Successfully reported this slideshow.

Silicon Cluster Optimization Using Extended Compact Genetic Algorithm

2,321 views

Published on

This paper presents an efficient cluster optimization algorithm. The proposed algorithm uses extended compact genetic algorithm (ECGA), one of the competent genetic algorithms (GAs) coupled with Nelder-Mead simplex local search. The lowest energy structures of silicon clusters with 4-11 atoms have been successfully predicted. The minimum population size and total number of function (potential energy of the cluster) evaluations required to converge to the global optimum with a reliability of 96% have been empirically determined and are O(n4.2) and O(n8.2) respectively. The results obtained indicate that the proposed algorithm is highly reliable in predicting globally optimal structures. However, certain efficiency techniques have to be employed for predicting structures of larger clusters to reduce the high computational cost due to function evaluation.

Published in: Economy & Finance, Education
  • Be the first to comment

Silicon Cluster Optimization Using Extended Compact Genetic Algorithm

  1. 1. Silicon Cluster Optimization Using Extended Compact Genetic Algorithm (ECGA) I'lli11a1‘a Sdl. s‘ll‘_' Gllaiiglilla. Xian Unive1‘sil_; ' of Illinois at. Urbaiia. -Cliainpaign {ksastry, gxiao}©uiuc. edu December I2, 2000 MatSE 385 ATOMIC SCALE SIMULATION
  2. 2. Si Cluster Optimization Using ECGA 1 Motivation Overview of ECGA l~Iarginal Product Models and BBS Silicon Potential Algorithm Implclnoiitation Results Conclusions nl. ... .. lnmlix -. .~-mill. .. I . l.. .4.. ., . , u. ... n.. .u .41.. .. . .-. .m. -. MatSE 385_ DC‘: 12_ 2mm I nlunlu .4 In--b< n Hutu a n. .u. ..l«_~. I ‘ 5 ALL ""'-A“-° ““ K. Sn. -stry. (2. Xino
  3. 3. Si Cluster Optimization Using ECGA 2 l/ Iotivation a Existing algorithms use “not-so-good" operators — Prop01't. iona. t.e selection — Single point. crossover % Increased interest in competent GAS — Solves hard problems quickly reliably and accurately as An interesting competent GA is ECGA (Harik, 1999) — builds models of good data as linkage groups -a Cluster optimization is a. l'P-hard problem (VVi1le and Vennik, 1985) m. ... .. Imulix -. .~-mm. .. I . ».. .4.. ., MatSE 385. Dec 12. ‘Z000 . _ Dunn-nu :1 («II Iqjluunnl I nnumu .1 In. -r H mm. 4 n. .u. .u«_~. K. Sn. -ztry. G . Xmo 1 L ; LJ_, um. -;n. o l‘
  4. 4. Si Cluster Optimization Using ECGA 3 Overview of EC GA Key Idea: Good probability distribution E Lmka_qe lea. rmIng Probability distribution: Mawjqmal Product Models ( . -"I P M ) Quantified based on M in. .'1I7r2.. mn De. s(:7"1Ipt*zI072, Length (. -*lDL) MDL Concept: Simpler distributions are better BJBCSE 385. DUE l2. 2000 K. Sn. -stry. (2. Xino
  5. 5. Si Cluster Optimization Using ECGA 4 Flowchart of Optimization Algorithm r _ T Initialize Perfonn 5v; ,|u; uc Have the N clusters . Nelder-Mead ; poiemia] energy : clusters randomly ‘ simplex Search / of each cluster C0IW€'Tg€d T {T Replace N*PC ~ Llizilriig iii Blind 1 loumumcnl . . . . . . Old ‘-‘l““°"-5' MPM mm‘ selection _ llli--L 1.: -mi: Ahnélhnv. I ah" flan) {'; _|; :'_: _"; ,'; ,f, ;j; ':: :-, -,_-; ;:; ';; ______ MatSE 386. Dec 12. 2000 lvlu-n. l|. bl8|| |'£~| K. 585“-y‘ (; _ xjao
  6. 6. Si Cluster Optimization Using ECGA 5 MPM and Building Blocks : I y. II x, | illli Ii lli ill I ILI I II 1l1r—- In | r—I-I I Illflrm : I:I———3]I——- I- I—I- — '[I- [I [I-I [III]I—— I[I: lI CIEI «-—IIII‘I III] IIl. - II‘-9 BII pnxlllull Ina. .. hnulu u-om-m. I ‘I-um. ..‘ ll: Ian. nu I V. ,- ‘ . . . I. .'f3T$.1'. ".. ... ... T£. ..3.'I. Z‘I‘§i. ... .u. “N55 355- D“ '~’- -'3‘"“’ lm-nn. Il. hl8|I (In K. 5n_. ;(ry_ (; , xino
  7. 7. Si Cluster Optimization Using ECGA 6 Marginal Product Models o Product of marginal distributions 011 a. partition of genes o Similar to CGA (Harik et. al;1998) and PBIL(Baluja;1994) o Represent. more than one gene iii a partition o Make exposition simpler a Gene partition maps to linkage groups in. (.1-ulia Ainélhnv» I ulnliul) "; _,: ;', _-, ;_"}, =;, §,; j;_'; f:-,7}; -;; ';; __‘___ MatSE 385. Dec 12. zone ""--”-"3" V5‘ K. Sastry. C}. Xiao
  8. 8. Si Cluster Optimization Using ECGA 7 Minimum Description Length Models o Hypothesis: Good distributions are those for which — Representation of the distribution is minimum (model conI. .ple; r-ity, Cm — R. epresentation of population compressed is minimum ((:0'rrl. p'I‘essed population. c: o7r1.ple:1:ity, Cp). o Penalize complex models o Penalize ina. ccurate models a Combined complexity, Cc = C", + C, , lllinnb huulia Kl-rilhlm I ml-ndan) {'; ,-; :;‘n-, ,,-; ,1,; _;j; ';: :-33;; -;; __m_ MatSE 386. Dec I2. 2000 lulu-n. l|. bl8|| |'£~ K, 585”-y‘ (; _ xjao
  9. 9. Si Cluster Optimization Using ECGA 8 Building MPM using MDL Uses a steepest ascent. search: 1. Assume all the genes t. o be independent ([l], [2], - - and compute CC. . Form all possible combinations (Ng, ;,(N; ,;, -1)/2) of merging two Subsets‘ eg-3 ' ' ' '9 ' ' . Select the set with minimum combined complexity . If CC > C; go to step 6. . Use the set with C: . as the current IVIPM and go to step 2. '. Merging is not possible, exit with set from step 2 as MPM. llli--L 1.: -ulin Ahwllhm. I ah" um) {’; _I; :'_: f_"j, 'j, f,: j;': ::-, ',_'; ;:; ';; _‘____ MatSE 386. Dec I2. 2000 IvIuun. l|. bl8|| |'! ~l K. 585“-y‘ (; _ xjao
  10. 10. Si Cluster Optimization Using ECGA 9 Generation of New Population o Transfer l', ,*(l-PC. ) best individuals to the next generation o The rest N, ,*P, A. indix-'idua. ls a. re generated as follows: Take each of the subset of MPl»'l from one of the individuals. Similar t. o Inultiple point crossover. NuI11l)e1' of crossover points = N; ,;, . Instead of two pa. rents we have NM, parents. llli--L (. uuIia Au-«In. -. I Inn: in") {'; _I; :;', ‘-; _"; ,'; ,f, ;j; _': ::-; ;;; ;'; ;______ MatSE 386. Dec 12. zone lvlu-A. l|. bl8II I'M K. Safitry, (2. Xiao
  11. 11. Si Cluster Optimization Using ECGA 10 Silicon Potential Gong Potential 0 Gong, X. G. Phys. Rev. B 47, 2329 (1993) o Empirical three body potential o Based 011 Stillinger Vel. )cr potential o Reflects both tetrahedral (~ 109°) and preferred bond angles (~ 60°) o Accurate for predicting structural properties IIII. .. (. uuIia Au-«In. .. I . .I. .. u. .. 3 M : *.*: .:: '.. :': .=: .:; :;:7:-. ::: :.": :., ... . Mass 386- Dec 12. 2000 "' '5‘ K. Sastry. (2. Xiao
  12. 12. Si Cluster Optimization Using ECGA 11 Gong Potential: Equations 11. TI. ZI. ».(z*. .')+ Z . ... (2<. I‘. Ic> i<j i. <j<k Iv-2(i. j) A (B‘r', :_j” — exp ((1', -_, - — a. )_1] . I. .I3(-i. _j. _ 1.1) h. (1'_, ,;. II‘k. .) + h (-I'M, r. ,-yj) + h (7‘. I.. rjr. ) I. (('r. ,- - . .—I ,7 (irjia rki) V I <2 (°0=‘9.‘-«II: + 5) [ 0 A = 7.0496, B = 0.6022. a : 1.8, p = 4. q = 0. 0 / = 2.7, "I = 1.2, C0 = -0.5. (:1 = 0.45 inrb huulia Kicrillnm I ml-ndan) "; .,: ;', ‘-, “-; ,1,; _;j; ';: :-,7;; ;;'; ;_v___ MatSE 386. Dec I2. 2000 lulu-n. l|. bl8|| |'£~ K, 585“-y‘ (; _ xjao
  13. 13. Si Cluster Optimization Using ECGA 12 Algorithm Implementation Variables are fixed-space Cartesian coordinates Each coordinate is encoded by 5-bit binary 25 independent runs, pc = 0.8, 4-11 atoms Nelder—lIIea(l simplex: Press et al Termination criteria: — Fitness variance 3 0.1 — Population variance 5 0.1 At most one failure allowed in. (.1-ulia Ainélnim I u| -tidal) "; _,: ;;-I, " 1,f_; j;_'_'_'(-,7}; -;; ';; ______ MatSE 385. Dec 12. zone ""-- -"3" W“ K. Sastry. (2. Xiao
  14. 14. Si Cluster Optimization Using ECGA 13 Results: Single GA Run MBCSE 385. DUE l2. 2000 K. Sn. -stry. (2. Xino
  15. 15. Si Cluster Optimization Using ECGA 14 Results: Single GA Run llli . ... .II. .. ..I. .I-. ..I. .., . : :ff'f";1I: :'”: 'm: f:: _v. .u Mats ‘ 85. Dec 12. )'2(000 . Safitry, (2. iao
  16. 16. Si Cluster Optimization Using ECGA 1.5 Results: Optimal Structures 9,-. . . 1" ~. .I ‘Ill ' I l . . = tl%K(il| I'l'i = r K _ _“_"7“&. ‘«i'ls Pelilugtilnll l1ip‘‘*r. 'InIiIl ' ' ' ‘ U 4m. J Unicappcd distorted . _. . . l. . I h‘ . ._ f ' 'IIII: ap KI'lI! )lI. I.l B}c: I.ppcd _tcIx: IguIial Pen , '_{Ii§l(, “‘m pnsm. 6:1 -1.7.5999 anupnsm. L. = - l5.b-$87 -uh («In in Ahcrfilim. Iulnuunu - tun-an ~ - - , '_; _f_"_“"; _'_, _'. ‘,'“, _____‘; ,__ Matsh. 386. Dec I2. 2000 -| '-H9" | '5‘~ K. Sastry, G. Xiao
  17. 17. Si Cluster Optimization Using ECGA 16 Results: Population Size . . . S 5 F 01 5 7 7 5 I‘ ll 5 NIImo-, -- ulal:1'| '|S. fl Average Case: O(In”'7), Worst Case: O(n). MBCSE 385. DUE l2. 2000 K. Sn. -stry. (2. Xino
  18. 18. Si Cluster Optimization Using ECGA 17 Results: Convergence Time 5 . No 0! aloms II in. » (mull: Ah-vilhnv» I Ialnuhny , ’;_, :'_; -"'_" §, ;'; I__(-3;; -;; ';; __, ___ MatSE 386. Dec I2. 2000 ‘'''--''-''’''l'“ K. Sastry. (2. Xiao
  19. 19. Si Cluster Optimization Using ECGA 18 Results: Function Evaluation .5,‘ I I I 3 -1 S 5 .7 II 9 10 N) cl alum’, r Average Case: O(n.1I3), Worst Case: O(n.1'7). imp (.1-ulia Aicwélhnv» I Ialnuhny , ';_, :;', _-, ;_"}, =;, §,; j;_'; f:-,7}; -;; ';; __, ___ MatSE 386. Dec 12. zone ""--”-"3" V“ K. Sastry. (2. Xiao
  20. 20. Si Cluster Optimization Using ECGA 19 Conclusions Optimal structures of small Si clusters found Results agree with literature (Iwamtsu, M. J. Chem. Phy. 112 (2000)) Convergence is very fast. (3 25 generations). Population size increases linearly with cluster size Polynomial increase of funct. ion evaluation Results t. o be confirmed with bigger clusters lllinm (.1-ulia Ainélnim I Inn. in. 3 {'; _I; :_': ;_" : .:""'_‘_', ";: _‘fj'; ‘:_w__ MatSE 386. Dec 12. zone ""--”-"3" '~ K. Sastry. (2. Xiao

×