220112_高次元データ同化とその発展

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November 21, 23

スライド概要

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1.

高次元データ同化とその発展 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 c 伊藤伸一 01/43

2.

自己紹介 学歴・職歴 2015.03 大阪大学 大学院理学研究科 博士(理学) ・破壊のパターン形成の数理 2015.04—2018.08 東京大学 地震研究所 特任研究員 2018.08—現在 東京大学 地震研究所 助教 2018.12—現在 東京大学 大学院情報理工学系研究科 助教(兼務) ・大規模4次元変分法データ同化の数理・応用 ・金属材料の粒成長パターン ・断層摩擦パラメータの空間パターン 2022.01.12 波多野研セミナー c 伊藤伸一 02/43

3.
[beta]
データ同化
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)

逆問題、…
天気予報

出典:Google Earth

2022.01.12 波多野研セミナー

<latexit sha1_base64="uqBeDhpsi6lxGlvwkHHdVQpc2sc=">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</latexit>

実験・観測デザインの
最適化

地震学

Kano et al. (2015)

材料科学

Ito et al. (2017)

c 伊藤伸一

03/43

4.
[beta]
データ同化
モデル
<latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit>

予測の高精度化、

事後分布

データ

p(xt | D)

D = {D1 , D2 , ...}

<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、…

最適化

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
・小さいモデルを使う?

→ 表現能力の低下

・評価回数を極力減らす?

→ 事後分布(or その統計量)の精度低下

→ モデルの規模・目的に適した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
2022.01.12 波多野研セミナー

c 伊藤伸一

04/43

5.
[beta]
本日の話題
アルゴリズムの話題を中心にお話しします。
・物理っぽい話はあまり出てきません。。(数学的には面白い)

・4次元変分法データ同化アルゴリズム
・4次元変分法における不確実性評価アルゴリズム

<latexit sha1_base64="GpdAVcHh9JDzrelSbAp+4vMUz30=">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</latexit>

p(✓)

Ito et al., Phys. Rev. E, 94, 043307 (2016)
Ito et al., STAM,18:1, 857-869 (2017)
・不確実性を厳密に評価するアルゴリズム

✓ˆ
<latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit>

Ito et al.,BIT numerical mathematics (2021)
Ito et al., arXiv:2109.13143

2022.01.12 波多野研セミナー

c 伊藤伸一

05/43

6.

4次元変分法データ同化 c 伊藤伸一 06/43

7.

データ同化アルゴリズムあれこれ 逐次データ同化 <latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit> 非逐次データ同化 p(xt | D) <latexit sha1_base64="YZ41l0LDB4d7yMkbgRl7814ENRI=">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</latexit> <latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit> p(x0 | D) p(xt | D) ・概形と時間発展を評価 事後分布をサンプル群の時間発展で評価 ・小中規模モデル向き たくさんのシミュレーションを 同時に走らせる必要がある(並列化可) ・EnKF, 粒子フィルタ など 2022.01.12 波多野研セミナー x0 <latexit sha1_base64="Jx+cW+SJxE+Pa7o4z0mLl2CNdgI=">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</latexit> ・概形も時間発展も評価しない 初期値の事後分布のMAP解だけを探索 ・大規模モデル向き シミュレーション1つに 計算資源を集中できるため ・4次元変分法 c 伊藤伸一 07/43

8.

次元の呪い 偏微分方程式 <latexit 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@y = r (cry) @t @c =0 @t (y1 , c1 ) (y2 , c2 ) 常微分方程式 <latexit 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... <latexit 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<latexit 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L <latexit 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空間方向 離散化 <latexit 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dx = F (x) dt x = (y1 , y2 , . . . , c1 , c2 , . . . ) > <latexit 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↑ 格子点上の物理量を全て並べたベクトル L 2 L 3 <latexit 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<latexit 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空間1方向に100格子点とると, 2次元問題 100x100 格子点 = O(10,000) 変数 ! 3次元問題 100x100x100 格子点 = O(1,000,000) 変数 !! <latexit 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p(x | D) の直接評価は 実質的に不可能 <latexit 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事後分布を欲しい情報を 効率よく評価するアルゴリズム 2022.01.12 波多野研セミナー c 伊藤伸一 08/43

9.

4次元変分法 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 Dデータと + !t t = h(xt )の関係 Dt = h(xt ) + !t シミュレーションモデル <latexit sha1_base64="yXpqNkT0w6ffWorbjXV9yQ9bbtU=">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</latexit> <latexit 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p(✓):事前分布 <latexit sha1_base64="WGN2PBPUYctwd9zIQsrpAM3Un4c=">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</latexit> ノイズ <latexit 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sha1_base64="e075dcW3ikGHqSQBkD9Kp8T8K60=">AAAOqniclZc9b9NAGMeflrcSIG1hQWKxSFs1Eo0uBUHF1JKqare+N7QuVexcEqtO7NpOyIvCB0DsDEwgMSA2BhbYWPgCDP0IiLFILAw8vpgkTS7JE1e1z889v///ucs5OWu2abgeY6cjoxcuXrp8Zexq6Nr1G+Hxicmbu65VdHS+o1um5SS1lMtNo8B3PMMzedJ2eCqvmXxPO074/Xsl7riGVdj2KjY/zKeyBSNj6CkPQ0cTM/as6uW4l1LzRlpZjiqq7Vi2Zyn8WW0uEfRF60cTERZj4lC6G/GgEYHgWLcmL38GFdJggQ5FyAOHAnjYNiEFLv4dQBwY2Bg7hBrGHGwZop9DHULIFjGLY0YKo8d4zuLdQRAt4L2v6QpaRxcT/x0kFZhmP9gHdsa+s4/sJ/vbU6smNPxaKnjVGiy3j8Zf3t76M5BS8JzCmnMtrm/VHmRgQVRrYPW2iPjj0Bt8qfr6bOvx5nRthr1jv3AEb9kp+4ZjKJR+6+83+OYbVJ/uWZMHZdS3xBy75yryaypgznMxS3lRdQE/lxrG/VnPikgZGT/yf0wWOvj3DkaUIO9FM1PF2TbwzgjcKB7+qGkejcxuj5BYTxzHqUIJrzrMQgRXUTRQ1VCxLNqaWA/pppMCUyJzCnXqHTplEl2WshUSW5GyVRJblbJFEluUsg6JdaSsRmI1KWuTWFvKnpDYEynrklhXyuZIbE7KrpLYVSmbILEJKbtMYpel7AqJXZGyayR2Tcruk9h9KZsksUkpu0Ril3o8vxyfEoukwKQKlvgFy2IeRUNty5evcyP4LqeptfJlan4exzNVrZUvUzPFb6aGEapeO9Fr9unVVfvUVsZ5oKn4mTKFYSrpXUee+K2uikyZQlbMGX0FtPLlo7LFGjHFroI6unZGprrdtq4aOhlk1bZ4N5MmuaelflttT0W7Xyteb+4teu1g/P2JJXrrzftZvOZwrI19mL8Di+H+YgFbecwy+vZGB+yYfH0Pe6wBfqzNT+bIhnTM4ZV3jLJbd67DeVA/xdvfuRbF7p3uXOrR/3CIMWfQpTTwc21pdnsO6+g/HeW+fnMDHOXzG8L3s3jn21h3Y3c+Fr8fm994EFl8ErypjcEduIt1xOERLOJ+ZB120OkVfIIv8DV8L7wZfho+aKSOjgTMLTh3hNP/ANbl42M=</latexit> コスト関数 C(✓) = <latexit sha1_base64="dGQ7FytXyMRO7Mw9F/722V4PPcI=">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</latexit> log p(✓) p(✓ | D) / e X C(✓) log q(Dt h(xt )) t2T obs 2022.01.12 波多野研セミナー c 伊藤伸一 09/43

10.

4次元変分法による初期値更新 x0 = ✓ C(✓) = <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">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</latexit> X t2T obs <latexit sha1_base64="ZS3Z0hDusosxrFqeLGFiyaBbqyI=">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</latexit> :データ ✓guess <latexit sha1_base64="YnLegehdSrKrSmzbu3/KDfP7jUU=">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</latexit> Time 適当に選んだ x0 <latexit sha1_base64="AqZRKhAp+ail3X3ypq29XDB6qtc=">AAACy3ichVFNSxtRFD0Zbf3oh6luBF2EhpSuwo0KSkEIunFTSExjBCPDzPQZByczw7yXEDNmI7jxD7hwZUFo6ar+BTcu3HbhTxCXKXTTRe9MBkor2jvMvPvOu+fe8+aYvmNLRXST0oaGnzwdGR0bf/b8xcuJ9KvJDem1AktULc/xgk3TkMKxXVFVtnLEph8Io2k6omburUbntbYIpO25H9S+L7abRsO1d2zLUAzp6dlMRw+pt1xXu0IZelhXoqPCRktI2evp6SzlKY7M/aSQJFkkUfLS16jjIzxYaKEJAReKcwcGJD9bKIDgM7aNkLGAMzs+F+hhnLktrhJcYTC6x98G77YS1OV91FPGbIunOPwGzMwgR9/pC/Xpir7SLf16sFcY94i07PNqDrjC1yeOpys//8tq8qqw+4f1qGaFHSzFWm3W7sdIdAtrwG93T/qVd+u58A19ojvWf0Y3dMk3cNs/rPOyWD99RI/JWqI/lnuwQqHD873YAYnIysK/xt1PNubyhfn8XHkhW1xJTB3FDF7jLTu3iCLWUEKV5xziM77hQnuvSa2rHQxKtVTCmcJfoR39BlCQpa4=</latexit> 2022.01.12 波多野研セミナー = ✓guess ではデータと全く合わないが,,, c 伊藤伸一 10/43

11.

4次元変分法による初期値更新 x0 = ✓ C(✓) = <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">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</latexit> X t2T obs <latexit sha1_base64="ZS3Z0hDusosxrFqeLGFiyaBbqyI=">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</latexit> :データ ✓ˆ <latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit> ✓guess <latexit 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Time 勾配法による最適化により データに適合する初期値 X C(✓) の勾配 = <latexit 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2022.01.12 波多野研セミナー x0 = ✓ˆ を探す <latexit sha1_base64="Lo0LEFFzTOuOLCRfu+MtakxyflQ=">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</latexit> の計算が必要 t2T obs c 伊藤伸一 11/43

12.

Adjoint法による勾配計算 勾配法によって最適化したいが,,, C(✓) = log p(✓) <latexit 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Adjoint法 L=C+ <latexit 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Z T X logq(D q(Dtt h(x h(xtt)) )) log t2T obs ✓ で直接微分ができない ✓ ◆ d T :終端時刻 > dt t f (xt ) xt t:Adjoint変数 dt <latexit 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<latexit sha1_base64="jRRHIl4aQaDCLy6WjaPD1Ye14kE=">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</latexit> <latexit 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0 <latexit sha1_base64="MTTozS6+KSrcznvpKqdNQgdB5ko=">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</latexit> (ラグランジュの未定乗数) 変分をとって係数比較 Adjoint モデル d dt t = (r x f ) > t <latexit sha1_base64="1kqcoTy2iTY3iSE4F+wj0lrXbFY=">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</latexit> + X t0 2T obs (t 0 t )r xt0 C <latexit sha1_base64="h5FlaEbyb5cnhAtKyCqna4vs1GU=">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</latexit> T =0 t=Tから時間後ろ向きに解くことで、目的の勾配が得られる 2022.01.12 波多野研セミナー c 伊藤伸一 12/43

13.
[beta]
時間後ろ向きに計算する
Adjoint モデル

d
dt

t

= (r x f )>

t +

X

t0 2T obs

<latexit sha1_base64="1kqcoTy2iTY3iSE4F+wj0lrXbFY=">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</latexit>

(t

t0 )r xt0 C
<latexit sha1_base64="h5FlaEbyb5cnhAtKyCqna4vs1GU=">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</latexit>

T

=0

t
<latexit sha1_base64="RWXuDuJHXJvlN8xJDO9R8AqtrUs=">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</latexit>
<latexit

Input

✓
<latexit sha1_base64="jRRHIl4aQaDCLy6WjaPD1Ye14kE=">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</latexit>

Output

<latexit sha1_base64="pbpm7LokrN6XiD4C/gJ6Clb8cQ0=">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</latexit>

dC
d✓

t
<latexit sha1_base64="RWXuDuJHXJvlN8xJDO9R8AqtrUs=">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</latexit>
<latexit

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T obs = {0, t1 , t2 , . . . , tn = T }

2022.01.12 波多野研セミナー

c 伊藤伸一

13/43

14.

4次元変分法 Adjoint モデル d dt t = (r x f )> X t + t0 2T obs <latexit 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(t t0 )r xt0 C <latexit 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T =0 4次元変分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 シミュレーションモデルを解く ② Adjoint モデルを時間後ろ向きに解く ✓guess <latexit sha1_base64="YnLegehdSrKrSmzbu3/KDfP7jUU=">AAAOmXiclZfNThNRFMcP+IVVC2hMTNg0FgwsaG7RKHEFQgi4AspHA0OaTnvbTpjODDPTOm1TH8AXcOEKExfK3hdw4wu44BGMS0zcuPDM7dgO7S09HcLMnXPP7/8/9/ZOe0e1dM1xGTsfGb12/cbNW2O3I3fu3ouOT0ze33PMip3juzlTN+20mnW4rhl819Vcnactm2fLqs731eMVv3+/ym1HM40dt2bxo3K2aGgFLZd1MZSZeKi4Je5mMw3F5Z7bKFa44zSbmYk4SzBxxHobyaARh+DYNCdvfgUF8mBCDipQBg4GuNjWIQsO/h1CEhhYGDuCBsZsbGmin0MTIshWMItjRhajx3gu4t1hEDXw3td0BJ1DFx3/bSRjMMN+sM/sgn1nZ+wn+9tXqyE0/FpqeFVbLLcy4+8epf4MpGJ4zmLNpQ53ZdUuFGBRVKth9ZaI+OPItfhq/f1F6uX2TOMJ+8h+4QhO2Tn7hmMwqr9zn7b49gdUn+lbkwse6ptijp1LFfk1GZjzRsxSWVRt4OfSwLg/60UR8ZDxI//HZKKDf29jJBbkvW1nKjjbGt5pgRvFwx81zaOV2esREeuJ4zgVqOI1B7MQx1U0F6iqqOiJtirWQ77tFINpkTmNOs0uHY9Ee1K2RmJrUrZOYutStkJiK1LWJrG2lFVJrCplLRJrSdkTEnsiZR0S60jZEoktSdl1ErsuZVdI7IqUXSWxq1J2jcSuSdkNErshZQ9I7IGUTZPYtJRdJrHLfZ5fjk+JSVJgUgVT/IIVMY+ioYTy5etcC77LaWqdfJman8fxTFXr5MvUdPGbqWKEqhcm+s0+vbr6FbV5OA80FT9TpjBMJf3rKBO/1RWRKVMoijmjr4BOvnxUllgjuthVUEcXZmSqO6F11dIpIKuE4r1MnuSel/qlQk9F2K8Tb7b3Fv12MP7+xBS9zfb9LF5LONbWPszfgSVwf7GIrTJmaVf2zg3YMfn6LvaYA/xYyE/myIZ0LOGVd42yV3e+y3lQP8Xb37lWxO6d7lzt0/98iDEX0KU68HPtaPZ6DuvoPx3elX7zAxzl8xvB97Nk99tYb2NvIZF8mljYehZfehW8qY3BFDzGOpLwApZwP7IJu+jUgFP4AmfRqehydD36upU6OhIwD+DSEU39A+IK3mc=</latexit> ✓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 ≒ シミュレーション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) 2022.01.12 波多野研セミナー c 伊藤伸一 14/43

15.
[beta]
4次元変分法の適用:反応拡散系
(x,
t)
の時間発展

Allen-Cahn モデル (phase- eld モデル)

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2つの相の時空間発展を記述

+ (1

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<latexit sha1_base64="3qMrFlKRdHaO/yZUGFY7VBmQIxc=">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</latexit>

@m
=0
@t

(x, t)
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1

(x, t) : Phase-

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)

1
+m
2

eld 変数

◆
相2

相1

200格子点

@
⌧
= ✏2 4
@t

✓

相1

(x, t)

0

<latexit sha1_base64="Z6IVwu0a+OQ2hYxc9Gak7CYeHIE=">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</latexit>

m : 相境界移動方向を決めるパラメータ

300格子点

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TDGL方程式、結晶成長モデル、延焼モデル、…

x=✏
時間微分はオイラー法で離散化、
タイムステップ: t = 0.1⌧
m = 0.1
ラプラシアンは中心差分で離散化、格子間隔:
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<latexit sha1_base64="YJJkoD+q5W19iUYhRhP1ccTq37A=">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</latexit>

問題設定

<latexit sha1_base64="RloqgP/JWQSPgqFK7OeGGGkNOFQ=">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</latexit>

Noisy なスナップショットの時系列データから

(x, 0)
の初期値
(x,
t)
・ m の値
・

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<latexit sha1_base64="Z6IVwu0a+OQ2hYxc9Gak7CYeHIE=">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</latexit>

<latexit sha1_base64="cvTWDqnGB0WWzH7k4gKGdgAcjgM=">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</latexit>

を4次元変分法によって推定(60,001次元)
詳細な設定
最適化法: L-BFGS-B 法

fi

fi

2022.01.12 波多野研セミナー

c 伊藤伸一

15/43

16.

4次元変分法の適用:反応拡散系 Ito et al., Phys. Rev. E, 94, 043307 (2016) (x, 0) m の推定 の推定 <latexit sha1_base64="cvTWDqnGB0WWzH7k4gKGdgAcjgM=">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</latexit> <latexit sha1_base64="2wtzSLXCd4Y9W9XjDFPJSpTb80M=">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</latexit> 真値 Iteration 2022.01.12 波多野研セミナー c 伊藤伸一 16/43

17.

4次元変分法の適用: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 粒構造推定(擬似データによる実験) fi 2022.01.12 波多野研セミナー 鋼構造材料の表面写真 c 伊藤伸一 17/43

18.

4次元変分法の広がり 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 地震学 ニューラルネットワーク Chen et al. (2018) 2022.01.12 波多野研セミナー c 伊藤伸一 18/43

19.

4次元変分法における不確実性評価アルゴリズム Ito et al., Phys. Rev. E, 94, 043307 (2016) Ito et al., STAM,18:1, 857-869 (2017) c 伊藤伸一 19/43

20.

4次元変分法による不確実性評価 4次元変分法の改善したいところ ・4次元変分法単体では事後分布の形状評価は原理的に不可能 ・分散くらいは評価できないか? → 推定の不確実性・信頼性評価 <latexit sha1_base64="aKB9kaQCOyKzH6yU8shDDgcB/1o=">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</latexit> ・サンプリング法との組み合わせ? ・ 計算回数をなるべく減らしたい ラプラス近似による不確実性評価 p(✓ | D) ✓ˆ ✓ <latexit sha1_base64="6L/sfM20+IbPGS/8f9cvkF/lmnE=">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</latexit> <latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit> 2022.01.12 波多野研セミナー c 伊藤伸一 20/43

21.

ラプラス近似 1 ˆ + (✓ C(✓) ⇠ C(✓) 2 ˆ > H✓ˆ (✓ ✓) ˆ ✓) > H✓ˆ = r✓ˆ r✓ˆ C <latexit sha1_base64="JRv58tOCfHzabxlnMrLO7yysm+M=">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</latexit> <latexit sha1_base64="aKB9kaQCOyKzH6yU8shDDgcB/1o=">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</latexit> p(✓ | D) ⇠ <latexit sha1_base64="A7asVjfc253gxyZu5hGmgX7o5lw=">AAAMRXicfdZNc9NGGAdwmZaSYlwCPfaiqZnBOSRjpQwwwwWSAOElIYHYDmTdzEpa2ZtoJUW7dpTsqN+tp36GfghuTK9FlkUeWY8GHZzd/T3/VayxdteOfC5Vt/tv49oPP17/6cbSzzebt1q/3F6+c7cvw0nssJ4T+mF8aFPJfB6wnuLKZ4dRzKiwfTawTzdnPpiyWPIwOFAXERsKOgq4xx2qsqHj5b+jDlFjpigR3DW3VkwiuTCzz7NYaeLF1NFEsURpl6mUPNFEUDWWnt5OjzUZ06wmT6fZpTvrJOIrf+rdrGMSlkTEZ546Wp1PY6V6PS1utlqOZgmiwij9zty1MRLz0VgNj5fb3bVufpm4YRWNtlFce8d3bvrEDZ2JYIFyfCrlkdWN1FDTWHHHZ2mLTCSLqHNKR+woawZUMDnU+bNOzbJqKqS8EHZaHZx9kco8yns81DyIJooFzmJAJV4YKJm2WiRg504oBA1cTWg8EjQpHksYaRILMxv7azZIfC74LIISPKhJZINXiRZxmUemzOm0rZWs1A4TTezQd2ch817bupdeFVHsdMZztbHaoA5WB9TF6oIyrAzUw+qBjrCOQMdYx6AcKwc9wXoCeor1FNTH6oMKrAI0wBqAhlhD0AhrBHqG9Qw0xhqDSqwSVGFVoBOsE9Ap1inoOdZz0ARrAnqB9QL0Eutl6WV4hvkZhDewboBuYt0E3cK6Bfoc63PQF1hfgL7E+hJ0G+s26Cusr0BfY30N+gbrG9C3WN+C7mDdAd3Fugv6Dus70D2se6D7WPdB32N9D/oB6wfQA6wHoD2sPdA+1j7oAOsA9BDrIehHrB9BP2H9VHoZLllcs/B0S8uSYKOa7YPk46VVhGebXE1ZPl5aTmZbfk1ZPl5aZbODj1tXN4fSq14/3eXCbEnNdkCS0oZQP8nCHKJmqSOitNiNqKh9APl46VaR5H5YsxOQQkq/tvxZZTWeSQ7mB6hvv9L8Yc8lb6fVM0cswoCls78dMpbZAYXpNesxEylf7K6k1Zw6Dyu5bp7ji/2a4DhmxS2valeLcLWP0152EK4PT0v9h7V39vi0+l3zyuliFwclTxZzq5Vg6f9tZedTq3oaxY3++pr1x9r6/oP2043ipLpk/Gb8bnQMy3hkPDW2jT2jZziN+42dRr8xaP7T/Nz80vxvXnqtUWR+NRau5v9fARDmeBM=</latexit> s det H✓ˆ exp N (2⇡) " 1 (✓ 2 : ヘッセ行列 <latexit sha1_base64="GH4BPXo9EmMbY4bo5jJTTNEfX4E=">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</latexit> ˆ✓)> H✓ˆ (✓ ˆ ✓) # p(✓ | D) H✓ˆ 1 <latexit sha1_base64="0lZJ2vqCh+UO6B0zkPKv0KowVfM=">AAALpnicbdZLV9pAFAfwaF9WSqvtshuOuMCFHmJ7Wpe+xfcLxMdYOwkTGM0kMTMgmJOu+2m6bb9Kv01DoN6Qm2yYub/5TyCHzIzh2Vyqcvnv2Piz5y9evpp4PZl7k3/7bmr6/Zl0277JaqZru/65QSWzucNqiiubnXs+o8KwWd24W+t7vcN8yV2nqnoeuxa06XCLm1RFpZupmSAggqqWtIJKeBOQFlUBUS2maBhd34J5PbyZKpYXyvFVwA192Chqw+voZnrSJg3XbAvmKNOmUl7pZU9dB9RX3LRZmCdtyTxq3tEmu4qaDhVMXgfxjwkLSQ2okLInjDBd7H/l1DzKWroOuOO1FXPM0YDqWq6jZJjPE4c9mK4Q1GkEhPpNQbvh4AG4XkB8UYhqP/pFYnPB+xGU4E5GIio+JfKkwSzSYWapqM9FQw23GxDDtRv9UGG2qM+GT4ModtrngRpYDVATqwnawNoAZVgZqIXVAm1ibYK2sLZAOVYOeov1FvQO6x2ojdUGFVgFqIPVAXWxuqAeVg/0Hus9qI/VB5VYJajCqkDbWNugHawd0AesD6BdrF3QHtYe6CPWx8TLsIJ5BcKrWFdB17Cuga5jXQfdwLoBuol1E3QL6xZoBWsFdBvrNugO1h3QXay7oHtY90D3se6DHmA9AD3Eegh6hPUI9BjrMegJ1hPQU6ynoFWsVdAa1hroGdYz0DrWOug51nPQC6wXoJdYLxMvwyPzMxaecmJZEqyZsX2QuJ5YRXi0yWUMi+uJ5aS/5WcMi+uJVTY6WTSyxg0g8apnT/c4Mls3Yzsg3cSGkD3JyBwiY6kjIrHYNanIfABxPXErT3LbzdgJyFAS/7b4WUVjrAKpDo5K//+l8cMeSNwO02cOX7gOC/ufJdKS0QGFBQv6EhMhH+3OhemcenBTuXKc46P9jGDLZ8NbPo2dH4bTfZy2opNmdriT6H/JvLPFO+nfGo/sjHZxUPLuaG4+FUx833x0PtXTp1HcOFtc0D8tLB5/Li6vDk+qE9pHbUYrabr2VVvWKtqRVtNM7af2S/ut/cmVcge5Wq4+GDo+Nsx80Eau3Pd/KCA4Lg==</latexit> ✓ˆ ✓ <latexit sha1_base64="6L/sfM20+IbPGS/8f9cvkF/lmnE=">AAALiHicbdbLctowFAZgp9c0lDZpl90wJYtkEQbSTpPuEnK/kwuXJGIyspFBwbIdW4DB4z5Dt+2b9W1qDJNjfOwN0vn0y+DBklTb4K4sFv/NvXj56vWbt/PvFjLvsx8+Li59qrlWz9FYVbMMy2mo1GUGN1lVcmmwhu0wKlSD1dXuztjrfea43DJv5NBmTUHbJte5RmVYqhHZYZI+LOaLhWJ05XCjNG3klelVeVhaMEjL0nqCmVIzqOvel4q2bPrUkVwzWJAlPZfZVOvSNrsPmyYVzG360dcNcnH1qXDdoVCDZFFQ2UnMI/XNps9NuyeZqc0GpKdbpnSDbJaYbKBZQlCz5RPqtAX1Ap+MZ7NsnzgiF9Z+jYvE4IKPIyjBzZREWHxOZEmL6aTPtJV8aTUcqlqeT1TLaI1DueV8aTl4HkSx0zFPVMWqgmpYNdAW1hYow8pAdaw6aBtrG7SDtQPKsXLQR6yPoF2sXVADqwEqsApQE6sJamG1QG2sNugT1idQB6sD6mJ1QSVWCdrD2gPtY+2DDrAOQD2sHugQ6xB0hHUUexm2MW9DuIy1DLqDdQd0F+su6B7WPdB9rPugB1gPQA+xHoIeYT0CPcZ6DHqC9QT0FOsp6BnWM9BzrOegF1gvQCtYK6CXWC9Br7BegV5jvQa9wXoDWsVaBa1hrYHWsdZBG1gboLdYb0HvsN7FXoYRc1IWnmJsWRKsnbJ9kKgeW0V4uMmlDIvqseVkvNOnDIvqsVU2PDu00sZNIPaqp083mpnNS9kOiBfbENInmZlDpCx1RMQWuzYVqQ8gqsduZbvcsFJ2AjKV2L8telbhGD1Hojb8S6OHPZGoHSTPHI6wTBaMP1dIxw0PKMwvlDaZCPhsdzVI5uTASuSKUY7P9lOCHYdNb/k8dm0aTvZxWg/Pkunhfqz/I/XOOu8nf2s0sj/bxUGXe7O5tUQw9n2z4fm0lDyN4kZtvVD6Vli//J7fKk9PqvPKF+WrsqKUlA1lSzlUKkpV0ZRH5bfyR/mbWcgUMxuZn5OhL+ammc/KzJUp/we42Ss6</latexit> <latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit> → ヘッセ行列/逆行列の大きさは (変数の数)2 → 直接法に基づいてヘッセ逆行列を計算しようとすると 計算コスト = (シミュレーション時間) x (変数の数)2 + (変数の数)3 要素計算 逆行列計算 ヘッセ行列/逆行列の直接計算は避けたい 2022.01.12 波多野研セミナー c 伊藤伸一 21/43

22.
[beta]
クリロフ部分空間法
ヘッセ逆行列要素を部分的に抽出することを考える

Hy = b

b = ei : 単位行列の列ベクトル に取れば
<latexit sha1_base64="dsIv2iRo2y2HFt6e3ayLnTAlzZ0=">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</latexit>

<latexit sha1_base64="Rm0o21olSgYXQrfkXukYcHF+pms=">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</latexit>

<latexit sha1_base64="XXSzyGhRQ03wISrqGMzgv7kc/cM=">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</latexit>

クリロフ部分空間法

⇣
⌘
yi = H✓ 1
2

<latexit sha1_base64="U97LctVi2kH3/765A/fd+0Zg4sw=">AAAL8XicbdbNU9NAFADwAH6gtQp69NKxHmAGmAYd5eIMH4Io8inlc5HZpJt2IZvE7LYEMuv/4c3x6p/jyT/Fm2na4SV5yaW77/feJtlpdtcKXC5Vo/F3ZHTszt1798cfPKw8qj5+MjH59ED63dBmTdt3/fDIopK53GNNxZXLjoKQUWG57NC6XOn7YY+FkvvevroO2JmgbY873KYqCZ1PCCKo6tjUjTf0eRzqqTgNSCde13rGmq69qxHFIhXLgHqaxNZMNsGaqWW7X+P5JERavpIz+Xg4a2qL6POJemOukV413DCHjboxvHbOJx+6yXB2VzBP2S6V8tRsBOospqHitst0lXQlC6h9SdvsNGl6VDB5FqfzomtZjamQ8lpYuhjsP2ZhHOUsnMXcC7qKeXa+QEWO7ympq1XisSvbF4J6rZjQsC1opAcv7QcxCUUtiX3vB4nLBe+XoArulVQkwduKKmkxh/SYPVU3p5NUy49iYvluq19Ue1k3X+rbJIqd9nmgFlYL1MZqg7awtkAZVgbqYHVA21jboB2sHVCOlYNeYL0AvcR6CepidUEFVgHqYfVAfaw+aIA1AP2G9RtoiDUElVglqMKqQLtYu6A9rD3QK6xXoBHWCPQa6zXoDdabzMewhHkJipexLoOuYF0BfY/1Pegq1lXQNaxroB+wfgBdx7oO+hHrR9BPWD+BbmDdAP2M9TPoJtZN0C2sW6DbWLdBd7DugO5i3QXdw7oH+gXrF9B9rPugTaxN0AOsB6CHWA9Bj7AegR5jPQY9wXqS+RhuWFiy8DQyy5Jg7ZLtg6TxzCrCk02uJC2NZ5aTDlNlaWk8s8omh5RWWd4AMp96+XA3udGiku2ARJkNoXyQ3BiiZKkjIrPYtakonYA0nrlVILnrl+wEZCiZf1s6V0mOUyNpG/6l6WQPJG3r4pkjFL7HdP93inSS45nN4jlzgQnN891pXaxTV36hrpHW8Xy/pLATsuEtb3Nnh8XFPq52kkNreXEv039TemeH94rvmmb28l1cKHmUr5stFGaet5qcT83iaRQ3DubnzFdz87uv64vLw5PquPHceGFMGabx1lg01o0do2nYxh/j38joyFhFVn5UflZ+DVJHR4Y1z4zcVfn9HzM8VZI=</latexit>

⇣
⌘
1
y j = H✓
は共分散
は分散、
i, j
i,i
<latexit sha1_base64="6VrghFAwfQEineCDUlE0dA92+Mo=">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</latexit>

・共役勾配(CG)法

r 1

Kr (H, b) = span{b, Hb, H b, . . . , H

b}

・共役残差(CR)法
・一般化最小残差(GMRES)法,...

CR法の疑似コード
r0
Hr 0 , p0 = r 0
Compute s0 = b
Set an initial guess
For k = 0, 1, 2, ...

↵k = (Hsk , sk ) / (Hsk , Hsk )
r k+1 = r k + ↵k pk
sk+1 = sk ↵k Hpk
k = (Hsk+1 , sk+1 ) / (Hsk , sk )
pk+1 = sk+1 + k pk
Hpk+1 = Hsk+1 + k Hpk

・ヘッセ行列- ベクトル積が高速に計算できれば
逆行列要素が高速に抽出可能になる

End For

2022.01.12 波多野研セミナー

c 伊藤伸一

22/43

23.

Second-order adjoint (SOA) 法 シミュレーションモデルおよび Adjoint モデルを各変数について線形化する シミュレーションモデル x0 = ✓ <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">AAACuXichVHLSsNQED2Nr1pfVTeCG7FUXJVpFRRFEN249FUVrJQkXm00TUJyW1pLf8AP0IULH+BC/Aw3Lty66CeISwU3LpykAVFRJyR37rlzZs7N0RzT8CRRI6K0tLa1d0Q7Y13dPb198f6BDc8uubrI6rZpu1ua6gnTsERWGtIUW44r1KJmik3tcNE/3ywL1zNsa11WHbFTVPctY8/QVclQrpKvUX0uJwtCqvl4glIUxMjPJB0mCYSxbMcfkMMubOgooQgBC5JzEyo8fraRBsFhbAc1xlzOjOBcoI4Yc0tcJbhCZfSQv/u82w5Ri/d+Ty9g6zzF5Ndl5giS9Eg39EL3dEtP9P5rr1rQw9dS5VVrcoWT7zseWnv7l1XkVaLwyfpTs8QepgOtBmt3AsS/hd7kl49OX9ZmVpO1MbqiZ9Z/SQ264xtY5Vf9ekWsnv2hR2Mt/h9L/lohUeH5duCAhzpbmf5u3M9kI5NKT6QyK5OJ+YXQ1CiGMYpxdm4K81jCMrI8x8EJznGhzCqqUlAOmqVKJOQM4kso3geL4p4m</latexit> Adjoint モデル d dt = (r x f ) t > t + X t0 2T obs <latexit sha1_base64="1kqcoTy2iTY3iSE4F+wj0lrXbFY=">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</latexit> <latexit sha1_base64="h5FlaEbyb5cnhAtKyCqna4vs1GU=">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</latexit> T (t 0 t )r xt0 C =0 2022.01.12 波多野研セミナー c 伊藤伸一 23/43

24.
[beta]
Second-order adjoint (SOA) 法
シミュレーションモデルおよび Adjoint モデルを各変数について線形化する

<latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit>

d
dt

Tangent linear (TL) モデル
t

= (r x f )

t

<latexit sha1_base64="nOOznfNHsRu/VgwTqdVSyfoIVns=">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</latexit>

0

=r

Second-order adjoint (SOA) モデル

d
⇠t = (r x f )> ⇠t + {(r x r x f ) t }>
dt

t +

<latexit sha1_base64="cWgBysKO1kJYknbs8RDoTHXoJrk=">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</latexit>

<latexit sha1_base64="EK7rs6L3wFhb50BjyIGqKd0OB7Q=">AAALi3icbdZbc5pAFAdw0msaa5u0j31xah6ShziYdNpMp53J/X4xiUaT4GQWXHQjCwRWRRn6JfrafrB+myI6OcCBF3f3d/6LMrK7qm0wV8jyv5lnz1+8fPV69s1c7m3+3fv5hQ/XrtVzNFrTLMNyGipxqcFMWhNMGLRhO5Rw1aB1tbs99nqfOi6zzKoY2rTJSdtkOtOICIduFI/d+9Xgp3w/X5RLcnQVcKM8bRSl6VW5X5gzlJal9Tg1hWYQ170ry7Zo+sQRTDNokFd6LrWJ1iVtehc2TcKp2/SjbxwU4uoT7rpDrgbpQU5EJzWP0NebPjPtnqCmlgwIT7dM4Qb5vGLSgWZxTsyWrxCnzYkX+Mp4Nsv2FYcXwrFf40HFYJyNIyjBzIxEOPiUyCstqit9qi0Vy8thqWp5vqJaRmscKiwWy4vBUxHBTsY8URWrCqph1UBbWFugFCsF1bHqoG2sbdAO1g4ow8pAH7A+gHaxdkENrAYox8pBTawmqIXVArWx2qCPWB9BHawOqIvVBRVYBWgPaw+0j7UPOsA6APWweqBDrEPQEdZR7GXYxLwJ4S2sW6DbWLdBd7DugO5i3QXdw7oHuo91H/QA6wHoIdZD0COsR6DHWI9BT7CegJ5iPQU9w3oGeo71HLSCtQJ6gfUC9BLrJegV1ivQKtYqaA1rDfQa6zVoHWsdtIG1AXqD9Qb0Futt7GUYUSdj4ZFjyxKn7YztQ4nGY6sICze5jLJoPLacdKjIKovGY6tseHxoZdVNIPaqZ083SszmZWwH4bEjtk1lTpKYg2csdQqPLXZtwjMfQDQeu5XtMsPK2AmUqcT+bdGzCmv0ghK14V8aPeyJRO0gfeZwuGXSYPy5pHTc8IBC/VJ5nfKAJbvLQTonBlYqJ0c5luxnBDsOnd7yqXZlGk73cVoPj5PZ4X6s/zXzzjrrp39rVNlPdnHQZV4yt5IKxr5vPjyfltOnUdy4Xi2V10qrF1+KG1vTk+qs9En6LC1JZembtCEdSBWpJmkSl35Lf6S/uXxuLfc992NS+mxmmvkoJa7c7n8qIiw7</latexit>

⇠T = 0

<latexit sha1_base64="lFXHXN5wHy1jjc38s9iNFpRQ4pg=">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</latexit>

⇠0 = H✓

0

X

t0 2T obs

(t

t0 )r xt r xt >C

t

= H✓ r

TLモデルとSOAモデルを組み合わせることで、
ヘッセ行列と任意のベクトルの積をベクトルとして出力することができる!
2022.01.12 波多野研セミナー

c 伊藤伸一

24/43

25.
[beta]
0

SOA 法によるヘッセ行列-ベクトル積の計算

<latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit>

d
dt
<latexit sha1_base64="nOOznfNHsRu/VgwTqdVSyfoIVns=">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</latexit>

TL モデル
t
0

= (r x f )
=r

SOAモデル

d
⇠t = (r x f )> ⇠t + {(r x r x f ) t }>
dt

t
<latexit sha1_base64="cWgBysKO1kJYknbs8RDoTHXoJrk=">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</latexit>

<latexit sha1_base64="EK7rs6L3wFhb50BjyIGqKd0OB7Q=">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</latexit>

⇠T = 0

<latexit sha1_base64="lFXHXN5wHy1jjc38s9iNFpRQ4pg=">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</latexit>

⇠0 = H✓

0

t +

= H✓ r

X

t0 2T obs

(t

t0 )r xt r xt >C

t

t
<latexit sha1_base64="RWXuDuJHXJvlN8xJDO9R8AqtrUs=">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</latexit>
<latexit

Input

0

=r

Output

= H✓ r

t
<latexit sha1_base64="RWXuDuJHXJvlN8xJDO9R8AqtrUs=">AAAKkXiclZbPTxNBFMcfiIrrD0AuJl4aC8ZTMzX+qqeihEC8UEqhgSVkdzttN2x3l91thTb4B+hV48GTJh6Mf4MnL/4DHvgTjEdMvHjwzexCV5nCY5tuZ95+P9958zqdjuk7dhgxtj80fG7k/IWLo5e0y1euXhsbn7i+EnrtwOIVy3O8oGoaIXdsl1ciO3J41Q+40TIdvmpuPRXPVzs8CG3PXY52fb7RMhquXbctI8JQKdocz7JcocAeFO5njjfyOSavLCTXojcx8gV0qIEHFrShBRxciLDtgAEhvtYhDwx8jG1AD2MBtmz5nMMeaMi2UcVRYWB0C+8N7K0nURf7wjOUtIWjOPgOkMzANPvOPrED9o19Zj/Yn4FePekhctnFTzNmub859vJG+fepVAs/I2j2qRNzjqAOj2SuNubuy4iYhRXzne7bg/LjpenebfaB/cT837N99hVn4HZ+WR9LfOkdugt/F6nncr4tmYGLFe5hXNSvISM76Cgih/l5OJroBxjJJLoXR0od62Zjz0ZtmNT9tDHEDGhjxMrjY2hyZXCsiQ4dzCP2MtEnbpvy+6wd+WdgCp9MIbv3H9smsW0lG5DYQMmaJNZUsj6J9ZXsNondVrIhiQ2VbJPENpXsPImdV7KzJHZWyc6R2Dklu0BiF5TsGoldU7JVEltVsjMkdkbJdrEVoJriwJQOntyPG6ijeOgpvXqt2sl+RnPr61VuQsfxTnXr61VujvwHMDFC9UsTg6pPz657Qm47WAeai1CqHM6SyeA8WsSdWZdKlUND1oy+Avp69ax8uUbEKcElzy7NqFyXU+sq9qkjq6fix5kaafSacrxy6leRHq8fjxkNT2uHR7LM4MbK3Vye5fKle9nik+TcNgo34RbcwbPZQyjinrwIFTwJcHgFr+GNNqkVtKKWaIeHEmYS/rm0Z38BNcAYCQ==</latexit>
<latexit

<latexit sha1_base64="ryuXTQ0mJCzzWoku/BcVPGMW6NI=">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</latexit>

T obs = {0, t1 , t2 , . . . , tn = T }

2022.01.12 波多野研セミナー

c 伊藤伸一

25/43

26.

SOA 法を用いた不確実性評価法 Hy = b <latexit sha1_base64="Rm0o21olSgYXQrfkXukYcHF+pms=">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</latexit> ⓪ y の初期値を設定 <latexit sha1_base64="aKB9kaQCOyKzH6yU8shDDgcB/1o=">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</latexit> p(✓ | D) H✓ˆ ① TL モデルを解く 1 <latexit sha1_base64="0lZJ2vqCh+UO6B0zkPKv0KowVfM=">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</latexit> ② SOA モデルを時間後ろ向きに解く ✓ˆ ③ クリロフ部分空間法で更新。①へ戻る。 b = ei に取れば <latexit sha1_base64="dsIv2iRo2y2HFt6e3ayLnTAlzZ0=">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</latexit> <latexit sha1_base64="XXSzyGhRQ03wISrqGMzgv7kc/cM=">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</latexit> ⇣ ⌘ yi = H✓ 1 i,i ✓ <latexit sha1_base64="6L/sfM20+IbPGS/8f9cvkF/lmnE=">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</latexit> <latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit> ⇣ ⌘ 1 は分散、 y j = H✓ <latexit sha1_base64="6VrghFAwfQEineCDUlE0dA92+Mo=">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</latexit> i, j は共分散 TL, SOAモデルの計算コスト ≒ シミュレーションモデルの計算コスト 逆行列の計算コスト = (シミュレーション時間) x (反復回数) x (変数の数) 1列のみの抽出であれば、 (シミュレーション時間) x (反復回数) c.f. 直接法 (シミュレーション時間) x (変数の数)2 + (変数の数)3 2022.01.12 波多野研セミナー c 伊藤伸一 26/43

27.
[beta]
不確実性評価法の適用
(x,
t)
の時間発展

Allen-Cahn モデル (phase- eld モデル)

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2つの相の時空間発展を記述

+ (1

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@m
=0
@t

(x, t)
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1

(x, t) : Phase-

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)

1
+m
2

eld 変数

◆
相2

相1

200格子点

@
⌧
= ✏2 4
@t

✓

相1

(x, t)

0

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m : 相境界移動方向を決めるパラメータ

300格子点

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TDGL方程式、結晶成長モデル、延焼モデル、…

x=✏
時間微分はオイラー法で離散化、
タイムステップ: t = 0.1⌧
m = 0.1
ラプラシアンは中心差分で離散化、格子間隔:
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<latexit sha1_base64="YJJkoD+q5W19iUYhRhP1ccTq37A=">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</latexit>

問題設定

<latexit sha1_base64="RloqgP/JWQSPgqFK7OeGGGkNOFQ=">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</latexit>

Noisy なスナップショットの時系列データから
・

m の不確実性 m
<latexit sha1_base64="cvTWDqnGB0WWzH7k4gKGdgAcjgM=">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</latexit>

<latexit sha1_base64="FE36sNuOQnl09xcmrmLUqJxOa+c=">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</latexit>

(60,001次元空間中の事後分布の標準偏差1つだけ)

のみをSOA法によって選択的に評価
詳細な設定
事前分布: 一様分布 ノイズ分布:

q(!) = N (0, 0.012 )
<latexit sha1_base64="dI48rqdGMKzo3Qy4xFrpWs/Z14Y=">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</latexit>

データ取得: 時間窓 [0.1⌧, 102.5⌧ ] クリロフ部分空間法:共役残差法
<latexit sha1_base64="hDPnxqQvgfmb4Q46gOzQT/F6bkQ=">AAAJKniclZa/b9NAFMdfW36Y8KMpLEhIKCItYkDROYBATBVUVbu1TdNGjaPIdi6JVce2/CNqE3Vj4h9gYAKJgR8b/AcszEgMESsLYiwSEmLg+WIR01zKw1F8z+/e53vfO1v2GZ5tBSFjg6npmRMnT51WzmTOnjt/YTY7d3ErcCPf5GXTtV2/YugBty2Hl0MrtHnF87neMWy+bew+jPu3u9wPLNfZDPc9XuvoLcdqWqYeYqqevarZvBlWWUHVQj26qbJi4U4cab7Vaoe1ejbPCkwcufFATYI8JMeaOzfzEzRogAsmRNABDg6EGNugQ4C/KqjAwMNcDfqY8zGyRD+HA8ggG2EVxwods7t4buFVNck6eB1rBoI2cRQb/z6SOVhgn9hLdsg+sNfsK/s1UasvNGIv+9gaQ5Z79dnHl0s//kl1sA2hPaKO9RxCE+4JrxZ690QmnoU55Lu9J4el+xsL/evsOfuG/p+xAXuPM3C6380X63zjqVBvINPEtgt7qKQJ124SG2INGn985WAee+ZR/+AIG5HYSMr6JNaXsgaJNaSsR2I9KRuQ2EDKtklsW8qukNgVKbtEYpek7DKJXZayqyR2VcrukNgdKVshsRUp28PIx2qKApMquOJd0sI6ioaWqpc/bxb2dchqo3qZWlzH8UxVG9XL1Gzx9jIwQ9VLE5NWn+6ud4y3PVwHmkpcKVP4HyeTfbTEjOn3b1Qv9+SJOxx/nxyytzQjU91MPRVDnSayWio/zjRIozek45VSz3R6vFF+yGRwn6Ae3RWMB1vFgnqrUFy/nV98kOwYFLgC1+AG7gruwiK+FdegjF/HR/AK3sI75Y3yURkon4el01MJcwn+OpQvvwGZ/Nk7</latexit>

fi

fi

2022.01.12 波多野研セミナー

c 伊藤伸一

27/43

28.

不確実性評価法の適用 データ取得間隔 <latexit sha1_base64="vVmVVIaLWXq2BIlNcVdjI92c70o=">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</latexit> T 102.5⌧ 0.1⌧ <latexit 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sha1_base64="3EirMri1ChkJhn6e12th86FFPoE=">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</latexit> m <latexit sha1_base64="FE36sNuOQnl09xcmrmLUqJxOa+c=">AAAJEniclZbLbtNAFIZPWy4mXHphg8QmIi1iFU1aJFBXFVRVu2ubpo1aV5Uvk8Sqb/IlahP1BVgiJBZlAxILxBuwQ92wYMuij4BYFgkJseB4bGHTTMrBUTzHZ873zz9jyx7dt60wYux0ZHTs0uUrV5Vrpes3bt4an5ic2gy9ODB4w/BsL2jqWshty+WNyIps3vQDrjm6zbf0/adJ/1aXB6HluRvRoc93Ha3tWi3L0CJMNVWT25FWdvYmKqzKxFEeDGpZUIHsWPUmx36CCiZ4YEAMDnBwIcLYBg1C/O1ADRj4mNuFPuYCjCzRz+EISsjGWMWxQsPsPp7beLWTZV28TjRDQRs4io3/AMkyzLAv7B07Y5/Ye/aV/Rqq1RcaiZdDbPWU5f7e+LM79R//pBxsI+jk1IWeI2jBY+HVQu++yCSzMFK+23t5Vp9fn+nfZ2/YN/T/mp2yE5yB2/1uvF3j68dC3USmhW0XDlBJFa69LNbFGph/fJVhGnumUf/oHBuT2FjKBiQ2kLI6idWlrE9ifSkbkthQynZIbEfKLpPYZSm7SGIXpewSiV2SsiskdkXKbpPYbSnbJLFNKdvDKMBqigKTKnjiXdLGOoqGWqiXP28W9jlktbxeppbUcTxT1fJ6mZot3l46Zqh6RWLY6tPd9S7wdoDrQFNJKmUK/+NkuI+2mDH9/uX1ck++uMPJ98kleysyMtWNwlOR6rSQVQv5QcYkjW5Kx6sXnunieHk+ZUq4T6id3xUMBpuz1dpcdXbtYWXhSbZjUOAu3IMHuCt4BAv4VlyFhtgNPIdjeKW8UD4oH5WTtHR0JGNuw1+H8vk3GkrQ6w==</latexit> m <latexit sha1_base64="FE36sNuOQnl09xcmrmLUqJxOa+c=">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</latexit> Time は4次元変分法による推定値 / 1/Ndata <latexit sha1_base64="hDUJnfh6dczxkkLGXk1C2Y1Snqo=">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</latexit> 高次元の事後分布からの選択的な不確実性抽出が可能に Ito et al., Phys. Rev. E, 94, 043307 (2016) 2022.01.12 波多野研セミナー c 伊藤伸一 28/43

29.

不確実性を厳密に評価するアルゴリズム Ito et al.,BIT numerical mathematics (2021) Ito et al., arXiv:2109.13143 c 伊藤伸一 29/43

30.

Question Adjoint モデルや SOA モデルを解いて得られる や の解析解は コスト関数を直接微分して得られるものと厳密に一致する。 しかし実際は、Runge—Kutta 法などの数値積分法を使って 時間方向に数値積分する必要がある → 離散化誤差や丸め誤差により厳密でなくなる。 Forward モデルは研究者自身が数理的・物理的背景から積分法を決める。 Q. 他のモデルはどうすべきか?誤差を抑える最適な積分法は存在するか? Forward model Adjoint model ? Tangent linear (TL) model <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> d dt ? t = (r x f ) t SOA model ? A. 離散化誤差を厳密にゼロにする最適な積分法のセットを構築できる。 Ito et al.,BIT numerical mathematics (2021) 2022.01.12 波多野研セミナー c 伊藤伸一 30/43

31.

厳密な勾配計算の重要性 勾配に誤差があると… 実問題では答えを知らないので、 最適化に成功/失敗しているかどうか 本来の勾配方向 判定する術がない… 誤差を含んだ勾配方向 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 2022.01.12 波多野研セミナー c 伊藤伸一 32/43

32.

厳密なヘシアン計算の重要性 ヘシアン-ベクトル積に誤差があると… Hy = b <latexit sha1_base64="pL0aX+5rH57c3v/xWA90izHgtac=">AAADWXicxVLNLgRBEP52x89a//YicRGbFadNjwgikQgXR3+LhI3MjF4mO39mepc12RfwAg4uSBzEY7h4AQfxBOJI4uKgpndEEBz1ZLqrvqqvurqqdM8yA8HYXSKpNDW3tKba0u0dnV3dPb19q4Fb8Q1eMFzL9dd1LeCW6fCCMIXF1z2fa7Zu8TW9PBfZ16rcD0zXWRE1jxdtbccxS6ahCYKKm7YmdoNSOF+vTetbPVmWZ3INfhfUWMgiXgtub2IAm9iGCwMV2OBwIEi2oCGgbwMqGDzCiggJ80kypZ2jjjRxK+TFyUMjtEz7DmkbMeqQHsUMJNugWyz6fWIOIsdu2SV7Yjfsij2w1x9jhTJGlEuNTr3B5d5W91H/8sufLJtOgd0P1q85C5QwKXM1KXdPItErjAa/enj8tDy1lAuH2Tl7pPzP2B27phc41WfjYpEvnfxT9JDqYst+5H70EDig+13Z3yDunUOWfdkXW1bKoUkICfdJ2yafSH6vYISFEo24aZo09etcfRdWR/PqeF5dHMvOzMYzl8IAhjBCczWBGcxjAQXKYA/HOMVZ8l5JKCkl3XBNJmJOBp+WknkDOLPEiA==</latexit> <latexit sha1_base64="9VG4wPKVj+Qlb4uDDnRUoY+3OVc=">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</latexit> ky をクリロフ部分空間法で解いた際の収束履歴 y exact k1 実問題では答えを知らないので、 Failed… 求解に成功/失敗しているかどうか 判定する術がない… 推定値の信頼性を与える不確実性 それ自体の信頼性がなくなってしまう Naive method H✓ˆ ⌃ = I <latexit sha1_base64="PwnF/nf4uHap9vKVMuzKoQmLtDs=">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</latexit> Our method Success! 正確な H の計算により 正しい不確実性の求解を Iteration step 2022.01.12 波多野研セミナー 保証できる! c 伊藤伸一 32/43

33.

勾配&ヘシアンの厳密計算 4つの方程式に内在するシンプレクティック性を考慮した数値積分法を提案 Forward, adjoint, TL, SOA 背後にある保存量を 数値的に厳密に保存 ヘシアンに含まれる誤差 Row Column 計算コスト &必要メモリ Naive method Naive method 2022.01.12 波多野研セミナー Row <latexit sha1_base64="pxJG11aDGXTtTjEJQ4gkNvHASXM=">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</latexit> Column Our method Our method c 伊藤伸一 31/43

34.
[beta]
不変量
Q. そもそもなぜ Adjointモデルは厳密な勾配を計算できるのか?
TL model

A. TL 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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t
dt

<latexit sha1_base64="5VoAAisYfRGuEXlwTh3SJBFLMJw=">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</latexit>

> >
t t+ tt

⇣

> >
= (r0x f ) 0

t

⌘

=0

<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>

d
dt

t

= (r x f )

Adjoint model

= r xt C(xt (✓)) と選べば、

h
i
>
>
>
>
>
((r
)
)
(r
)
r
C(x
)
=
x
r
C(x
)
=
x
r
C(x
)
=
x
t
✓
t
0
x
t
✓
t
x
t
t
t
t
t
0
0 r✓ C(xt )
h
i
>
>
>
xt ) 0 ) r xt C(xt ) = 0 (r✓ xt ) r xt C(xt ) = >0 r✓ C(xt ) * xt (✓ + ✏ 0 ) = xt (✓) + ✏
<latexit sha1_base64="RIOX2w7GPDwnqp/feo71pdtvvc8=">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</latexit>

t

<latexit sha1_base64="8BTUDzm/wyz/8vwkHxSEzxrAjAM=">AAAMknichdZbc5NAFABg4t2aqFXffGFMH9KHdhJ11HHGGdt46f2eXuzGzkKWZC0LCJuUlsEf4qv+Kf+NQGgP4ewoD8lyvnMOKYXdNTybB7LZ/FO5dv3GzVu379ydulet3X/wcPrRfuAOfZN1TNd2/UODBszmDutILm126PmMCsNmB8ZpO/WDEfMD7jp78txjXUH7Dre4SWUSOpmu1EiP2ZKeRDL+GhHperFOHGrYSSRMg3G7kX3P6u90YjNLNvLPPInIAZM0znKIz/sDOXvZsXkZyBur+04lfa8KLjPTSxz/90L//sHjpK6u6K+XmuqXNScP6835ZnboeNDKB3UtP7ZOpqds0nPNoWCONG0aBMetpie7EfUlN20W18gwYB41T2mfHSdDhwoWdKPsHxfrRY2oCIJzYcTloKByUOojrTfdiDveUDLHnCyQoeU6MohrNeKwM9MVgjq9iFC/L2gYRyTt5noR8YWexH6kQWJzwdMSVMEdRUUSvKpInx6LjJjZqLdmk1TDDSNiuHYvLdJn6q2Z+CqJYqcpj9XAaoCaWE3QHtYeKMPKQC2sFmgfax90gHUAyrFy0G9Yv4GeYj0FtbHaoAKrAHWwOqAuVhfUw+qBfsf6HdTH6oMGWANQiVWCDrEOQUdYR6BnWM9AQ6wh6DnWc9ALrBeFl2EB8wIUL2JdBG1jbYN+wPoB9CPWj6CfsH4C/Yz1M+gS1iXQZazLoCtYV0BXsa6CrmFdA13Hug66gXUDdBPrJugW1i3QbazboDtYd0B3se6C7mHdA+1g7YDuY90HPcB6AHqI9RD0COsR6BesXwovwwXzFRNPszAtCdZXLB8kixdmEZ4scoq0LF6YTtI1X5GWxQuzbLKL6qnyxlB41dXtLia6hYrlgISFBUHdZKKHUEx1RBQmuz4VyhuQxQuX8gJuu4qVgORSeNqye5XkWDrJxvCUZjd7LNk4Lu85fOE6LE6/G2QQJBsUFs233jAR88nT2bhcJ8/cUl0zq+OT54rCgc/yS17lzuXF5XNcbSW7anXxqHD+Snlli4/Kf2uWOZo8xYUBDyfr5kqFhd9bS/anrfJuFA/2n8+3Xsw/335Zf7+Y71TvaE+1Z1pDa2mvtffakraldTSzIis/K78qv6tPqm+rC9X2OPVaJa95rE0c1bW/mX6SGw==</latexit>

<latexit sha1_base64="9mYRG8P8ZwG0ANbOb2VGgFwnR6I=">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</latexit>

<latexit sha1_base64="uHi8RSX+U+ihrsaQk1DuOjM9kyw=">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</latexit>

<latexit sha1_base64="rKWJROh4xk2V1BJcmlKpQl2zse4=">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</latexit>

0

= r✓ C(xt (✓))

<latexit sha1_base64="Un/ZtOy/PcZf2C6b3qHd6JAnbQU=">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</latexit>

t

2
+
O(✏
)
t

= (r✓ xt )

0

→ Adjointモデルへは、不変量を保存する数値積分法を採用すべき
e.g,. Symplectic partitioned Runge—Kutta (SPRK) method
(Sanz-Serna, 2016)
2022.01.12 波多野研セミナー

c 伊藤伸一

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35.

ヘシアンの厳密計算に向けて 不変量を保存する数値積分法をadjointモデルに適用することによって 厳密な勾配を計算できる (Sanz-Serna, 2016) では、ヘシアン-ベクトル積では?? Forward model Tangent linear (TL) model <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> d dt t ? = (r x f ) t User-de ned scheme Adjoint model Invariant-preserving scheme fi 2022.01.12 波多野研セミナー SOA model ? c 伊藤伸一 34/43

36.

ヘシアンの厳密計算に向けて アイデア: Adjoint モデルと SOA モデルの数学的等価性 <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> 拡大ベクトル 、 d dt t = (r x f ) t を定義すると 4つのモデルは 1つの大きなForward-adjoint 系を成す → 勾配と同様、invariant-preserving scheme により 厳密なヘシアン-ベクトル積が計算可能 2022.01.12 波多野研セミナー c 伊藤伸一 35/43

37.
[beta]
Inhomogeneous wave equation
検証:1次元不均質媒質波動方程式を用いた速度構造の推定問題

∂2U(z, t)
∂
∂
=
E(z)
U(z,
t)
]
∂t 2
∂z [
∂z
(空間方向には有限体積法で離散化)

時間積分法:
Heun method

(2-stage 2nd-order Runge—Kutta )

と

の初期状態は既知と仮定

速度構造

z

は未知(推定対象)

<latexit sha1_base64="pXaCb5Kmd0Ygt3EbZVOqC2SHUuI=">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</latexit>

コスト関数:
where

2022.01.12 波多野研セミナー

:

in the above movie

c 伊藤伸一

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38.

Time integrator Augmented forward model discretize Heun method Augmented adjoint model Naive discretization Heun method pn = pn+1 + h⇣ 2 ⌘ l¯n,1 + l¯n,2 , l¯n,2 = rqG(qn+1 )> Pn,2 , l¯n,1 = rqG(qn )> Pn,1 , Our method ⌘ h ⇣¯ pn = pn+1 + ln,1 + l¯n,2 , 2 l¯n,2 = rqG(Qn,2 )> Pn,2 , l¯n,1 = rqG(qn )> Pn,1 , Pn,2 = pn+1 , <latexit sha1_base64="CBreDbPCesB9QaOp1LDWarNY89c=">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</latexit> Pn,2 = pn+1 , Pn,1 = pn+1 + hl¯n,2 . <latexit sha1_base64="+B1N3A2zEwHO1V2IZxk8UhQuGJg=">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</latexit> 2022.01.12 波多野研セミナー Invariantpreserving Our discretization Pn,1 = pn+1 + hl¯n,2 . c 伊藤伸一 37/43

39.

ヘシアンに含まれる誤差 :非対称度( が正確に計算できていればゼロになる量) Our method Heun method Heun 法に起因する離散化誤差 O(h2) :最適推定値 機械誤差(丸め誤差など)が 累積した誤差のみ 提案手法は離散化誤差を一切含まない形でヘシアンを計算可能 2022.01.12 波多野研セミナー c 伊藤伸一 38/43

40.

Importance of exact Hessian computation Hy = b <latexit sha1_base64="pL0aX+5rH57c3v/xWA90izHgtac=">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</latexit> Convergence behavior when solving by a Krylov subspace method <latexit sha1_base64="9VG4wPKVj+Qlb4uDDnRUoY+3OVc=">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</latexit> ky y exact k1 実問題では答えを知らないので、 Failed… Naive method H✓ˆ ⌃ = I Our method <latexit sha1_base64="PwnF/nf4uHap9vKVMuzKoQmLtDs=">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</latexit> Success! 求解に成功/失敗しているかどうか 判定する術がない… 正確な H の計算により 正しい不確実性の求解を 保証できる! Iteration step 信頼性のある不確実性評価が可能になった. 任意の大規模シミュレーションモデルへ適用が可能. 2022.01.12 波多野研セミナー c 伊藤伸一 39/43

41.
[beta]
Model
Bungo Channel LSSE model
Hirahara and Nishikiori (2019)

{
<latexit sha1_base64="H6hgW2UPuyTeXlurJ5jRNw1vdhc=">AAADwHicxVI9TxtBEH3mkmAgCTg0SDQolqNU1hohgqgQNJRgMCABQnfHGk6+r9ytTcyJP4BoEwoqkCgQP4OGP5CCn4AoiZQmBe/WF0WEAGXmdLszb+fNzO6MFbpOrIS4ynUZL16+6s739Pa9fvO2f6DwbikOmpEta3bgBtGKZcbSdXxZU45y5UoYSdOzXLlsNWbS8+WWjGIn8BdVO5TrnrnlO3XHNhWh6lqyMVAUZaFl5KFSyZQiMpkLCrlFrGETAWw04UHCh6LuwkTMbxUVCITE1pEQi6g5+lxiD73kNukl6WESbXDdorWaoT7tNGas2TazuPwjMkdQEt/FmbgVl+JcXItfj8ZKdIy0ljZ3q8OV4Ub//tDCz2dZHneF7T+sJ2tWqGNC1+qw9lAj6S3sDr+1e3i7MFktJR/Eibhh/cfiSlzwBn7rh306L6tH/yl6wnfxdD9Kj3oofGH+QPc3znrn82RH98XTL+VzEhLiEa1N+qT67xdMsUSjT3Mt5rnPtJg30ehzzPY/me0Ok7Nd+XuSHypLo+XKeHlsfqw4NZ1NeR7DeI+PnORPmMIs5lBj9joO8BXfjGlj2wiMzx3XrlzGGcQ9MXbvAGnY2Q8=</latexit>

Rate-and-state dependent friction
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⌧ (x, t) = A(x) log v(x, t) + B(x) log ✓(x, t)

Notation

x = (X, Y )>
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Aging law

v : Slip velocity
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✓ : State variable
K, k : Green’s functions

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˙ t) = 1
✓(x,

v(x, t)✓(x, t)
L(x)

Balance equation Z
⌧˙ (x, t) = ⌘v(x, t) + dx0 [K(x, x0 ) {v(x0 , t)

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⌘
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vpl , vlock
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: Damping parameter
: Speeds of plates

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vpl )]

Slip velocity

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Parameters of interest

v
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Y (km)

vpl } + k(x0 ) (vlock

A(x), B(x), L(x) : Spatially dependent frictional parameters
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Previous study assumed patch-like-constrained elds.

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X(km)
→ Fields in high-resolution need to be investigated to understand the physics.

Aim

* Obtain hi-res uncertainty elds of frictional parameters
* Uncover the relation between the uncertainty elds and slip motion

c 伊藤伸一
fi

fi

fi

2022.01.12 波多野研セミナー

40/43

42.
[beta]
Result
Obtained uncertainty elds

Problem setting
True parameters:

(a)
100km

35km

(b)

120km

Data : Time evolution of eld
time window (a)—(d)

v
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(c)

Cross-section view on X=0 line

(d)

The results say
* Underlying uncertainty elds
show very complex patterns
despite simple parameter elds.
* Seismic activity reduces
the magnitude of uncertainty eld.
→ Feedback to observational design

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2022.01.12 波多野研セミナー

Ito et al., arXiv:2109.13143

c 伊藤伸一

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ありがとうございました。 c 伊藤伸一 43/43