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October 01, 24
スライド概要
2024年9月30日 解析力学ミニ研究会 https://sites.google.com/view/watanabegroup/AMbook/amworkshop にて講演した内容
研究してるひと https://www.eri.u-tokyo.ac.jp/people/ito/
変分データ同化の保存量と対称性 伊藤伸一 東京大学地震研究所 東京大学大学院情報理工学系研究科 解析力学ミニ研究会 伊藤伸一 2024年9月30日 1 /29
自己紹介 伊藤伸一 複雑系+データ同化 2015.3 大阪大学 大学院理学研究科 博士(理学) 破壊・亀裂パターンの統計則(物理) 2015.4 東京大学 地震研究所 特任研究員 大規模データ同化の数理(数学) 粒成長予測(応用) 2018.8 東京大学 地震研究所 助教 2018.12 東京大学 大学院情報理工学系研究科 助教(兼務) データ同化の数値解析への展開(数学) スロー地震(応用) 最近 破壊・亀裂と深層学習(応用・数学) 解析力学ミニ研究会 伊藤伸一 2024年9月30日 2 /29
本日の内容 データ同化、特に変分データ同化法に関連するお話をします。 ・変分データ同化法の概要 ・変分データ同化法の数理構造 — 保存量と高精度化(と対称性?) 解析力学ミニ研究会 伊藤伸一 2024年9月30日 3 /29
変分データ同化法の概要 解析力学ミニ研究会 伊藤伸一 2024年9月30日 4 /29
データ同化とは
モデル
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事後分布
データ
p(xt | D)
D = {D1 , D2 , ...}
予測の高精度化、
<latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit>
p(xt | D)
逆問題、…
天気予報
出典:Google Earth
解析力学ミニ研究会 伊藤伸一 2024年9月30日
<latexit sha1_base64="uqBeDhpsi6lxGlvwkHHdVQpc2sc=">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</latexit>
実験・観測デザインの
最適化
地震学
材料科学
Kano et al. (2015)
Ito et al. (2017)
5 /29
データ同化とは
モデル
<latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit>
事後分布
データ
p(xt | D)
D = {D1 , D2 , ...}
予測の高精度化、
<latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit>
p(xt | D)
<latexit sha1_base64="uqBeDhpsi6lxGlvwkHHdVQpc2sc=">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</latexit>
実験・観測デザインの
逆問題、…
・事後分布の全体評価には O(eN)回の評価が必要
最適化
N:モデル変数ベクトル xt の次元
・事後分布(に比例した量)を1点評価するのに1回のモデルの時間発展計算が必要
→ 大規模系において事後分布の全体評価はほぼ不可能
→ モデルの規模・目的に適したデータ同化アルゴリズムの選択が大切
現実的な計算資源・時間内で欲しい情報をいかに精度よく抽出するか
解析力学ミニ研究会 伊藤伸一 2024年9月30日
6 /29
変分データ同化 モデルの初期値に関する事後分布から事後確率最大解(MAP解)を抽出する手法 Forward モデル Dデータと + !t t = h(xt )の関係 Dt = h(xt ) + !t <latexit sha1_base64="yXpqNkT0w6ffWorbjXV9yQ9bbtU=">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</latexit> <latexit sha1_base64="yXpqNkT0w6ffWorbjXV9yQ9bbtU=">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</latexit> 初期値 x0 = ✓ h:観測演算子 <latexit sha1_base64="oWi9GfktOIkXj88PGSyzQpA7glc=">AAAO03iclZfPTxNBFMcf+AurFtCLiZeNgGkPkCkaJZ5AGgIn+Q2BxabbDu2G7e6yu60LTT0YTyaePXjSxIPxrP+AF/8BD8SrHoxHTLx48M10aQud0scSdmffvM/3+2Y6284armX6AWMHPb3nzl+4eKnvcuzK1Wvx/oHB66u+U/ZyfCXnWI63bmR9bpk2XwnMwOLrrsezJcPia8bOtOhfq3DPNx17Odhz+VYpW7DNbTOXDTCUGXjsJvSgyIOsXjLzWjqp6a7nuIGjHcXrkXymGuimrS0/qeoBD4OqY/i1Wk3bTaSxpzZaTITimkxmBobYGJOH1t5IRY0hiI55Z/DiZ9AhDw7koAwl4GBDgG0LsuDj3yakgIGLsS2oYszDlin7OdQghmwZszhmZDG6g+cC3m1GURvvhaYv6Ry6WPjvIanBCPvGPrBD9pV9ZL/Yv45aVakhatnDq1FnuZvpf3lz6W9XSsNzFmsuNrlTqw5gGyZktSZW78qIGEeuzlf2Xx8uPVwcqd5h79hvHMFbdsC+4Bjsyp/c+wW++AbVRzrWFECI+o6cY/9YRaImG3Oeylkqyapt/FyqGBezXpCREBkRORqTgw7i3sOIFuU9a2TqONsm3pmRG8VDjJrmUc9s94jJ9cRxnDpU8JqDBAzhKkpGqgYqhrJtyPWQbzhpMCwzh1GndkInJNGhkt0jsXtKdp/E7ivZMoktK1mPxHpK1iCxhpJ1SayrZHdJ7K6S9Umsr2SLJLaoZGdJ7KySnSax00o2TWLTSnaGxM4o2TkSO6dkN0jshpJdJ7HrSnaKxE51eH45PiUOSYEpFRz5C1bAPIqG3pKvXudm9F1OU2vmq9REHsczVa2Zr1Kz5G+mgRGqXivRafbp1e2fUluI80BTEZkqhbNU0rmOEvFbXZeZKoWCnDP6Cmjmq0flyjViyV0FdXStjEp1uWVd1XW2kdVb4u1MnuSeV/ottTwVrX7NeK2xt+i0gxH7E0f21hr3CbwWcaz1fZjYgY3h/mICWyXMMk/tTXbZMQn9AHucLn6sxU/lyM7oWMQrPzHKdt3RE87d+ineYudalrt3unOlQ//9M4x5G10qXT/Xpma751kdxdMRnuo32sVRPb8xfD9LnXwba2+sjo+l7o6NL9wbmnwUvan1wS24jXWk4AFM4n5kHlbQ6RN8hx/wM74Sr8afx1/UU3t7IuYGHDvir/4DH+D0sg==</latexit> ∝ <latexit sha1_base64="BVEjgQf0iX35/BBzir1DM3XQaXQ=">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</latexit> x 尤度関数 Y p(✓ | D) / p(✓) q(Dt 事前分布 T obs:観測時刻のセット !t ⇠ q(•) (ベイズの定理) h(xt )) t2T obs p(✓):事前分布 <latexit 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sha1_base64="9gygmdn+QQW5JH9b9JNHqm5Ff9o=">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</latexit> <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">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</latexit> 事後分布 i.i.d MAP解 <latexit sha1_base64="SMmXtD5Kzifn6YlK+hRImb/o8g8=">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</latexit> <latexit sha1_base64="e075dcW3ikGHqSQBkD9Kp8T8K60=">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</latexit> コスト関数 C(✓) = <latexit sha1_base64="dGQ7FytXyMRO7Mw9F/722V4PPcI=">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</latexit> log p(✓) p(✓ | D) / e C(✓) X t2T obs 解析力学ミニ研究会 伊藤伸一 2024年9月30日 log q(Dt h(xt )) 7 /29
Adjoint法による勾配計算 勾配法によって最適化したいが,,, C(✓) = <latexit sha1_base64="dGQ7FytXyMRO7Mw9F/722V4PPcI=">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</latexit> Adjoint法 log p(✓) X logq(D q(Dtt h(x h(xtt)) )) log t2T obs ✓ で直接微分ができない ✓ ◆ Z T d > L=C+ dt t f (xt ) xt dt 0 X <latexit 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<latexit sha1_base64="jRRHIl4aQaDCLy6WjaPD1Ye14kE=">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</latexit> d > t = (r x f ) 変分をとって係数比較 dt <latexit 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0 + (t t )r xt0 C :ラグランジュの未定乗数 t t0 2T obs <latexit sha1_base64="1kqcoTy2iTY3iSE4F+wj0lrXbFY=">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</latexit> Adjoint モデル d > (r ) t = xf dt t + <latexit 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X t0 2T obs (t 0 t )r xt0 C t=Tから時間後ろ向きに解くことで、目的の勾配 解析力学ミニ研究会 伊藤伸一 2024年9月30日 <latexit 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T =0 が得られる 8 /29
変分データ同化による初期値推定 d > (r ) = f t x dt <latexit 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X t + t0 2T obs (t t0 )r xt0 C <latexit 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T =0 変分データ同化法による初期値最適化 C(✓) <latexit 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⓪ 初期推定値を設定 ① Forward モデルを解く ② Adjoint モデルを時間後ろ向きに解く ✓guess <latexit 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✓ˆ <latexit sha1_base64="e8y7Sn9k5Dcdy/FYFy9xaHHxaEg=">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</latexit> ✓ <latexit sha1_base64="jRRHIl4aQaDCLy6WjaPD1Ye14kE=">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</latexit> ③ 勾配に従って少し動かす。①へ戻る。 Adjointモデルの計算コスト ≒ Forward モデルの計算コスト 全体の計算コスト = (シミュレーション時間) x (反復回数) c.f. 数値微分 (シミュレーション時間) x (変数の数) x (反復回数) 解析力学ミニ研究会 伊藤伸一 2024年9月30日 9 /29
変分データ同化の活用例:金属結晶粒成長 Multi-phase- eld モデル シミュレーション結果 Ito et al., STAM,18:1, 857-869 (2017) Grain ID i=1,…,n :ドメイン(粒)の通し番号 平均粒サイズの時間発展 粒構造推定(擬似データによる実験) fi 解析力学ミニ研究会 伊藤伸一 2024年9月30日 鋼構造材料の表面写真 10 /29
変分データ同化の活用例:マントル対流 秋田大学 中尾篤史先生との共同研究 Nakao et al., Geophys. J. Int., 236, 1, 379–394, (2024) 高粘度の熱対流方程式 { <latexit sha1_base64="Hs9tI3Q4/Zvja9wBLxjAytApQZ0=">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</latexit> 過 去 現在の熱構造(地震波トモグラフィーなど)と 現在の表面の流れ(GPS観測など)から 過去の内部の流れ・熱構造を推定する問題 (a)真の解の時間発展 (b)推定初期解からの時間発展 (c)(a)と(b)の差 現 在 解析力学ミニ研究会 伊藤伸一 2024年9月30日 11 /29
変分データ同化の広がり 気象・海洋 材料科学 時空間発展するモデルの最適化・逆問題 固体地球科学 ニューラルネットワーク Chen et al. (2018) 解析力学ミニ研究会 伊藤伸一 2024年9月30日 12 /29
変分データ同化法の数理構造 — 時間不変量と厳密性 — シンプレクティック解法 解析力学ミニ研究会 伊藤伸一 2024年9月30日 13 /29
Forward モデルとAdjointモデル(再喝) Forward モデル x0 = ✓ <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">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</latexit> Adjoint モデル <latexit sha1_base64="smlwvWBW+Zlr7b/UAOrbSI/01sM=">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</latexit> 0 = r✓ C, T = rxT C これらが持つ数理構造とその応用の話をします。 解析力学ミニ研究会 伊藤伸一 2024年9月30日 14 /29
Adjointモデルの厳密性の数値的再現 Adjoint モデルを解いて得られる勾配の解析解は コスト関数を直接微分して得られる勾配と厳密に一致する。 しかし実際は、Runge—Kutta 法などの数値積分法を使って Forwardモデル・Adjointモデルともに時間方向に数値積分する必要がある。 → 離散化誤差により厳密性が壊れる。 理想 Forward 解析解 勾配 解析解 Adjoint 解析解 Forward 数値解 勾配 数値解 Adjoint 数値解 現実 解析力学ミニ研究会 伊藤伸一 2024年9月30日 15 /29
Forward 数値解 現実 解析力学ミニ研究会 伊藤伸一 2024年9月30日 勾配 数値解 ⇡ <latexit 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Forward 解析解 ⇡ <latexit sha1_base64="73zbkx78eqDyLahfYjUsSx/k7e8=">AAAOiXictZfPb9RGFMcfPwo00CWBC1IvK7apeulqNiEh4gTJJSgIQSAJKCHIdmazZr22ZTubbKz8A9w4IZVTkTgg7vwDvfQf6IE/AXEEqRcOfOfZTuyw3nEPXWvtmTfv8+a9N+OZsek7dhgJ8eHEyVOnfzhz9tyPY+cv/FS7OD5xaTX0dgJLrlie4wWPTCOUju3KlciOHPnID6TRMx25ZnYXVPtaXwah7bkPo4Evn/SMbddu25YRQbS2Yfh+4O09HW+I5pSYm5mZqX9faDUF/xqU/u55E2fe0wZtkUcW7VCPJLkUoeyQQSGudWqRIB+yJxRDFqBkc7ukAxoDuwMtCQ0D0i7u26itp1IXdWUzZNpCLw7+Acg6TYp/xFvxWfwt3omP4muprZhtKF8GeJoJK/2nF59fefCvlurhGVHniNISbXiochByfJOl2plllZse7XEPYyMzEsH2HGfCRg8+S1SOrMS7/v7Lzw9uLE/Gv4rX4hOy86f4IP5Cftz+F+vNfbn86n+3Xh5thAiLmSnTNJGNLBfKngvpLo99j3PmYrbFkAeobUFTlbNcKlnM0oM0l2W0yRnPsyb8i1l6oCEHQ8mBltR7PJrNPFYZqnNNR+wXiP0KhFkgTA2xhRgS/TZoNY5lsQ2XR9q4Hxf8eazRXh06MqvaXm4N5W5pufmh3LyWW0qJYfSSlu7wihjyCprnQ+YXtfwdXIpMWBPte2mO28jxHeZHW7g7gr9bgV8awS9V4Lu4yi10K1i4PYK/XYH3uCX8bgRNntstLb8PeQBpuReighd97lXibqSe5Os60sAKnFFJWbc6RNh/8+/jBjy2K3AGys+OkQ7vlyZadL4GvK96hzFmFqrEqdhBgRtoR9bk3nweITWuyYnjyMJihT53C8RuBaJdINoViE6B6FQg+gWiryFsXNl+5UFb5SPkrCdzRmL0Iz4vNPl/tNKHqPcq7GkStWxG5d+hmJa19Bb3keyJbiXdZB70KulmEST71Cj9KJ0xxSgsjiymh5BvHstXpq+LMMRT9b5O03wuy++yMTV4ldmkKS5PpbnP5Oo5nT7z9c1Dn/3KHqjzoMN3mfNmmAeKSE7+nf/syejzmgEtn1dMn5njJwrJ2VbrmpV+g6jZGsNmndcKD7lrcwxJSxZPiN3Chk21qxos3YY82WWVvkO/wN8G7huwHKT2Ept5f5JW5b93zE6X+3ORH593/Ozt0BO/HzLZNYbvs+wjrF5eWJ1qtmabs/evNW7Op19q5+hnukq/YQSu002sYPdohU/dL+gPelU7X2vV5mo3EtWTJ1LmMhV+tYVvkY/9PA==</latexit> ⇡ <latexit sha1_base64="73zbkx78eqDyLahfYjUsSx/k7e8=">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</latexit> Adjointモデルの厳密性の数値的再現 Adjoint モデルを解いて得られる勾配の解析解は コスト関数を直接微分して得られる勾配と厳密に一致する。 しかし実際は、Runge—Kutta 法などの数値積分法を使って Forwardモデル・Adjointモデルともに時間方向に数値積分する必要がある。 → 離散化誤差により厳密性が壊れる。 理想 勾配 解析解 Adjoint 解析解 ⇡ <latexit sha1_base64="73zbkx78eqDyLahfYjUsSx/k7e8=">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</latexit> ←精度の良い積分法 Adjoint 数値解 一致することはない ↑精度を上げても 16 /29
厳密な勾配計算の重要性 勾配に誤差があると… 通常、勾配のノルムが 十分小さくなった時に最適化を止める (コスト関数の最小値は予め知り得ない) 本来の降下方向 → 誤差を含んだ勾配が小さくても 本来の勾配は小さくないかもしれない 誤差を含んだ降下方向 → 正しくない”最適値”が得られるかもしれない 確率的には収束するかもしれないが、 無駄な勾配計算が発生し、 採用した最適化法本来のパフォーマンスが発揮できない 解析力学ミニ研究会 伊藤伸一 2024年9月30日 17 /29
Adjointモデルの厳密性の数値的再現 Adjoint モデルを解いて得られる勾配の解析解は コスト関数を直接微分して得られる勾配と厳密に一致する。 しかし実際は、Runge—Kutta 法などの数値積分法を使って Forwardモデル・Adjointモデルともに時間方向に数値積分する必要がある。 → 離散化誤差により厳密性が壊れる。 理想 Forward 解析解 勾配 解析解 Adjoint 解析解 Forward 数値解 勾配 数値解 Adjoint 数値解 現実 両者の積分法をうまく設計すると↑を 厳密にイコールにできる! 解析力学ミニ研究会 伊藤伸一 2024年9月30日 18 /29
Adjoint systemの時間不変量
Q. なぜ Adjointモデルの解析解は厳密な勾配を計算できるのか?
A. 接線形モデルと Adjoint モデルの間に時間不変量が存在する。
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接線形(TL) モデル
> >⇣
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t t+ tt = (r0x f ) 0 t = 0
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Adjoint モデル
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xt ) 0 )> r xt C(xt ) = >0 (r✓ xt )> r xt C(xt ) = >0 r✓ C(xt )
*
xt (✓ + ✏ 0 ) = xt (✓) + ✏ t + O(✏ 2 )
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<latexit sha1_base64="uHi8RSX+U+ihrsaQk1DuOjM9kyw=">AAAL5XicbdZPU9NAFADwgP+QWgU9eslYDu0wMC06ysUZ/oPyXykUWOxs0k26kE1Csi2BTPwG3hyvfhw/gx/Cq15N08JL85JLd9/vvU2702ye5lrcl9Xq75HRe/cfPHw09ni88KT49NnE5PND3+l4OqvrjuV4DY36zOI2q0suLdZwPUaFZrEj7WK550dd5vncsQ/ktcvOBDVtbnCdyjjUnKBBM5QRsZghy0S2maTThLk+txybtJglaTOsRsTjZltW1Pdqkj1IrKjT6m2uepssozi6W76Nf5mrNCdK1dlqcql4UBsMSsrg2mtOjluk5egdwWypW9T3T2tVV56F1JNct1hUJB2fuVS/oCY7jYc2Fcw/C5O9iNS0hlT4/rXQomxQUNnOrCON+bOQ225HMlsfLpCB4djSj4pFYrMr3RGC2q2QUM8UNIhC0lvNcUPiCTWOfe0FicUF75WgCm7nVMTBu4pivJUG6TK9XKpV4lTNCUKiOVarV6ROlWpT0V0SxU573FcNqwaqY9VBW1hboAwrAzWwGqAmVhO0jbUNyrFy0HOs56AXWC9ALawWqMAqQG2sNqiD1QF1sbqgl1gvQT2sHqiP1QeVWCVoB2sHtIu1C3qF9Qo0wBqAXmO9Br3BepN6GBYxL0LxEtYl0GWsy6ArWFdAV7Gugq5hXQNdx7oOuoF1A/QD1g+gH7F+BN3Eugm6hXULdBvrNugO1h3QXay7oHtY90D3se6DfsL6CfQz1s+gB1gPQOtY66CHWA9Bj7AegTawNkCPsR6DnmA9ST0MN8zLOXiqqWNJMDPn9UGSeOoU4fFLLictiaeOk15DkJOWxFOnbNyYtPLy+pB61POXuxlaLch5HZAg9ULIX2RoDZFz1BGROuxMKnI3IImnbtXvd/Ju15fUvy3ZqzjHUEkyhn9pstl9ScZRtufwhGOzqPdZJm0/blBYOFubZyLiw9NKlK2TV06mrprU8eF5TmHbY4Nb3uXODIqzc1xtxI1qfnE3NX+be2eDd7O/NcnsDk9xoc+D4bqZTGHq+xbj/rSW7Ubx4HButvZ6dm7/TWlhadCpjikvlVdKWakp75QFZUPZU+qKrvxS/ih/lX8Fs/Ct8L3wo586OjKoeaEMXYWf/wFnKFAC</latexit>
t = r xt C(xt (✓)) と選べば、 0 = r✓ C(xt (✓))
<latexit sha1_base64="Un/ZtOy/PcZf2C6b3qHd6JAnbQU=">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</latexit>
t = (r✓ xt ) 0
<latexit sha1_base64="rKWJROh4xk2V1BJcmlKpQl2zse4=">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</latexit>
<latexit sha1_base64="RIOX2w7GPDwnqp/feo71pdtvvc8=">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</latexit>
勾配の厳密性=時間不変量の存在
→ 離散化した後でも時間不変量を保存するような数値積分法を選ぶのが良い
(J.M. Sanz-Serna, SIAM, 2016)
解析力学ミニ研究会 伊藤伸一 2024年9月30日
19 /29
Symplectic(シンプレクティック)解法 ハミルトン系に対して”エネルギー”を保存する時間積分法 1 2 Example: H(pt , qt ) = pt + qt2 2 <latexit sha1_base64="p+a+wauRCeWDpkOnEC8depKhvO0=">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</latexit> <latexit 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<latexit 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{ <latexit sha1_base64="Lv8vpJqMT4xZ4440fPeB4UkQhpA=">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</latexit> { <latexit 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sha1_base64="YC9yE44u2b7fZ79lZeFB4rpLSl4=">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</latexit> d @H qt = = pt dt @pt 離散化 { qn+1 = qn + hpn ① (陽的 Euler 法) pn+1 = pn hqn <latexit sha1_base64="4okFWa5Q5x/wULE8HlLbp08Rv8E=">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</latexit> <latexit sha1_base64="VacRNugzFmpkZGZyf1W95X2Fcqo=">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</latexit> d pt = dt @H = @qt qt <latexit sha1_base64="powYlVB2SEKp9KbjCaewzcoFJ0A=">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</latexit> pt 陽的 Euler Exact orbit qn+1 = qn + hpn ② (Verlet 法) pn+1 = pn hqn+1 <latexit sha1_base64="JDPO2nWIKuAD0WUC5vSRVJnX/sk=">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</latexit> <latexit sha1_base64="powYlVB2SEKp9KbjCaewzcoFJ0A=">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</latexit> pt h : 時間刻み幅 <latexit sha1_base64="9gygmdn+QQW5JH9b9JNHqm5Ff9o=">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</latexit> Exact orbit ② qt qt 解析力学ミニ研究会 伊藤伸一 2024年9月30日 <latexit sha1_base64="2edWOKWsfq2QEOi1KFuAe6SqI0o=">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</latexit> Verlet ① <latexit sha1_base64="ecUeeQfMKRNVpqTA3+1ZTjMwH5c=">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</latexit> p n = p tn pn+1 = ptn +h <latexit 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<latexit sha1_base64="4okFWa5Q5x/wULE8HlLbp08Rv8E=">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</latexit> <latexit sha1_base64="Lv8vpJqMT4xZ4440fPeB4UkQhpA=">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</latexit> <latexit sha1_base64="ONu2zsLqiyIG/o8+ND3qVGisdz8=">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</latexit> q n = q tn qn+1 = qtn +h <latexit sha1_base64="ecUeeQfMKRNVpqTA3+1ZTjMwH5c=">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</latexit> 20 /29
Runge—Kutta (RK) 法
h : 時間刻み幅
s-段 Runge—Kutta (RK) 法
<latexit sha1_base64="9gygmdn+QQW5JH9b9JNHqm5Ff9o=">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</latexit>
aij , bi : 重み係数
<latexit sha1_base64="zOx5lxzu5LRG/Eu0uRV0c9/20CY=">AAACuXichVHLSsNAFD3GV3226kZwI5aKCymTKii6Ed249FUttKUkcWpH0yQk02It/QE/QBcufIAL8TPcuHDrwk8QlwpuXHiTBkTFeofJvXPmnnvP5OqOKTzJ2FOb0t7R2dUd6ent6x8YjMaGhrc9u+IaPG3Ypu1mdM3jprB4Wgpp8ozjcq2sm3xHP1jx73eq3PWEbW3JmsPzZW3PEkVhaJKgnFaoi/3GtE6uUYjFWZIFNv47UMMgjtDW7NgDctiFDQMVlMFhQVJsQoNHKwsVDA5hedQJcykSwT1HA73ErVAWpwyN0AP67tEpG6IWnf2aXsA2qItJ2yXmOBLskd2wV3bPbtkz+/izVj2o4WupkdebXO4Uosejm+//ssrkJUpfrJaaJYqYD7QK0u4EiP8Ko8mvHp2+bi5sJOqT7Iq9kP5L9sTu6AVW9c24XucbZy306KTF/2OJPzMkDqm/HUzAgz9K9efgfgfbqaQ6k0ytz8aXlsOhRjCGCUzR5OawhFWsIU19HJzgHBfKoqIpJWW/maq0hZwRfDPF+wTu5J5R</latexit>
{
<latexit sha1_base64="Lv8vpJqMT4xZ4440fPeB4UkQhpA=">AAADTnicxVI9TxtBEH0+EjAmwUAaJBqEZURljRECRIVIk9I2GCMBQnfnNZx8X7pbm48TfyBKm6RIFSSKKD+DhpImBT8hSmmUNERhbn0IgQUp2dPtzryZNzs7M4ZvW6EkukxpfS9e9g+kBzNDr14PZ0dGx9ZDrxWYomp6thdsGHoobMsVVWlJW2z4gdAdwxY1o/k2ttfaIggtz12Th77YdvRd12pYpi4ZqmxFOyM5KpBak71CMRFySFbJG01NYAt1eDDRggMBF5JlGzpC/jZRBMFnbBsRYwFLlrILHCPD3BZ7CfbQGW3yvsvaZoK6rMcxQ8U2+Rab/4CZk8jTD/pGHTqn7/STrh+NFakYcS6HfBpdrvB3su/HV//8l+XwKbF3x3oyZ4kGFlWuFufuKyR+hdnlt48+d1aXKvlomk7oF+f/lS7pjF/gtq/M07KofHmm6BHXxVH9yD/qIXHA93uqv2HSO5ct+6ovjqqUy5MQMR6wVmefWL6tYIxFCo25GZ604sO56hXWZwvF+QKV53LLK8nMpTGBKczwXC1gGe9QQpUzaOADPuKTdqH91v5q/7quWirhvMG91Ze+AR+lwrY=</latexit>
離散化
古典的4段4次RK法
Butcher 配列
陽的Euler法(1段1次) Runge—Kutta-Fehlberg 法 (6段4次5次)
4次公式
5次公式
Runge—Kutta法のメリット
・高次公式はそれなりに安定性が良い
・低次と高次を効率的に同時に計算できる公式が存在する。
→ 精度を監視して適応的に時間刻みを変化させることができる。
解析力学ミニ研究会 伊藤伸一 2024年9月30日
21 /29
時間不変量を保存するRunge—Kutta法の設計 ・Forwardモデル,TLモデルについては、とあるs段RK法を適用する。 n+1 = <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">AAALv3icbdZbb5swFABg2t26Ztna7XEv0dKH9KFR0k1bpWlS7/d7m15dRQYMcYOBgpOQIPZf9mv2uj3u34zQqIdw4CX2+c4xwQLbqmtxX9Zq/yYmnz1/8fLV1Ovpwpvi23czs+8vfKfjaayhOZbjXanUZxa3WUNyabEr12NUqBa7VNtrQ7/sMs/njn0u+y67E9S0ucE1KuNQc+Y7MTyqhXoU6jIiOrMkbYYyKv0oEYsZskJsqlpxKIgM4nGzJechqTlTrlVryVXCjfqoUVZG13FzdtoiuqN1BLOlZlHfv63XXHkXUk9yzWJRkXR85lKtTU12GzdtKph/FyZPGZXSGlLh+32hRtmgoLKVGUcaS3cht92OZLY2XiADw7GlHxWLxGY9zRGC2npIqGcKGkQhGY7muCHxRCmO/RwGicUFH5agCm7nVMTBp4piPHUG6TKtUq7Px6mqE4REdSx9WFSaK9fnoqckip0O+VFVrCqohlUD1bHqoAwrAzWwGqAmVhO0hbUFyrFy0Hus96BtrG1QC6sFKrAKUBurDepgdUBdrC7oA9YHUA+rB+pj9UElVgnawdoB7WLtgvaw9kADrAFoH2sfdIB1kPoYVjCvQPEq1lXQNaxroOtY10E3sG6AbmLdBN3CugW6jXUbdAfrDugu1l3QPax7oPtY90EPsB6AHmI9BD3CegR6jPUY9ATrCegp1lPQM6xnoOdYz0EbWBugF1gvQC+xXoJeYb0CvcZ6DXqD9Sb1MQyYl7Pw1FLLkmBmzvZBknhqFeHxJpeTlsRTy0mLyby0JJ5aZeMjh56X9wipTz1/uMHYaEHOdkCC1IaQP8jYGCJnqSMitdiZVOROQBJP3cr1ueXk7ARkJKm3LZmrOMcokaQNb2ky2Y+StKPsmcMTjs2i4W+FtPz4gMLCan2JiYiPd+ejbJ3sOZm6WlLHx/s5hS2PjW75lLswKs72cbURH0Hzi7up/tfcOxu8m33WJLM73sWFPg/G6xYyhan/W4zPp/XsaRQ3Lhar9c/VxZMv5eXV0Ul1SvmofFIqSl35piwr28qx0lA05ZfyW/mj/C2sFMyCXXAfUycnRjUflLGr0P8PqrVCbw==</latexit> n+h s X bi dn,i , i=1 d t = (r x f ) t dt <latexit sha1_base64="9iBL9qrnU1tN47ptGOOSw/na9R4=">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</latexit> dn,i = r x f (Xn,i )Dn,i (i = 1, . . . , s) , s X Dn,i = n + h ai j dn, j (i = 1, . . . , s) . j=1 ・Adjointモデルに対しては異なる係数を持つs段RK法を適用する。 は ただし、2つの重み係数の組 <latexit sha1_base64="56Y7u9cyHM3ItBECtCsjHh8yGmo=">AAALw3icbdbLUtswFAZgQ2+UNC20y24yDQuYDpmYdlo2nXK/X8IlJIBSRnbkRMSyja0Eg8d9mz5Nt+2ib1PHZDi2j72JpO/8cuKJJWmOyT1Zrf4bG3/y9NnzFxMvJwuviq/fTE2/PfPsvquzum6bttvUqMdMbrG65NJkTcdlVGgma2i91aE3Bsz1uG2dyjuHtQTtWNzgOpXR0NXUd9JmpqRXgfVRDX8ERNpOSMwo3x6Nlb6VHktwQXg1Va5WqvFVwg111Cgro6t2NT1pkrat9wWzpG5Sz7tUq45sBdSVXDdZWCR9jzlU79EOu4yaFhXMawXxLw1LSQ2o8Lw7oYXZQUFlNzOPNBZbAbecvmSWng5I37At6YXFIrHYrW4LQa12QKjbEdQPAzKczXYC4opSNPZzOEhMLvgwghLcyklEg4+JYvQwDTJg+mxZnYtKNdsPiGab7WGoNFNWZ8LHIoqdDvlBNawaqI5VB21jbYMyrAzUwGqAdrB2QLtYu6AcKwe9xnoN2sPaAzWxmqACqwC1sFqgNlYb1MHqgN5gvQF1sbqgHlYPVGKVoH2sfdAB1gHoLdZbUB+rD3qH9Q70Hut94mVYxrwM4RWsK6CrWFdB17Cuga5jXQfdwLoBuol1E3QL6xboNtZt0B2sO6C7WHdB97Duge5j3Qc9wHoAeoj1ELSGtQZ6hPUI9BjrMegJ1hPQU6ynoHWsddAzrGegDawN0CbWJug51nPQC6wXiZfhnrk5C081sSwJ1snZPkg8nlhFeLTJ5ZTF44nlpMtkXlk8nlhl41NBTt0DJF71/OnuU7P5OdsB8RMbQv4kqTlEzlJHRGKx61CR+wDi8cStHI+bds5OQEaS+LfFzyqqMUokbsO/NH7YDxK3w+yZwxW2xcLh5yzpetEBhQUVdZGJkKe7c2E2J2/tTK4a53i6nxPsumx0y8fa+VE428dpIzqG5ocHif6X3DsbfJD9rXHlIN3FQY/76dx8Jpj4vsXofKpmT6O4cbZQUT9VFo4+l5dWRifVCeW98kGZVVTlq7KkbCk1pa7oyi/lt/JH+VtYL/QKbkE+lI6PjTLvlNRVCP8DHDxDzA==</latexit> > n+1 n+1 = 解析力学ミニ研究会 伊藤伸一 2024年9月30日 > n n が満たされるように選ぶ。 22 /29
Symplectic partitioned RK (SPRK)法 > n+1 n+1 = <latexit sha1_base64="56Y7u9cyHM3ItBECtCsjHh8yGmo=">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</latexit> bi = Bi > n n (i = 1, . . . , s) , bi Ai j + B j a ji = Bi b j (i, j = 1, . . . , s) . <latexit sha1_base64="uQosLXqtBeyP+HxIWi313hFwsSg=">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</latexit> TLのRK 証明 n+1 = n+h s X bi dn,i , i=1 S ( t, t) = <latexit sha1_base64="XmSl2xLXaWwqKtiG7O7vgQ/nH08=">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</latexit> <latexit sha1_base64="ewM5nHiBj4l+WrVe3LplPT2PlN8=">AAALu3icbdbJUtswGAdwQzdKmhbaYy+ZhkOYlkxMO4VLZ9j3HbIAymRkR04Elm1sZSEe9036NL22D9C3qeNk+Bx/9iXS99NfTjyxJM0xuSdLpX9T08+ev3j5aub1bOZN9u27ufn3Fc/uuDor67ZpuzWNeszkFitLLk1Wc1xGhWayqna/OfRql7ket60r+eiwuqAtixtcpzIsNeZWLgukyUxJG771WQ2+EDPMNse9xaWYxi1YbMzlS8VSdOVwQx038sr4OmvMz5qkaesdwSypm9TzbtWSI+s+dSXXTRZkScdjDtXvaYvdhk2LCubV/egXBrm4+lR43qPQgmRRUNlOzCON1brPLacjmaVPBmTfsC3pBdkssVhPt4WgVtMn1G0J2g98MpzNdnziilxY+zksEpMLPoygBLdSEmHxKZENn6NBukwv5NXFcKhm932i2WZzGMot5NWF4GkQxU6HPFINqwaqY9VBm1iboAwrAzWwGqAtrC3QNtY2KMfKQe+w3oHeY70HNbGaoAKrALWwWqA2VhvUweqAPmB9AHWxuqAeVg9UYpWgHawd0C7WLmgPaw+0j7UP+oj1EXSAdRB7GdYxr0N4A+sG6CbWTdAtrFug21i3QXew7oDuYt0F3cO6B7qPdR/0AOsB6CHWQ9AjrEegx1iPQU+wnoCeYj0FPcN6BnqO9Rz0AusF6CXWS9ArrFegZaxl0ArWCmgVaxW0hrUGeo31GvQG603sZRgwN2XhKcWWJcFaKdsHieqxVYSHm1zKsKgeW07aTKYNi+qxVTY6FqSMG0HsVU+fbjAxWz9lOyD92IaQPsnEHCJlqSMitti1qEh9AFE9divH46adshOQscT+bdGzCscYORK14V8aPeyRRO0geeZwhW2xYPhZIG0vPKAwv6iuMhHwye5ikMzJnp3IlaIcn+ynBNsuG9/yaezSOJzs47QRHj/Tw91Y/3vqnQ3eTf7WaGR3souDHu9P5pYSwdj3zYbnUzV5GsWNynJR/VpcPv+WX9sYn1RnlI/KJ6WgqMqKsqbsKWdKWdGVX8pv5Y/yN/Mjo2fuMuZo6PTUOPNBmbgynf8v5T7C</latexit> > t t と置くと、 S ( n+1 , n+1 ) =h <latexit sha1_base64="9MyBerPi8WEfLAfKxmB7LM1wbI8=">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</latexit> s X S ( n, n) Bi S ( n , ln,i ) + h i=1 s X s X i=1 bi S (dn,i , n ) + h2 s X i, j=1 s X <latexit 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Bi b j S (dn,i , ln, j ) s X <latexit 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=h <latexit 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s X i=1 i=1 (bi i, j=1 ⇣ > ⌘ 2 Bi ) S Dn,i , r x f (Xn,i ) ⇤n,i + h =0 <latexit 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解析力学ミニ研究会 伊藤伸一 2024年9月30日 s ⇣ X i, j=1 j=1 AdjointのRK s s X X h2 bi Ai j S (dn,i , ln, j ) + h2 Bi b j S (dn,i , ln, j ) Bi S (Dn,i , ln,i ) h2 Bi ai j S (dn, j , ln,i ) + h bi S (dn,i , ⇤n,i ) i=1 i, j=1 i=1 i, j=1 s s s X X X Bi ai j S (dn, j , ln,i ) + h bi S (dn,i , ⇤n,i ) h2 bi Ai j S (dn,i , ln, j ) + h2 Bi b j S (dn,i , ln, j ) =h dn,i = r x f (Xn,i )Dn,i (i = 1, s X Dn,i = n + h ai j dn, j (i = i, j=1 i, j=1 Bi b j bi Ai j =0 <latexit 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⌘ B j a ji S (dn,i , ln, j ) 23 /29
SPRK法による厳密勾配の計算 ・Forwardモデル、TLモデルについてはとあるs段RK法を適用する。 n+1 = <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> n+h s X bi dn,i , i=1 d t = (r x f ) t dt <latexit sha1_base64="9iBL9qrnU1tN47ptGOOSw/na9R4=">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</latexit> dn,i = r x f (Xn,i )Dn,i (i = 1, . . . , s) , s X Dn,i = n + h ai j dn, j (i = 1, . . . , s) . j=1 ・Adjointモデルに対しては異なる係数を持つs段RK法を適用する。 bi = Bi (i = 1, . . . , s) , bi Ai j + B j a ji = Bi b j (i, j = 1, . . . , s) . <latexit sha1_base64="uQosLXqtBeyP+HxIWi313hFwsSg=">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</latexit> Adjoint数値解 = コスト関数にForward数値解を代入,微分して得る勾配 解析力学ミニ研究会 伊藤伸一 2024年9月30日 24 /29
簡単な例:ロジスティック方程式 ・簡単な具体例を使って、時間不変量を考慮することの威力を見る。 ・ロジスティック方程式 <latexit 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{ <latexit 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⇣ 生物の個体数変化などを表す数理モデル ⌘ x d t 厳密解 xt = rxt 1 Kert dt K xt = rt e + K/✓ x0 = ✓ r, K :既知の定数とする <latexit 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<latexit sha1_base64="jE+fZ3CPQLhK/ZBx+Hab7JPdsWU=">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</latexit> <latexit sha1_base64="XtWADnMbGUXitAx1pYo1TFyWGSE=">AAANinictZdLb9NAEICHAqWUhrZwQeISEYoQiLBJKU8htYUDVaqq9EnVl2x305g6dmRv06ZW/gAnbhw4gcQBcecKEhf+AIf+BMQRJC4cmB3bbQxx1hywZe/ueL7Zmdn1rq3XLNMTjO0f6Tp67Hj3iZ6Tvaf6Mqf7BwbPLHjOtmvwecOxHPeJrnncMm0+L0xh8Sc1l2tV3eKL+tYD+Xyxzl3PdOw50ajx1aq2aZtl09AEitYHruyu+6KZvZ9dKbua4Zf4mu+KZtMPyqul6yuiwoWWvVZorg/kWL7Ibo+MjGT/rhTyjI4chMe0M9h9EVZgAxwwYBuqwMEGgXULNPDwXIYCMKihbBV8lLlYM+k5hyb0IruNWhw1NJRu4X0TW8uh1Ma2tOkRbWAvFl4uklkYYl/YW/adfWbv2Ff2K9GWTzakLw0s9YDltfX+Z+dmfyqpKpYCKoeUkiijhzIHHsU3lKgdWZa5qcIu9dDbMSMCbd+mTJjYQ40kMkdG4F1978X32bszQ/4l9pp9w+y8YvvsE+bHrv8w3jzmMy//u/XkaAVGGM9MkqaO2YhyIe3ZKN2hsa9SzmycbT7KXWxtoKasR7mUMp+kzTCXSbROGW9ldfTPJ2lTQTbakg0lqfa4Mxt5LDOUpZaK2IsReykIPUboCmIDYwj0y0jLcUyKrb1cKONeivmzpNBeaDsyC8pextpyY0puvC03ruRKIdGOLinpCq2IHq2grbxH/CMlP4mnJANWx+e7YY7LmONJ4jtbmOrAT6XgSx34Ugp+C89kC1spLEx04CdS8A498f4aQZ3mdkHJ76HcRWmyFyyFF3XqleNdCz1pbatIDVfgiArqqtVB4P7b+j6uoMdmCk7D+tM/SIv2Sx2fqHx1aV91DmKMLKSJU7KNGNdQjqxOvdVohOS4Bl8chxYepehzJ0bspCDKMaKcgqjEiEoKoh4j6grCxDParxzUlvnwKOvBnOE4+oK+F/J0Ha70HrarKfY0jq1oRrW+Qz7MKOkN6iPYE+1UusE8qKbSjSII9qlO+iKcMfEoDIrMhzmUr/2Rr0hfFaGHpex9GYbpu6x1l/UhR6vMGhSpXgxzH8llORyWre21A59rqT2Q34MW3XmLN+08kETw5V/5Z0968d8j+sHIJlcWivnCzfzI4xu50fHwL6QHzsMFuIzWb8Eovp3TMI+xPIf38AE+ZvoyxcydzL1AtetIyJyF2JF5+BuRYtIM</latexit> 1 rt <latexit sha1_base64="XtWADnMbGUXitAx1pYo1TFyWGSE=">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</latexit> ・問題設定 時刻 t = T の時に個体数 y が観測できたとして、 <latexit sha1_base64="ZktdUSrt6i/6WS43YrRLImiPCOc=">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</latexit> <latexit sha1_base64="nH0k4fTQVLex0pXms/x4cNj6PDY=">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</latexit> Ke xt = rt e + K/✓ r, K <latexit sha1_base64="jE+fZ3CPQLhK/ZBx+Hab7JPdsWU=">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</latexit> 時刻 t = 0 での個体数 ✓ を推定する問題を考える <latexit sha1_base64="m4Q/RACRWLpESSiBxhZMf0PwogQ=">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</latexit> <latexit 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1 C = (y 2 y <latexit 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<latexit 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コスト関数 xT ) 2 @C 勾配 @✓ をAdjointモデルの数値計算で求める。 <latexit 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解析力学ミニ研究会 伊藤伸一 2024年9月30日 ✓ <latexit 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t=0 <latexit 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<latexit sha1_base64="y/Plp0ktl7rB/m241nYI03RnXO8=">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</latexit> T 25 /29
簡単な例:ロジスティック方程式 Forwardモデル ✓ dxt = rxt 1 dt <latexit sha1_base64="FDXjdxmDdUYgdFuaXALHrHfC1lY=">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</latexit> x◆ /h <latexit sha1_base64="ot+D/1Asf0a7tEkEGW4fY6SsUts=">AAAOi3ictZfNbtRIEMcLWNhsFoavC9JeRgxBXBh1EhIQ4gBBK4GCEAQSiBKCbKcnY8Zfsp0JEysvwHUPewAOu9IeEPd9gb3wAnvgEVYcQeLCgX+X7cQO42lz2LHG7q6uX3VVdbu7bQaOHcVCvD9w8NAPh4/8OPbT+M9HjzWOnzh5ainyN0NLLlq+44ePTSOSju3JxdiOHfk4CKXhmo58ZPZuqvZHfRlGtu89jAeBfOIaG57dsS0jhmh5NQj9IPab3acnWqI9Ja7MzMw0vy1MtgX/WpT97vknj/xNq7ROPlm0SS5J8ihG2SGDIlwrNEmCAsieUAJZiJLN7ZJ2aBzsJrQkNAxIe7hvoLaSST3Ulc2IaQu9OPiHIJs0If4Vb8RH8U68Ff+JL5W2ErahfBngaaasDJ4ef3HmwWct5eIZU3eP0hIdeKhyEHF8E5XauWWVG5eecw/jIzMSw/YVzoSNHgKWqBxZqXf97d8/Pri6MJGcF3+KD8jOH+K9+Af58fqfrL/uy4WX/7v16mhjRFjOTJWmiWzkuVD2PEi3eOxdzpmH2ZZAHqK2Dk1VznOpZAlLd7JcVtEmZ7zImvAvYemOhhwMJQdaUu/xaDb3WGWoyTUdsV0itmsQZokwNcQ6Ykj1O6DVOFbFNlwea+NeLvmzrNFeGjoyS9pebgzlbmi5uaHcnJabz4hh9LyW7vKKGPEKWuQj5m9p+Tu4FJmyJtqfZznuIMd3mB9t4e4I/m4Nfn4EP1+D7+GqttCrYeH2CP52Dd7nluibETR5bk9q+W3IQ0irvRA1vOhzrxJ3I/OkWNeRBlbgnErLutUhxv5bfB9X4bFdgzNQfraPdHi/NNGi8zXkfdXfjTG3UCdOxQ5K3EA7sib3FvAIqXFNTxx7Fm7V6HOrRGzVIDololOD6JaIbg2iXyL6GsLGle9XPrRVPiLOejpnJEY/5vNCm/97K32EultjT5Oo5TOq+A4ltKCl17mPdE/0aumm88CtpZtHkO5To/TjbMaUo7A4soQeQr62L1+5vi7CCE/V+wpN87msuMsm1OJVZo2muDyV5T6Xq+d09izW13Z9Dmp7oM6DDt9lwZthHigiPfl3v9uT0ec1A1oBr5gBM/tPFJKzrdY1K/sGUbM1gc0mrxU+ctfhGNKWPJ4Iu4UNm2pXNVi6AXm6yyp9h87B3xbuq7AcZvZSm0V/0lblv7/PTo/785CfgHf8/O3QExd3mfwax/dZ/hHWrC4sTbUnZ9uz9y+1rs9lX2pj9AudpQsYgct0HSvYPVpE/y79Rq/odeNYY7pxtXEtVT14IGNOU+nX+PUrCU394g==</latexit> t K Euler法 ✓ <latexit sha1_base64="D+wvfsjJPUVoHkglJaJ8ZWjm7t0=">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</latexit> xn+1 = xn + hrxn 1 Adjointモデル <latexit sha1_base64="qY52YJQfcd1QBMoAjMF5/PQdTpM=">AAAC1nichVG/axRBFP6yGo3nj5zaCDZLjkgscrw7cjEIQjCNYpMfXnKQDcfu3txlyNzuMjt3JC5rJ4FU6SysFCzEwj/AMo3/QIo0WotlBBuLvNs9UZHEN8zMm++9780387xIydgQHY1Y586PXrg4dqlw+crVa+PF6zdW47CnfVH3QxXqhufGQslA1I00SjQiLdyup8Sat7UwiK/1hY5lGDw1O5HY6LqdQLal7xqGmsXH005bu37SchSTWm4zMWmatExqP7C1o0TbTNmVaTtPqm7n4Sepo2Vn09z9g1RoFktUrtJcrVaz/3UqZcqshKEthsWPcNBCCB89dCEQwLCv4CLmsY4KCBFjG0gY0+zJLC6QosDcHmcJznAZ3eK1w6f1IRrweVAzztg+36J4ambamKRDekfH9Ine01f6eWqtJKsx0LLDu5dzRdQc37u18uO/rC7vBpu/WWdqNmhjLtMqWXuUIYNX+Dm//+zl8cr95cnkDr2hb6z/NR3RAb8g6H/33y6J5Vdn6DHY5uph9r8xUm7Ur27Ypzur1XJltjy7NFOafzhs2RhuYwJT3Jd7mMcjLKLO93zAIT7ji9WwnlsvrN081RoZcm7iL7P2TwDr47Cp</latexit> d t =r 1 dt 2xt K ! <latexit sha1_base64="tAY73VOS3HYF5Qbib8fZBqKsY1M=">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</latexit> n 1 = <latexit sha1_base64="WR15xH3gMkxr1XZmwvy1SMXuECQ=">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</latexit> n + hr SPRK法 ② n 1 = n Forward数値解代入・微分 K で得られる勾配との相対誤差 t Euler法 ① 離散化 ◆ x 相対誤差 離散化 n + hr 1 1 2xn K ! 2xn 1 K r=1,K=1,T=5,θTrue=0.1 ,θ=0.01 解析力学ミニ研究会 伊藤伸一 2024年9月30日 ① Euler法 ② SPRK法 n ! n 時間刻み幅 h 26 /29
Adjointモデルの厳密性の数値的再現 ・ForwardとAdjointの数値積分法を適切に構成することで厳密性を再現できる。 Forwardの積分法に対応したAdjointの積分法を機械的に作れる。 Forward <latexit sha1_base64="3fO7aisTMCaQyJPtfbvSetXq8Kw=">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</latexit> > t t = const. 厳密勾配を 実現する積分法 … Runge—Kutta 線形多段法 Symplectic Adjoint ・勾配(1階微分)の厳密計算 J.M. Sanz-Serna, SIAM, 2016 ・ヘッセ行列(2階微分)の厳密計算 S. Ito, T. Matsuda, and Y. Miyatake, BIT Numerical Mathematics, 2021 解析力学ミニ研究会 伊藤伸一 2024年9月30日 27 /29
Adjoint system の時間不変量たち 1. > t t = const. <latexit sha1_base64="3fO7aisTMCaQyJPtfbvSetXq8Kw=">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</latexit> d ⇣ dt <latexit sha1_base64="ZdJ1nHvI1QrWP2Fww7mLmtHNAys=">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</latexit> 2. > t t ⌘ 厳密な勾配計算 d = t dt !> t + > t f (xt ) = const. <latexit 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Forward model > d t dt t = ((r x f ) t ) > t + > t ⇣ > t (r x f ) f (xt ) + = 1変数系の場合 TL model 0 f (x0 ) = t ⌘> f (xt ) + d t = (r x f ) t dt d > xt t (r x f ) dtAdjoint model > t (r x f ) f (xt ) = 0 <latexit 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<latexit 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=0 <latexit 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⌘ ⇣ d ⇣ > d > > d (r ) f (x ) + f (x ) = f (x ) = f t t t t x t t dt dt dt !> ⇣ ⌘> d > > d > f (xt ) + t f (xt ) = (r x f ) t f (xt ) + t (r x f ) xt t dt dt <latexit 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t ⌘ ??? !> <latexit 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(r x f ) > T f (xT ) n. 他にもあるかも..? 解析力学ミニ研究会 伊藤伸一 2024年9月30日 f (xT ) ←途中の xt に依存しない。 0 = T f (x0 ) 始点と終点の情報だけで 勾配が求まる。 28 /29
まとめと open problems まとめ ・変分データ同化法における時間不変量と勾配計算の厳密性について紹介した。 ・時間不変量を保存するように数値積分法を設計することで、 数値計算レベルでも勾配の厳密性を担保できる。 Open problems ・時間不変量は何を意味しているのか? ・背後にどんな対称性があるか? ・保存されることに実用性はあるか? 解析力学ミニ研究会 伊藤伸一 2024年9月30日 29 /29
ありがとうございました。 解析力学ミニ研究会 伊藤伸一 2024年9月30日 30 /29