1 / 42

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

tiger
Download Presentation

Global Optimization Techniques in Computational Electromagnetics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 …

More Related