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Particle Swarm Optimizationによる結像光学系の最適化

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(1)社団法人 情報処理学会 研究報告 IPSJ SIG Technical Report. 2006−MPS−58(7)   2006/3/16. 2CTVKENG5YCTO1RVKOK\CVKQP ߦࠃࠆ⚿௝శቇ♽ߩᦨㆡൻ ߦࠃࠆ⚿௝శቇ♽ߩᦨㆡൻ ૒⮮ ஜ. ຠ㊁ ാᴦ ᧲੩ㄘᎿᄢቇ. ⚿௝శቇ♽ߣߪ㧘ࠞࡔ࡜߿㗼ᓸ㏜ߩࠃ߁ߦ௝ࠍ↢ᚑߔࠆߎߣࠍ⋡⊛ߣߒߚశቇ♽ߢ޽ࠆ㧚ߎࠇࠄ ߩశቇ♽ࠍ⥄േ⊛ߦ⸳⸘ߔࠆ᦭ലߥᚻᴺߪᧂߛߦ㐿⊒ߐࠇߡ߅ࠄߕ㧘శቇ⊛ߥ⍮⼂߿శቇ♽⸳⸘ ߩ⚻㛎ࠍ↪޿ߚ⹜ⴕ㍲⺋ߩ➅ࠅ㄰ߒ߆ࠄᣂߒ޿శቇ♽߇⸳⸘ߐࠇߡ޿ࠆ㧚ߘߎߢᧄ⎇ⓥߢߪ㧘శ ቇ♽ߩ⥄േ⸳⸘ࠍᦨ⚳⊛ߥ⋡ᮡߣቯ߼㧘ᧄⓂߦߢߪߘߩ৻ㇱߣߒߡశቇ♽⸳⸘ߦኻߔࠆ Particle Swarm Optimization ߩㆡᔕࠍⴕߞߚ㧚. Optimization of the optical image forming system by Particle Particle Swarm Optimization Ken Sato. Yuji Shinano. Tokyo University of Agriculture and Technology The optical image forming system is included many optical systems, such as a camera and microscope. Design of the optical image forming system still much depends on experiences of designer, and few part of design process is automated. The aim of our research is to automate design process of the optical image forming system㧚In this abstract, we apply technique of particle swarm optimization to design of the optical image forming system㧚 㧝 ߪߓ߼ߦ ߪߓ߼ߦ. ో૕௝ࠍㅒ⸃ᨆ⊛ߦ᳞߼ࠆߎߣߪ㕖Ᏹߦ࿎㔍.  శቇ♽ߣߪ㧘‫৻ޟ‬ᨎ㧘߹ߚߪⶄᢙᨎߩ࡟ࡦ࠭. ߢ޽ࠆ㧚. ࠍ⚵ߺวࠊߖߡ৻ቯߩᯏ⢻ࠍᜬߚߖߚ߽ߩ‫ߣޠ‬.  ⃻࿷ߢ߽㧘శቇ♽ࠍ⥄േ⸳⸘ߢ߈ࠆ᦭ലߥᦨ. ቯ⟵ߐࠇࠆ㧚ᧄ⎇ⓥߢߪ㧘౉኿ߒߚశࠍ♖ᐲ⦟. ㆡൻᚻᴺߪ㐿⊒ߐࠇߡ߅ࠄߕ㧘ታ㓙ߩశቇ♽⸳. ߊ㓸శߔࠆ⚿௝శቇ♽ߩ⸳⸘ࠍኻ⽎ߣߔࠆ㧚శ. ⸘ߦ߅޿ߡߪ㧘శቇߦㅢߓߚࡌ࠹࡜ࡦߩ⸳⸘⠪. ቇ♽ߩ⚿௝ᕈ⢻ߪ㧘ฦ㕙ߦ߅ߌࠆᦛ₸࡮㕙㑆〒. ߩ⹜ⴕ㍲⺋ߦࠃࠆ᭎ᒻ᳿ቯߣ㧘⸘▚ᯏߦࠃࠆᓸ. 㔌࡮ࠟ࡜ࠬߩ᧚⾰㧔ዮ᛬₸㧕╬ߦࠃߞߡ᳿߹ࠆ㧚. ⺞ᢛߩ➅ࠅ㄰ߒ߇ⴕࠊࠇߡ޿ࠆ㧚⃻⁁ߢߪ㧘᭎. ߎࠇࠄࠍ⺞ᢛߒ⚿௝ᕈ⢻ࠍะ਄ߐߖߡ޿ߊߎ. ᒻߩ᳿ቯᲑ㓏ߪ㧘⸳⸘⠪ߩశቇ⊛⍮⼂ߣ⚻㛎ߦ. ߣ߇㧘శቇ♽ߩᦨㆡൻߢ޽ࠆ㧚శቇ♽ߩ⸳⸘ߦ. ࠃࠆߣߎࠈ߇ᄢ߈޿[2][3]㧚. ߪ㧘ઁߩᯏེ⸳⸘ಽ㊁ߦߪߥ޿⸳⸘ࠍ࿎㔍ߣߔ ࠆ․ᓽ⊛ߥ㕙߇ᄙߊሽ࿷ߔࠆ㧚․ߦ㧘శቇ♽ߪ. 㧞 శቇ♽ߩ⸳⸘Ꮏ⒟ శቇ♽ߩ⸳⸘Ꮏ⒟ߦ߅ߌࠆᦨㆡൻ Ꮏ⒟ߦ߅ߌࠆᦨㆡൻ ߦ߅ߌࠆᦨㆡൻ. ో૕ߣߒߡߩߺᯏ⢻ߒ㧘ߘࠇߙࠇߩㇱಽߩᯏ⢻.  LENS Design[1]ߦࠃࠆߣ㧘శቇ♽ߩ⸳⸘Ꮏ. ࠍಽ㔌ߔࠆߎߣߪߢ߈ߥ޿ὐ߇⸳⸘ࠍ࿎㔍ߦ. ⒟ߦ߅ߌࠆᦨㆡൻߪ৻⥸⊛ߦߪ 3 Ბ㓏ߦᄢ೎. ߒߡ޿ࠆ㧚శቇ♽ࠍ⸳⸘ߔࠆ㓙ߦ୥⵬ߣߥࠆ᭴. ߐࠇࠆߣ޽ࠆ㧚╙㧝Ბ㓏ߪశቇ♽ߩ᭎ᒻ᳿ቯᲑ. ᚑⷐ⚛ߩ߽ߟࡄ࡜ࡔ࠲ߩ⚵วߖߪ⩨ᄢߥᢙߣ. 㓏ߢ㧘ㆊ෰ߦ․⸵ߢ޽ߞߚశቇ♽߿㧘⸳⸘⠪⥄. ߥࠆ߇㧘ㇱಽߩಽ㔌߇ߢ߈ߥ޿ߣ޿߁․ᓽ߆ࠄ㧘 りߩశቇ⊛ߥ⍮⼂࡮⚻㛎߆ࠄశቇ♽ߩᄢ߹߆ߥ. −23− 1.

(2) ᒻࠍ᳿ቯߔࠆ㧚ߎߩᲑ㓏ߢߪ⚿௝ᕈ⢻ࠃࠅ߽㧘. Ꮕ࡮௝㕙ḧᦛ࡮ᱡᦛߩ 5 ⒳ߦಽ㘃ߐࠇࠆ㧚. 㓸శߐࠇࠆߴ߈శ✢߇శቇ♽ߩㅜਛߢో෻. ෼Ꮕߩ⊒↢㊂ࠍⷞⷡ⊛ߦ⴫⃻ߔࠆ႐ว㧘࿑㧝. ኿࡮᜛ᢔߔࠆߎߣߥߊ௝㕙߹ߢ೔㆐ߢ߈ࠆ߆߇. ߩࠃ߁ߥࠬࡐ࠶࠻࠳ࠗࠕࠣ࡜ࡓ߇↪޿ࠄࠇࠆ㧚. ㊀ⷞߐࠇࠆ㧚╙ 2 Ბ㓏ߢߪ㧘╙ 1 Ბ㓏ߢ↢ᚑ. ࠬࡐ࠶࠻࠳ࠗࠕࠣ࡜ࡓߪ㧘శቇ♽ߦછᗧߩⷺᐲ. ߐࠇߚశቇ♽ߦኻߒߡᦨㆡൻ߇ⴕߥࠊࠇࠆ㧚ߎ. ߆ࠄశ✢⟲ࠍ౉኿ߐߖߚ႐วߦ㧘ฦశ✢߇௝㕙. ߩ╙ 2 Ბ㓏ߢߩᦨㆡൻߢߪ㧘శቇ♽ߩ઀᭽ࠍḩ. ߩߤߩ૏⟎ߦ೔㆐ߒߚ߆ࠍࠪࡒࡘ࡟࡯࠻ߒߚ. ߚߒߟߟ㧘௝ߩࡏࠤ࡮ߦߓߺ࡮ࠁ߇ߺࠍ㒰෰ߔ. ߽ߩߢ޽ࠆ㧚. ࠆ੐ߦኾᔨߐࠇࠆ㧚╙ 3 Ბ㓏ߢߪ㧘⃻ታ⊛ߥ↪. (+'.& 215+6+10. ㅜࠍ⠨߃ߚ਄ߢశቇ♽ߩዪᚲ⊛ߥᦨㆡൻ߇ⴕ ߥࠊࠇࠆ㧚╙ 3 Ბ㓏ߢߩᦨㆡൻߢߪ㧘╙ 2 Ბ  &). 㓏ߢะ਄ߐࠇߚ⚿௝ᕈ⢻ߪߢ߈ࠆߛߌ⛽ᜬߒ ߥ߇ࠄ߽㧘ߘߩઁߩ⃻ታ⊛ߥ᧦ઙߦᴪ߁ࠃ߁ߥ శቇ♽߳ߣᦨㆡൻߐࠇࠆ㧚ౕ૕⊛ߦߪ㧘଀߃߫ ⚿௝ᕈ⢻߇ૐਅߔࠆߣߒߡ߽㧘ࠟ࡜ࠬߩ✚⾰㊂ ࠍᷫዋߐߖࠆߎߣ߇ߢ߈ࠇ߫㧘శቇ♽ߩ㊀㊂ߣ.  &). ⵾ㅧࠦࠬ࠻ߩ㕙ߢߪᦨㆡߦߥࠆ㧚ߎߩࠃ߁ߥ࠻ ࡟࡯࠼ࠝࡈߪᄙ᭽ߦሽ࿷ߒᓧࠆ߇㧘ᦨ⚳⊛ߦߪ ⸳⸘⠪߇್ᢿߒᦨ⚳⊛ߥశቇ♽߇⸳⸘ߐࠇࠆ㧚  ᧄ⎇ⓥߢߪ㧘శቇ♽⸳⸘ߩోᎿ⒟ࠍ฽߻⥄േ ⸳⸘ࠍᦨ⚳⊛ߥ⋡ᮡߣቯ߼㧘ోᎿ⒟ࠍᰴߩࠃ߁.  &). ߦಽഀߔࠆ㧚߹ߕ㧘╙ 1 Ბ㓏ߪశቇ♽ߩ᭎ᒻࠍ ᳿ቯߒ㧘ߘߩ᭎ᒻߩ▸࿐ౝߢߩⶄᢙߩᄙ᭽ߥశ. '. ቇ♽⟲ߩ↢ᚑࠍⴕ߁ೋᦼ⸃↢ᚑᲑ㓏ߣߒ㧘╙ 2. &'(1%75+0). //. . Ბ㓏ߪ╙ 1 Ბ㓏ߢ↢ᚑߐࠇߚೋᦼ⸃⟲ࠍ↪޿. %QQMG6TKRNGVH. ߚᦨㆡൻᲑ㓏ߣߔࠆ㧚╙ 2 Ბ㓏ߢߪⶄᢙߩశቇ. ࿑ 1㧦ࠬࡐ࠶࠻࠳ࠗࠕࠣ࡜ࡓ. ♽߆ࠄ㧘઀᭽ࠍḩߚߒߟߟ௝⾰ࠍะ਄ߐߖߚ 1 ߟߩశቇ♽ࠍ↢ᚑߔࠆ㧚╙ 3 Ბ㓏ߪ㧘ᧄ᧪ߩ╙. 㧠 ໧㗴⸳ቯ ໧㗴⸳ቯ. 3 Ბ㓏ߣห᭽ߢ޽ࠆ㧚.  ᧄ⎇ⓥߢߪ㧘໧㗴ࠍ◲නߦߔࠆߚ߼㧘޿ߊࠄ. ᧄⓂߢߪ㧘ߎߩቯ⟵ߒߚ⸳⸘Ꮏ⒟ਛ߆ࠄ㧘╙ 2 Ბ㓏ߦ޽ߚࠆᦨㆡൻࠍᛒ߁㧚. ߆ߩ໧㗴⸳ቯࠍⴕߞߡ޿ࠆ㧚߹ߕ㧘ዋߥ޿ᨎᢙ ߢ෼Ꮕࠍ㒰෰ߔࠆ㕖⃿㕙࡟ࡦ࠭ߪᛒࠊߥ޿㧚㕖 ⃿㕙ߩ࡟ࡦ࠭ࠍᛒ߁႐ว㧘ࡄ࡜ࡔ࠲ߩᢙ߇῜⊒. 㧟 శቇ♽ߩ⹏ଔ శቇ♽ߩ⹏ଔ. ⊛ߦჇ߃ߡߒ߹߁ߎߣ߆ࠄ㧘࡟ࡦ࠭㕙ߪ⃿㕙ߩ.  శቇ♽ߩ↪ㅜ߿᳞߼ࠄࠇࠆᕈ⢻ߩ࡟ࡌ࡞ߦ. ߺࠍᛒ߁߽ߩߣߒߚ㧚. ࠃߞߡ㧘శቇ♽ߦߪ᭽‫⹏ߥޘ‬ଔၮḰ߇޽ࠆ㧚ߘ. ᰴߦ㧘ᧄ⎇ⓥߦ߅ߌࠆ⹏ଔ㑐ᢙߪ௝㕙਄ߢߩ. ߩၮᧄߣߥࠆ⹏ଔၮḰߪ㧘ℂᗐ⊛ߥ⚿௝૏⟎ߣ. ฦశ✢ߩ⚿௝ὐߣℂᗐὐߩ࠭࡟ߩ✚๺ߢ⴫⃻. ታ㓙ߩ⚿௝૏⟎ߣߩ࠭࡟ߢ㧘ߎࠇߪ‫ޟ‬෼Ꮕ‫ߣޠ‬. ߒߚ㧚ߎࠇࠍᦨዊൻߔࠆߎߣߢ㧘5 ⒳ߩ෼Ꮕߩ. ๭߫ࠇࠆ㧚෼Ꮕߪ⃿㕙෼Ꮕ࡮ࠦࡑ෼Ꮕ࡮㕖ὐ෼. ߁ߜ 4 ⒳ߩ෼Ꮕࠍᛥ߃ࠆߎߣ߇น⢻ߢ޽ࠆ㧚ߎ. −24− 2.

(3) ߩ 4 ⒳ߪ㧘 ‫ޟ‬௝ߩࡏࠤ‫⋧ߦޠ‬ᒰߔࠆ෼Ꮕߢ޽ࠅ㧘. ⸃ᖱႎ㧔Global Best㧕ࠍㅦᐲࡌࠢ࠻࡞ߦട๧. ௝ߩ⾰ࠍ⠨߃ߚ႐วߦߪ߽ߞߣ߽ᩮᧄ⊛ߥⷐ. ߔࠆߎߣߢ⸃ߩᦝᣂࠍⴕ޿㧘Best Solution ࠍ. ⚛ߢ޽ࠆ㧚ᱷࠆ 1 ⒳ߪᱡᦛ෼Ꮕߢ޽ࠆ߇㧘ߎࠇ. ⊒⷗ߔࠆᚻᴺߢ޽ࠆ㧚PSO ߩ⸃ߩᦝᣂᑼࠍએ. ߪ‫ޟ‬௝ߩࠁ߇ߺ‫⋧ߦޠ‬ᒰߔࠆ෼Ꮕߢ޽ࠆ㧚ᧄ᧪. ਅߦ␜ߔ㧚. v ′ = wv + c1 r1 (PL − x) + c 2 r2 (PG − x) x′ = x + v ′. ߪࠁ߇ߺ߽ᛥ߃ࠆߴ߈ߛ߇㧘5 ⒳ߩ෼Ꮕࠍ৻ᐲ ߦᛥ߃ࠆߎߣߪ࿎㔍ߢ޽ࠆߚ߼㧘߹ߕߪࡏࠤߩ ߺࠍኻ⽎ߣߒߡ⎇ⓥࠍⴕ߁ߎߣߦߒߚ㧚 ⎇ⓥߩߚ ߼ߩశ ቇ♽ߩ⋡ᮡ઀᭽ࠍ. F୯. ߎߎߢ㧘x ߪ⸃ࡌࠢ࠻࡞ࠍ␜ߒ㧘v ߪㅦᐲࡌࠢ ࠻࡞ࠍ␜ߔ㧚PL ߪ୘૕ߏߣߩᥳቯ⸃ᖱႎࠍ଻. : 4.5. ᜬߔࠆࡌࠢ࠻࡞ߢ޽ࠅ㧘PG ߪ⸃㓸࿅ో૕ߢߩ. ὶὐ〒㔌: 50mm. ฝߩࠃ߁ߦቯ߼ߚ㧚 ↹ⷺ ᧄ⎇ⓥߢߪ㧘᳢↪⊛. ᥳቯ⸃ᖱႎࠍ଻ᜬߔࠆࡌࠢ࠻࡞ߢ޽ࠆ㧚r1㧘r2. : 46q. ߪ[0,1]ߩੂᢙ㧘w㧘c1㧘c2 ߪត⚝ߩㅦᐲࠍ⺞ᢛ. ߥశቇ♽ߩᦨㆡൻࠍ⋡ᮡߣߒߡ޿ࠆߚ߼㧘઀᭽. ߔࠆߚ߼ߩࡄ࡜ࡔ࠲ߢ޽ࠆ㧚. ߪ৻⥸⊛ߥࠞࡔ࡜࡟ࡦ࠭ߩ઀᭽ߣ߶߷ห᭽ߥ.  න⚐ߥ GA ߩࠝࡍ࡟࡯࠲ߪ࡜ࡦ࠳ࡓᕈ߇ᒝ. ߽ߩߣቯ߼ߚ㧚ߥ߅㧘F ୯ߪ௝ߩ᣿ࠆߐࠍ⴫ߔ. ߊ㧘శቇ♽ߩ⸳⸘໧㗴ߦ߅޿ߡߪታⴕਇน⢻ߥ. ᢙ୯ߢ㧘F ୯߇ዊߐ޿߶ߤ௝߇᣿ࠆ޿ߎߣࠍ␜. ⸃߇↢ᚑߐࠇ߿ߔ޿㧚৻ᣇߢ PSO ߪ㧘Global. ߔ㧚↹ⷺߣߪ㧘శቇ♽ߦ౉኿ߔࠆశ✢⟲ߩⷺᐲ. ߥត⚝ⓨ㑆ߦ߅ߌࠆጊ⊓ࠅᴺߣߺߥߖࠆ㧚ߚߛ. ߩ▸࿐ࠍ␜ߔᢙ୯ߢ޽ࠆ㧚. ߒ㧘ᘠᕈࠍ⠨ᘦߔࠆߩߢ෩ኒߦหߓߢߪߥ޿㧚 ߎࠇߪ㧘ㆡಾߥࡄ࡜ࡔ࠲ࠍਈ߃ࠆߎߣ߇಴᧪ࠇ. 㧡 ᦨㆡൻᚻᴺ ᦨㆡൻᚻᴺ. ߫㧘ฦೋᦼ⸃ߩ๟ㄝࠍត⚝ߒߟߟࠃࠅఝ⦟ߥೋ. శቇ♽ߩ⸳⸘ߢߪ㧘ฎౖ⊛ߦߪ㧘ᷫ⴮ᦨዊ⥄. ᦼ⸃ߩᣇะ߳෼᧤ߒߡ޿ߊߎߣ߇น⢻ߢ޽ࠆ. ਸ਼ᴺ㧔DLS : Damped Least Square㧕߇↪޿. ߣ⠨߃ࠄࠇࠆ㧚ߘߎߢ㧘ᧄ⎇ⓥߢߪ PSO ߦࠃ. ࠄࠇߡ߈ߚ㧚ߒ߆ߒ㧘ㄭᐕ㧘ࡔ࠲ࡅࡘ࡯࡝ࠬ࠹. ࠆᦨㆡൻࠍ⹜ߺߚ㧚 . ࠖ࠶ࠢࠬߩㆡ↪଀߽Ⴧ߃ߡ޿ࠆ㧚଀߃߫㧘ᢥ₂ [4]ߢߪ㧘ᄢၞ⊛ᦨㆡൻߣߒߡታᢙ୯ GA㧔ㆮવ. 㧢 ታ㛎߅ࠃ߮⠨ኤ ታ㛎߅ࠃ߮⠨ኤ. ⊛ࠕ࡞ࠧ࡝࠭ࡓ㧕ࠍ↪޿ߡ޿ࠆ㧚߹ߚᢥ₂[5]. శቇ♽⸳⸘ߩ╙ 1 Ბ㓏ߦ޽ߚࠆೋᦼ⸃⟲ߩ. ߪ㧘GP㧔Genetic Programming㧕ߦࠃࠆᦨㆡ. ↢ᚑߦ㑐ߒߡߪ㧘ᥳቯ⊛ߦ࡜ࡦ࠳ࡓߦ⸃ࠍ↢ᚑ. ൻࠍⴕ޿㧘᧼࡟ࡦ࠭⟲߆ࠄ․⸵ߣߥߞߡ޿ࠆశ. ߒߚ㧚PSO ߢߪᦨㆡൻਛߢ࡟ࡦ࠭ᨎᢙ߇ᄌൻ. ቇ♽ߣห╬ᕈ⢻ߩశቇ♽ߩ↢ᚑߦᚑഞߒߚߣ. ߔࠆ੐ߪήߊ㧘࡟ࡦ࠭ 3 ᨎߩ࡜ࡦ࠳ࡓߦ↢ᚑߐ. ႎ๔ߒߡ޿ࠆ㧚. ࠇߚೋᦼ⸃⟲ߢᦨㆡൻࠍⴕ߁႐ว㧘ᦨ⚳⊛ߥ⸃.  ᧄ⎇ⓥߢߪ㧘ᦨ ㆡൻᚻᴺߣߒ ߡ Particle. ߽࡟ࡦ࠭ 3 ᨎߣߥࠆ㧚ᢙ୯ታ㛎ߪ࡟ࡦ࠭ 3 ᨎ㧘. Swarm Optimization(PSO㧕[6]ࠍណ↪ߒߚ㧚. 4 ᨎ㧘5 ᨎߩߘࠇߙࠇߩ႐วߦ߅޿ߡታ㛎ߒߚ㧚. PSO ߣߪ㧘㠽߿㝼ߩ⟲ࠇ߇ⴕേߔࠆ᭽ሶࠍᮨ. ೋᦼ⸃⟲ߩ୘ᢙߪ 100 ୘૕ߣߒߚ㧚. ୮ߒߚㅴൻ⊛ࠕ࡞ࠧ࡝࠭ࡓߩ৻⒳ߢ޽ࠆ㧚ⶄᢙ.  PSO ࠍㆡᔕߒߚ⚿ᨐߪ㧘⚕㕙ߩㇺว਄࡟ࡦ. ୘ߩ୘૕㧔Particle㧕ࠍ↪޿ߡ⸃㓸࿅ࠍ↢ᚑߒ㧘. ࠭ 5 ᨎߩ႐วߦߩߺ࿑ 2 ߦ␜ߔ㧚. ฦ୘૕ߪㅦᐲࡌࠢ࠻࡞ࠍᜬߟ㧚୘૕ߏߣߩᥳቯ ⸃ᖱႎ㧔Local Best㧕ߣ㧘⸃㓸࿅ో૕ߢߩᥳቯ. −25− 3.

(4) ㆡൻߦኻߒߡ PSO ߦࠃࠆᦨㆡൻࠍ⹜ߺߚ㧚ᧄⓂ. 㪧㪪㪦. ߢ␜ߒߚታ㛎ߢߪ㧘࡜ࡦ࠳ࡓߦ↢ᚑߒߚೋᦼ⸃ࠍ ↪޿ߚ㧚ߒ߆ߒ㧘╙㧝Ბ㓏ߦ߅޿ߡ㧘ㆡᒰߥᄙ᭽ ⹏ଔ㑐ᢙ୯. ᕈࠍ߽ߞߚ⸃㓸࿅ߩ↢ᚑ߇ታ⃻ߢ߈ࠇ߫㧘ߘࠇࠄ ࠍᧄⓂߢ␜ߒߚ PSO ߩೋᦼ⸃ߣߒߡਈ߃ࠆߎߣ ߢ㧘ᦝߥࠆലᨐࠍ޽ߍࠄࠇࠆ߽ߩߣ⠨߃ࠆ㧚  ੹ᓟߩ⺖㗴ߪ㧘ᄙ᭽ᕈࠍᜬߞߚೋᦼ⸃㓸࿅↢ᚑ 㪇. 㪈㪇㪇. 㪉㪇㪇 㪊㪇㪇 䉟䊁䊧䊷䉲䊢䊮࿁ᢙ. 㪋㪇㪇. ࠕ࡞ࠧ࡝࠭ࡓߩ㐿⊒ߣ㧘ߘࠇࠄߦኻߔࠆ PSO ߢ. 㪌㪇㪇. ߩᦨㆡൻߩലᨐᬌ⸽ߢ޽ࠆ㧚߹ߚ㧘PSO ߣ޿߁ᚻ ᴺߪᲧセ⊛ᣂߒ޿ᚻᴺߢ޽ࠅ㧘⃻࿷߽ᣣ‫ޘ‬ᣂߒ޿. ࿑㧞㧦PSO ታⴕߩ᭽ሶ㧔࡟ࡦ࠭ 5 ᨎ㧕. ⹜ߺ߇ߥߐࠇߡ޿ࠆ㧚ᰴ‫⊒ߣޘ‬᩺ߐࠇࠆᡷ⦟ဳ  ᧄታ㛎ߢߪ㧘⚳ੌ᧦ઙࠍ 500 ࠗ࠹࡟࡯࡚ࠪࡦ. PSO ߩㆡ↪ߦࠃࠆലᨐᬌ⸽߽ᔅⷐߢ޽ࠆ㧚. ߣߒߡ⸳ቯߒߚ㧚࿑ 2 ߢߪ㧘ᡷༀߩᄙߊߪ 100 ࠗ ࠹࡟࡯࡚ࠪࡦએౝߦ↢ߓߡ޿ࠆ㧚ߘࠇએᓟ߽ࠊߕ. ෳ⠨ᢥ₂ ෳ⠨ᢥ₂. ߆ߦᡷༀߪ↢ߓߡ޿ࠆߎߣ߇ࠊ߆ࠆ㧚. [1] Gregory Hallock Smith : “LENS Design”㧘 Willmann-Bell, Inc.㧘Virginia㧘1998 [2] 㜞ᯅ෹ಷ : “࡟ࡦ࠭⸳⸘”㧘᧲ᶏᄢቇ಴ ␠㧘᧲ ੩㧘1994 [3] ᧻੗ศ຦ : “࡟ࡦ࠭⸳⸘ᴺ”㧘 ౒┙಴ ᩣᑼળ␠㧘 ᧲੩㧘1972 [4] I. Ono, S. Kobayashi, K. Yoshida, ”Optimal lens design by real-coded genetic algorithms. //. . 5ECNG. . using UNDX”, Computer methods in applied. -O(GD. ࿑㧟㧦PSO ታⴕ⚿ᨐ㧔࡟ࡦ࠭ 5 ᨎ㧕. mechanics and engineering, , 2000, pp.483-497, [5] Koza, John R., Al-Sakran, Sameer H., and.  ࿑ 3 ߩశቇ♽ߪ㧘ᧄታ㛎ߦ߅޿ߡᦨ߽⹏ଔ㑐ᢙ. Jones, Lee W., “Automated re-invention of six. ୯ߩ⦟߆ߞߚశቇ♽ࠍឬ↹ߒߚ߽ߩߢ޽ࠆ㧚࡜ࡦ. patented optical lens systems using genetic. ࠳ࡓߦ↢ᚑߐࠇߚೋᦼ⸃߆ࠄᲧߴࠆߣᄢ߈ߊᡷༀ. programming”,. ߐࠇߡ޿ࠆ㧚 ᭽‫੹ߥޘ‬ᓟߩ⺖㗴ߪ޽ࠆ߇㧘ᧄታ㛎. Computation. ߢߪ㧘╙ 2 ࠬ࠹࠶ࡊߦኻߔࠆࠕ࡞ࠧ࡝࠭ࡓߩലᨐ. Washington DC, USA, June 25 - 29, 2005.. ߣߒߡ PSO ߇᦭ᦸߥశቇ♽ߩᦨㆡൻᚻᴺߣߥࠆ. Proceedings, pp.25-29.. น⢻ᕈࠍ␜ߒߚ㧚. [6]Kennedy, J. and Eberhart, R., “Particle. Genetic Conference. And. Evolutionary. (GECCO. 2005),. Swarm Optimization.” Proceedings of IEEE 㧣 ߅ࠊࠅߦ ߅ࠊࠅߦ. Conference.  ᧄⓂߢߪ㧘శቇ♽ߩ⸳⸘ߦኻߒߡᦨㆡൻᚻᴺࠍ. Australia, 1995, pp. 1942-1948.. ㆡ↪ߔࠆߚ߼ߩ⸳⸘Ꮏ⒟ࠍቯ⟵ߒߚ㧚߹ߚ㧘ቯ⟵ ߒߚ⸳⸘Ꮏ⒟ߦ߅ߌࠆ 3 Ბ㓏ߩਛߢ╙ 2 Ბ㓏ߩᦨ. −26− 4E. on. Neural. Networks,. Perth,.

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Keywords: continuous time random walk, Brownian motion, collision time, skew Young tableaux, tandem queue.. AMS 2000 Subject Classification: Primary:

Kilbas; Conditions of the existence of a classical solution of a Cauchy type problem for the diffusion equation with the Riemann-Liouville partial derivative, Differential Equations,

It turns out that the symbol which is defined in a probabilistic way coincides with the analytic (in the sense of pseudo-differential operators) symbol for the class of Feller

Then it follows immediately from a suitable version of “Hensel’s Lemma” [cf., e.g., the argument of [4], Lemma 2.1] that S may be obtained, as the notation suggests, as the m A

In order to be able to apply the Cartan–K¨ ahler theorem to prove existence of solutions in the real-analytic category, one needs a stronger result than Proposition 2.3; one needs