  31. Steepest descentanalytical approach 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 ITSS 2007, Pforzheim Global optimization techniques …

  32. 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 ITSS 2007, Pforzheim Global optimization techniques …

  33. Newton methoddirection, step ITSS 2007, Pforzheim Global optimization techniques …

  34. 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 ITSS 2007, Pforzheim Global optimization techniques …

  35. Steepest descent vs. Newtoncomparison Steepest descent Newton method • Step length ak = 1 all the time • Properly chosen step length ak • Convergence for Rosenbrock: 7000 steps • Convergence for Rosenbrock: 3 steps ITSS 2007, Pforzheim Global optimization techniques …

  36. 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 ITSS 2007, Pforzheim Global optimization techniques …

  37. GPS wire antennaGA v. PSO (1) 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. ITSS 2007, Pforzheim Global optimization techniques …

  38. GPS wire antennaGA v. PSO (2) 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. ITSS 2007, Pforzheim Global optimization techniques …

  39. GPS wire antennaGA v. PSO (3) 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. ITSS 2007, Pforzheim Global optimization techniques …

  40. GPS wire antennaGA v. PSO (4) 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. ITSS 2007, Pforzheim Global optimization techniques …

  41. GPS wire antennaGA v. PSO (5)

  42. 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 ITSS 2007, Pforzheim Global optimization techniques …

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