ゼミ発表 (2019年9月)

Seminar Presentation (September, 2019)

Transcript

1. θϛൃද ۚઘେհ (ؙ໺ݚڀࣨ, ਫ਼౓อূ෇͖਺஋ܭࢉάϧʔϓ OB) 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ, ਺ֶԠ༻਺ཧઐ߈ ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 1 / 16
2. ຊൃදͷྲྀΕ 1 ਺஋ੵ෼ؔ࿈ͷ࣭໰ 2 ODE/PDE ؔ࿈ͷ࣭໰ 3 ॳาతͳ࣭໰ 4 JSIAM 2019 ೥ձͷ౔࢈࿩ ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 2 / 16
3. ਺஋ੵ෼ؔ࿈ͷ࣭໰ ਺஋ੵ෼ؔ࿈ͷ࣭໰ ਺஋ੵ෼ݚڀऀ͕ͨͪ௚໘͍ͯ͠Δ೉఺ ࣍ݩͷढ͍ (curse of dimensionality) ႈܕಛҟ఺, আڈՄೳಛҟ఺ → ෦෼ੵ෼Ͱରॲ ແݶ۠ؒੵ෼ ৼಈੵ෼ (Fourier ܕੵ෼, Laplace ܕੵ෼ͳͲ)→Ooura-Mori etc. ৼಈੵ෼ʹಛԽͨ͠ਫ਼౓อূͬͯߦΘΕͯ·͔͢ ? (͋·Γฉ͔ͳ͍ؾ͕͢Δ) Fourier جఈΛ࢖͏ͱ͖ʹ Fourier ܕੵ෼͕ग़͖ͯͨΓ͠ͳ͍ΜͰ͔͢ ? ෦෼ੵ෼͸ େֶडݧͰ͸࠶ؼܕੵ෼ (ݮਰৼಈͳͲ) ʹର͢Δఆੴͱͯ͠࢖ΘΕ͍ͯΔ ෳૉղੳͰ͸ৗ౟खஈͱͯ͠࢖ΘΕ͍ͯΔ (cf: Ablowitz-Fokas chapter 6) ฏࢁ߂ઌੜ͕গͳ͘ͱ΋ 2000 ೥୅͔Β਺஋ੵ෼ʹԠ༻͍ͯ͠Δ , → ͜ΕΒΛ೦಄ʹਫ਼౓อূͰ෦෼ੵ෼Λ࢖͏ண૝ʹࢸͬͨͷͰ͔͢ ? ฏࢁ߂. (1997). ೔ຊԠ༻਺ཧֶձ࿦จࢽ, 7(2), 131-138. ฏࢁ߂, ؗ໺༟จ, & ฏ໺রൺݹ. (2004). ਺ཧղੳݚڀॴߨڀ࿥ 1395: 190-195. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 3 / 16
4. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ T ઌੜ: 3 ࣍ݩଟ༷ମ্Ͱ PDE Λਫ਼౓อূ͍ͨ͠ ! A ͘Μ: ٿମ, ପԁମ্Ͱ Henon eq. Λਫ਼౓อূ͍ͨ͠ ! 3 ࣍ݩଟ༷ମ্Ͱ ODE/PDE Λਫ਼౓อূͨ͠ͱ͍͏࣮ྫͬͯ͋Γ·͔͢ ? 3 ࣍ݩଟ༷ମ্ͰجఈΛ࡞ͬͨΓ, Newton-Kantorovich Λద༻Ͱ͖·͔͢ ? (i.e. ؔ਺্ۭؒͰͷٞ࿦Λଟ༷ମʹ֦ுͰ͖·͔͢ ?) ʮجఈΛுΔʯ Fourier جఈΛுΔ = Fourier ڃ਺Ͱؔ਺ۙࣅ͢Δ Legendre جఈΛுΔ = Legendre ଟ߲ࣜͰଟ߲ࣜۙࣅ͢Δ ͱ͍͏ೝࣝͰ͋ͬͯ·͔͢ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 4 / 16
5. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ Fourier ม׵͸ (ਫ਼౓อূͰ͸) Fourier جఈͰ׆༻͞Ε͍ͯΔ͕, Laplace ม׵Λ ਫ਼౓อূͰ׆༻͢ΔࣄྫΛฉ͔ͳ͍ؾ͕͢Δ. Laplace ม׵ͷ׆༻ྫ Burgers eq. ↓ Cole-Hopf ม׵ (Laplace ม׵͔Βಋग़Ͱ͖Δ) ↓ heat eq. cf: Ablowitz-Fokas chapter 6 Laplace ม׵ͰجఈΛுΔྲّྀͬͯ͋ͬͨΓ͠·͔͢ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 5 / 16
6. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ ؔ਺ղੳతͳख๏Λ࢖Θͳ͍ ODE ͷਫ਼౓อূ๏͕ 2 λΠϓʹ෼ྨ͞ΕΔ͜ͱΛ ઌ೔ͷ JSIAM ͰॳΊͯ஌Γ·ͨ͠. Taylor ڃ਺ܕ →Lohner ๏, kv (PSA), · · · (৭ʑ͋ΔΒ͍͠) spectral method ܕ →Urabe-Galerkin1, radii polynomial approach2 → ղ๏ͷ࢖͍෼ཱ͚ͬͯ֬͞Εͯ·͔͢ ? ദ໦ઌੜ͸ PSA Λ਺஋ੵ෼, ODE ͷਫ਼౓อূʹԠ༻͍ͯ͠·͕͢, ௐ΂ͯΈͨ Βฏࢁ߂ઌੜ͸গͳ͘ͱ΋ 2000 ೥୅͔Β Taylor ڃ਺Λ਺஋ੵ෼, ODE ͷղ๏ʹ Ԡ༻͍ͯͯ͠, ฒྻԽ΋Ͱ͖͍ͯΔΑ͏Ͱ͢. → ദ໦ઌੜ͸ฏࢁઌੜ (΋͘͠͸ଞͷํʑ) ͷӨڹΛड͚ͯͨΓ͢Δͷ͔΋ ? stiff eq. ʹରͯ͠ spectral method Λద༻͢ΔͱͲ͏ͳΔ ? stiffness ʹ͍͍ͭͯΖ͍Ζڭ͍͖͑ͯͨͩ, ํఔࣜͷඇઢܗੑͱؔ܎͋Δͷ͔΋ ͠Εͳ͍ͱ͍͏͜ͱ͕෼͔ͬͨ. ߴ֊ eq. (ߴ֊ Painleve ౳) ͩͱ stiff ʹͳΓ΍͍͢ͱ͍͏܏޲ͬͯ͋Γ·͔͢ ? 1େੴઌੜͷඇઢܗղੳೖ໳Λࢀর 2Hungria, A., Lessard, J. P., & Mireles-James, J. D. (2014). Math. Comp. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 6 / 16
7. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ ઢܕ ODE ʹ͓͍ͯ stiffness ͸ߦྻͷ৚݅਺Ͱܾ·Δ લॲཧ (precondition) ͸ߦྻͷ৚݅਺ΛԼ͛ΔͨΊʹ࢖ΘΕΔ લॲཧΛ࢖ͬͯ stiff ͳઢܗ ODE Λղ͖΍͘͢͢Δ͜ͱ͕Ͱ͖ΔͷͰ͸ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 7 / 16
8. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ JSIAM ೥ձͰ 3 ࣍ݩ Allen-Cahn eq. ͷਫ਼౓อূʹ͍ͭͯ࿩Λ͓ฉ͖·͕ͨ͠, ϝϞϦ͕େมͦ͏ͳͷͰฒྻԽ͕ඞཁͳͷ͔ͳͱࢥ͍·ͨ͠. தඌઌੜͷ΍Γํ΍༏ղྼղ๏ͬͯฒྻԽʹରԠ͍ͯ͠·͔͢ ? JSIAM ೥ձͷ 40 ෼ߨԋͰ (Laplace, Maxwell ͷΑ͏ʹ࣌ؒൃల͠ͳ͍) PDE ͷ ݻ༗஋໰୊͕ѻΘΕ·ͨ͠ (ট଴͞ΕΔ΄ͲͳͷͰੈքతʹྲྀߦ͍ͬͯΔ͸ͣ). ࣌ؒൃల͢Δํఔࣜ (KdV, mKdV ͳͲ) ͷݻ༗஋໰୊͸Ͳ͏೉͘͠ͳΓ·͔͢ ? (࣌ؒൃల͢Δํఔࣜͩͱ֤࣌ؒ͝ͱʹۙࣅղΛٻΊΔඞཁ͕ग़͖ͯͯਫ਼౓อূ ͕೉͘͠ͳΔͱ͍͏ͷ͸ཧղͰ͖·͢) ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 8 / 16
9. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ ༗ݶཁૉ๏ (FEM) ͷϥΠϒϥϦ͕ཚཱ͍ͯ͠Δؾ͕͠·͢ (ݸਓͷײ૝). FreeFem++3 (JSIAM ೥ձͰߨशձ͕͋Γ·ͨ͠) FEnics Project4 FEMhub5 Ͳ͏΍ͬͯࠩผԽͯ͠Δͷ ? ࠷ۙ͸ʮFEM× ػցֶशʯΈ͍ͨͳݚڀ΋͋ΔΒ͍͠Ͱ͢ 6. → ͬͪ͜ํ໘ͷԠ༻΋͋ΔͷͰ͸ ? 3༗ݶཁૉ๏ͰֶͿݱ৅ͱ਺ཧʕFreeFem++਺ཧࢥߟϓϩάϥϛϯάʕ, ೔ຊԠ༻਺ཧֶձ ؂ मɾେ௩ ްೋɾߴੴ ෢࢙ஶ, ڞཱग़൛. 4Logg, A., Mardal, K. A., & Wells, G. (Eds.). (2012). Automated solution of differential equations by the finite element method: The FEniCS book. Springer Science & Business Media. 5http://femhub.com/ 6https://www.ricos.co.jp/tech-0101 ml fem geometry/ ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 9 / 16
10. ODE/PDE ؔ࿈ͷ࣭໰ ODE/PDE ؔ࿈ͷ࣭໰ ղ๏ͷ൚༻ੑ FDM < FEM < FVM ࢖͑ΔͳΒΈΜͳ FVM Λ࢖͍͍ͨ͸ͣ. FEM ͷޡࠩධՁʹ͍ͭͯ͸ݚڀ͕ଟ͍ (େੴઌੜͷຊ) FVM ͷޡࠩධՁʹؔ͢Δݚڀ͸͋·Γฉ͔ͳ͍ؾ͕͢Δ → ԿͰͩΖ͏ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 10 / 16
11. ॳาతͳ࣭໰ ॳาతͳ࣭໰ ਺ֶؔ਺ͷਫ਼౓อূ΍ Type II PSA ͸ Taylor ڃ਺ͱͦͷ৒༨߲ʹج͍͍ͮͯΔ ͕, ৒༨߲ʹ΋͍Ζ͍Ζ͋Δ. Peano ܕ Lagrange ܕ Cauchy ܕ Bernoulli ܕ Roche-Schlomilch ܕ ੵ෼ܕ → ৒༨߲ͷ࢖͍෼ཱ͚ͬͯ֬͞Εͯ·͔͢ ? (Մੵ෼ܥͷઌੜʹฉ͔Ε·ͨ͠) ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 11 / 16
12. ॳาతͳ࣭໰ ॳาతͳ࣭໰ ਺஋ઢܗ୅਺΍ܭࢉزԿֶʹ͸ʮ෼ׂ౷࣏๏ʯͱ͍͏֓೦͕͋Δͦ͏Ͱ͢. ෼ׂ౷࣏๏ʢdivide-and-conquer methodʣ: ͦͷ··Ͱ͸ղܾͰ͖ͳ͍ େ͖ͳ໰୊Λখ͞ͳ໰୊ʹ෼ׂ͠, ͦͷશͯΛղܾ͢Δ͜ͱͰ࠷ऴతʹ ࠷ॳͷ໰୊શମΛղܾ͢Δͱ͍͏໰୊ղܾͷख๏ (wikipedia) Schur ิݩ, block CG ܥ ∈ ෼ׂ౷࣏๏ ׳शతʹ͸͜͏͍͏ݴ͍ํ͸͠ͳ͍ؾ͕͠·͕͢, ͜ͷղऍ͋ͬͯ·͔͢ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 12 / 16
13. ॳาతͳ࣭໰ ॳาతͳ࣭໰ ʮ࠷దԽʯ (Optimization) ͱ໊ͷͭ͘ݴ༿͕࠷ۙ૿͍͑ͯΔؾ͕͠·͢. ྫ: Search Engine Optimization (ݕࡧΤϯδϯ࠷దԽ, ݕࡧΤϯδϯͰαΠτΛ্Ґʹදࣔͤ͞Δཪٕ) → ਺ཧ࠷దԽ͕Ԡ༻͞ΕͯͨΓ͢ΔͷͰ͔͢ ? (਺ཧ࠷దԽͱؔ܎ͳ͍ͳΒ, ͜ͷจ຺Ͱͷʮ࠷దԽʯͬͯͲ͏͍͏ҙຯͰ͔͢ ?) ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 13 / 16
14. ॳาతͳ࣭໰ ॳาతͳ࣭໰ kv ϥΠϒϥϦ͸ 2019 ೥ 9 ݄ݱࡏ Version 0.4.48 ʹͳ͍ͬͯΔ. Version 1 ͸͍ͭग़ΔͷͩΖ͏͔ ? Version ͷ൪߸͸Ͳ͏΍͚ͬͯͭͯΔ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 14 / 16
15. JSIAM 2019 ೥ձͷ౔࢈࿩ JSIAM 2019 ೥ձͷ౔࢈࿩ ߴ҆-ੴໟ: ”Gauss ͷ௒زԿඍ෼ํఔࣜͷϞϊυϩϛʔߦྻʹର͢Δਫ਼౓อূ” Ϟϊυϩϛʔߦྻɿઢܕ ODE Λಛ௃͚ͮΔߦྻ → ͜Ε͕෼͔Ε͹ ODE ͷղ͕ղੳ઀ଓʹΑΓͲ͏มΘΔ͔Λ׬શʹ೺ѲͰ͖Δ. ͜ͷ࿩͸߹ྲྀܕ௒زԿ, Heun eq. (௒زԿͷ֬ఆಛҟ఺ 4 ͭ ver.) ʹ΋ద༻Մೳ ͩͦ͏Ͱ͢. (Fuchs type ͳΒԿͰ΋͍͚Δ͔΋) ࠓޙ͸ଟม਺௒زԿ (Picard-Fuchs type eq.) ΁ͷద༻Λ໨ࢦ͍ͯͯ͠, ௒ݭཧ࿦ ͰݱΕΔ Calabi-Yau eq. Λ೦಄ʹஔ͍͍ͯΔΑ͏Ͱ͢. → linear stiff eq. Β͍͠Ͱ͢ (܎਺ߦྻͷ৚݅਺Λௐ΂ͯΈΔͱ໘ന͍͔΋) ਆอಓ෉. (2003). ෳૉؔ਺ೖ໳. ؠ೾ॻళ. ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 15 / 16
16. JSIAM 2019 ೥ձͷ౔࢈࿩ JSIAM 2019 ೥ձͷ౔࢈࿩ ϙελʔൃදͰ dLV (ݻ༗஋໰୊ʹର͢ΔՄੵ෼ΞϧΰϦζϜ) ͷ࿩͕͋ͬͨͷ Ͱ, ৭ʑڭ͑ͯ΋Β͍·ͨ͠. dLV Ͱݻ༗ϕΫτϧ·Ͱ͸ٻ·Βͳ͍ ݻ༗ϕΫτϧΛٻΊΔʹ͸ Wielandt ͷٯ൓෮͔ Jacobi ๏Λ࢖͏͔͠ͳ͍ Α͏Ͱ͢. ݻ༗஋ɾݻ༗ϕΫτϧ͕͢΂ͯಉ࣌ʹٻ·Δ఺Ͱ, ͜ͷํ๏ (Jacobi ๏) ͸ࣺ͕͍ͯͨ (by ࢁຊઌੜͷ਺஋ղੳೖ໳, 2003, p105) ࠃ֎Ͱ Jacobi ๏ͷ҆ఆੑ͕࠶ධՁ͞Ε͍ͯΔ (by ࢁຊઌੜͷ਺஋ղੳ ೖ໳, 2003, p105) ࠓͰ΋ Jacobi ๏ϒʔϜͬͯ͋Γ·͔͢ ? ۚઘେհ (ؙ໺ݚڀࣨ OB) θϛൃද 2019 ೥ 9 ݄, ˏദ໦ݚڀࣨ 16 / 16