210304_Adjoint-based exact Hessian computation

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November 21, 23

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地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 2021.03.04 Adjoint-based exact Hessian computation Shin-ichi Ito1,2 Today’s talk is based on BIT Numerical Mathematics (2021) Joint work with Takeru Matsuda3 and Yuto Miyatake4 1. Earthquake Research Institute, The University of Tokyo 2. Graduate School of Information Science and Technology, The University of Tokyo 3. RIKEN Center for Brain Science 4. Cybermedia Center, Osaka University ©Shin-ichi Ito 01/30

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One-sentence summary We developed an adjoint-based method that enables exact Hessian computations up to a given oating point limit. Error in a Hessian matrix Naive method Row Column Our method 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito fl Row Column 01/30

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Outline Introduction Adjoint-based gradient and Hessian computations Sanz-Serna’s idea and exact Hessian computation Numerical veri cations fi ©Shin-ichi Ito 02/30

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Introduction Adjoint-based gradient and Hessian computations ©Shin-ichi Ito 03/30

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Adjoint method We consider an autonomous ordinary differential equation(ODE): Forward model and a minimization problem of a cost function de ned by the nal state via an gradient method using a gradient Adjoint method calculates Guess exactly by solving Optimum Adjoint model fi ©Shin-ichi Ito fi fi fi 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 05/30

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Second-order adjoint method Adjoint method is available for a rst-order derivative computation, while second-order adjoint (SOA) method is available Curvature = for a second-order derivative (Hessian matrix ) computation. Hessian matrix <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> d dt t = (r x f ) t Tangent linear model SOA model * These model are derived from the rst-order perturbation of the forward and adjoint models. SOA provides an exact Hessian-vector product for arbitrary vectors. SOA is available for optimization methods, uncertainty quanti cation, … ©Shin-ichi Ito fi fi fi 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 06/30

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Application
・Fast optimization
・Newton method

where

: Descent vector

・Nonlinear conjugate gradient method
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: Conjugate vector
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⌘ : Learning rate

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・Uncertainty quanti cation (UQ)
Solve

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C(✓)

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

<latexit sha1_base64="e075dcW3ikGHqSQBkD9Kp8T8K60=">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</latexit>

C(✓) =
<latexit sha1_base64="DD9Q1NVqN6uVEoKB6w8UDTBwSkI=">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</latexit>

H✓ˆ ⌃ = I

log p(✓)

Ito et al., (2016)

⌃ = H✓ˆ

<latexit sha1_base64="PwnF/nf4uHap9vKVMuzKoQmLtDs=">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</latexit>

via Krylov subspace methods

1

<latexit sha1_base64="FBPodAcjHCT9IQYGEkv/C2CyG0g=">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</latexit>

✓ˆ

✓
<latexit sha1_base64="6L/sfM20+IbPGS/8f9cvkF/lmnE=">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</latexit>

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

fi

地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」

©Shin-ichi Ito

07/30

8.

Question Adjoint and SOA methods give exact gradient and Hessian-vec. prod. if all of the models are solved analytically. In practice, all of the models are solved numerically, using time integral schemes, e.g., Runge—Kutta methods. → inexact due to discretization errors, machine errors Forward model Tangent linear (TL) model <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> Adjoint model ? d dt t ? = (r x f ) t SOA model ? For a given forward scheme, are there "optimal schemes” for other models? 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 08/30

9.

Question Adjoint and SOA methods give exact gradient and Hessian-vec. prod. if all of the models are solved analytically. In practice, all of the models are solved numerically, using time integral schemes, e.g., Runge—Kutta methods. → inexact due to discretization errors, machine errors Forward model Tangent linear (TL) model <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> Adjoint model d dt t ? = (r x f ) t SOA model Sanz-Serna (2016) ? For a given forward scheme, are there "optimal schemes” for other models? → Yes. There exists an optimal scheme for the adjoint model that can avoid any discretization errors. 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 09/30

10.

Question Adjoint and SOA methods give exact gradient and Hessian-vec. prod. if all of the models are solved analytically. In practice, all of the models are solved numerically, using time integral schemes, e.g., Runge—Kutta methods. → inexact due to discretization errors, machine errors Forward model Tangent linear (TL) model <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> Adjoint model d dt t = (r x f ) t SOA model Sanz-Serna (2016) Ours For a given forward scheme, are there "optimal schemes” for other models? → Yes. There exists a set of optimal schemes for all of these models that can avoid any discretization errors. 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 10/30

11.

Importance of exact Hessian computation * Numerical details will be explained later. Error in a Hessian matrix Row Column Row Naive method Column Our method Our method allows exact Hessian computation up to a given oating point limit. fl 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 11/30

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[beta]
Importance of exact Hessian computation
Convergence behavior when solving Hy
<latexit sha1_base64="Rm0o21olSgYXQrfkXukYcHF+pms=">AAALlnicbdbLctowFAZgp9c0lDZJN53philZJIswkHbabNrJPeROLgSSmMnIRgYFy3YsAQaP+yjdtq/Ut6kxTI7xsTfI+s4vXwZL0hyTCVks/pt59vzFy1evZ9/MZd5m372fX1i8FnbX1WlVt03brWtEUJNZtCqZNGndcSnhmklrWmd75LUedQWzrSs5cGiDk5bFDKYTGXbdzy/6KieyLQy/HAS5Qe5HTrufzxcLxejI4UZp0sgrk6NyvzBnqk1b73JqSd0kQtyVio5s+MSVTDdpkFW7gjpE75AWvQubFuFUNPzo5oNcXH3ChRhwLUh2ju4xMY401hs+s5yupJY+HZCeYVtSBNmsatG+bnNOrKavErfFiReMn9h2fNXlubDv16hTNRlnowhKMCslEXY+JbJqkxpqj+rL+dJKWKrZnq9qttkchXJL+dJS8FREsJMRj1XDqoHqWHXQJtYmKMVKQQ2sBmgLawu0jbUNyrAy0AesD6AdrB1QE6sJyrFyUAurBWpjtUEdrA7oI9ZHUBerCyqwClCJVYJ2sXZBe1h7oH2sfVAPqwc6wDoAHWIdxj6GTcybEN7CugW6jXUbdAfrDugu1l3QPax7oPtY90HLWMugB1gPQA+xHoIeYT0CPcZ6DHqC9QT0FOsp6BnWM9AK1groOdZz0AusF6CXWC9Br7BegVaxVkGvsV6D1rDWQOtY66A3WG9Ab7Hexj6GIXVTJp5ibFritJWyfKhRf2wWYeEil1IW9cemkzaVaWVRf2yWDXcSzbS6McQ+9fThhlOjeSnLgerFFoT0QabG4ClTncpjk12L8NQXEPXHLuUIZtopK4E6kdi/LXpXYY2RU6M2/Eujlz2WqB0k9xwuty0ajH6X1bYINyjUL5TWKQ/Y9OlKkMzJvp3IFaMcmz5PCbZdOrnkU+3qJJw8x2kj3Fmmh3ux82+pVzZYL/msUWVv+hQHBfOmc6uJYOx+s+H+tJTcjeLG9Vqh9KWwdv41v7E12anOKp+Uz8qyUlK+KxtKWakoVUVXPOW38kf5m/mY+ZnZzeyPS5/NTDIflKkjU/kPgrYwAA==</latexit>

=b

Failed…

Our method
Naive method

Success!

Iteration step
When using an incorrect scheme,
・Optimization may not converge to correct solution.
・Uncertainty quanti cation may fail.

fi

地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」

©Shin-ichi Ito

12/30

13.

Sanz-Serna’s idea and exact Hessian computation ©Shin-ichi Ito 13/30

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> Invariant Why can the analytical solution of the adjoint model provide the exact gradient ? A product of TL and adjoint variables is time-invariant d dt t !>⇣ ⇣ ⌘ d > > d > d ⌘ >> = ((r x tf ) t t )= t + t t t (r 0 x f ) t )> + x ft ) tt == ((r dt dt dt t > t + <latexit 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<latexit 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> t t = ⇣ TL model (r x f ) > ⌘ d t = 0 t = (r x f ) dt t <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> Adjoint model > 0 0 = r xt C(xt (✓)) , we obtain h i > > > > > ) = x (✓) + ✏ + O(✏ 2 ) (✓ x + ✏ C(xtt) = 0 r0✓ C(xt )t t t r xt C(xt ) = ((r✓ xt ) 0 ) r xt C(xt ) = 0 (r✓ xt ) r xt* h i (r ) = x > > > > t ✓ t 0 xt ) 0 ) r xt C(xt ) = 0 (r✓ xt ) r xt C(xt ) = 0 r✓ C(xt ) If we choose <latexit 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t <latexit 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0 = r✓ C(xt (✓)) 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 14/30

15.

Runge—Kutta family s-stage Runge—Kutta (RK) method h : stepsize Forward model <latexit sha1_base64="9gygmdn+QQW5JH9b9JNHqm5Ff9o=">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</latexit> aij , bi : weights Discretize ・Classical 4-stage RK (4) ・Explicit Euler (1) <latexit sha1_base64="zOx5lxzu5LRG/Eu0uRV0c9/20CY=">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</latexit> ・Heun (2) ・Adaptive stepsize RK: Bogacki–Shampine(2-3), Runge-Kutta-Fehlberg(4-5), Dormand-Prince(4-5), … 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito * (…) : order 15/30

16.

Sanz-Serna’s idea Solve forward models via a s-stage Runge—Kutta (RK) method n+1 <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> d dt = n +h s X bi dn,i , i=1 t = (r x f ) t <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 and solve adjoint model via another s-stage RK method Sanz-Serna (2016) provided a relation between so that <latexit sha1_base64="56Y7u9cyHM3ItBECtCsjHh8yGmo=">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</latexit> > n+1 n+1 = > n n 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 and exactly. ©Shin-ichi Ito 16/30

17.

Sanz-Serna’s idea > n+1 n+1 <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> = (i = 1, . . . , s) , bi = Bi > n n bi Ai j + B j a ji = Bi b j (i, j = 1, . . . , s) . <latexit sha1_base64="uQosLXqtBeyP+HxIWi313hFwsSg=">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</latexit> Proof n+1 > t t S( S ( n, <latexit sha1_base64="XmSl2xLXaWwqKtiG7O7vgQ/nH08=">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</latexit> n+1 , n+1 ) =h <latexit sha1_base64="9MyBerPi8WEfLAfKxmB7LM1wbI8=">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</latexit> n +h bi dn,i , i=1 S ( t, t) = <latexit sha1_base64="ewM5nHiBj4l+WrVe3LplPT2PlN8=">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</latexit> = s X s X Bi S ( n , ln,i ) + h i=1 s X n) s X i=1 bi S (dn,i , n) + h2 s X i, j=1 s X <latexit sha1_base64="9iBL9qrnU1tN47ptGOOSw/na9R4=">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</latexit> Bi b j S (dn,i , ln, j ) s X 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) . s X j=1 s X Bi S (Dn,i , ln,i ) h2 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 ) i=1 i, j=1 i=1 i, j=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 <latexit sha1_base64="2/pyQZVUGvbLqo7R/e2Rfwk/YKY=">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</latexit> =h <latexit sha1_base64="HwlIQueQQLu5JSIRyKq9jKTQSng=">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</latexit> 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 sha1_base64="czOaDA1r+iwrCCHadqX2wGcLVsc=">AAALhHicbdZJc5swFAdwkm5pXLdJe+zFU+eQzDQZk3TJpZ3s++IsXpLgyQgsbMUICMg2NkM/Qa/td+u3KcaePODBxZJ+7y9sxkhSbYO5olT6NzX97PmLl69mXs/m3uTfvpubf191ra6j0YpmGZZTV4lLDWbSimDCoHXboYSrBq2pne2R13rUcZllXouBTRuctEymM42IcOjyR+l+rlhaKUVXATfkSaMoTa7y/fysoTQtrcupKTSDuO6dXLJFwyeOYJpBg7zSdalNtA5p0buwaRJO3YYffdWgEFefcNcdcDVID3Ii2ql5hL7e8JlpdwU1tWRAeLplCjfI5xWT9jWLc2I2fYU4LU68wFdGs1m2rzi8EI79Gg0qBuNsFEEJZmYkwsGnRF5pUl3pUW2xKC+Fparl+YpqGc1RqLBQlBeCpyKCnYx4rCpWFVTDqoE2sTZBKVYKqmPVQVtYW6BtrG1QhpWBPmB9AO1g7YAaWA1QjpWDmlhNUAurBWpjtUEfsT6COlgdUBerCyqwCtAu1i5oD2sPtI+1D+ph9UAHWAegQ6zD2MuwiXkTwltYt0C3sW6D7mDdAd3Fugu6h3UPdB/rPugB1gPQQ6yHoEdYj0CPsR6DnmA9AT3Fegp6hvUM9BzrOWgZaxn0AusF6CXWS9ArrFeg11ivQStYK6BVrFXQGtYaaB1rHfQG6w3oLdbb2MswpE7GwlOKLUuctjK2DyUaj60iLNzkMsqi8dhy0qYiqywaj62y4bmhmVU3htirnj3dMDGbl7EdKF5sQ8ieJDEHz1jqFB5b7FqEZz6AaDx2K9tlhpWxEygTif3bomcV1ugFJWrDvzR62GOJ2kH6zOFwy6TB6HNRabvhAYX6K/I65QFLdpeCdE70rVSuFOVYsp8RbDt0csun2uVJON3HaT08R2aHe7H+t8w766yX/q1RZS/ZxUGXecnccioY+7758Hwqp0+juFFdXZHXVlYvvhQ3tiYn1Rnpo/RJWpRk6bu0IR1IZakiaZIu/Zb+SH9zL3Ofc2u5r+PS6alJ5oOUuHI//wOCACkN</latexit> s ⇣ X i, j=1 i, j=1 Bi b j bi Ai j ⌘ B j a ji S (dn,i , ln, j ) =0 <latexit sha1_base64="czOaDA1r+iwrCCHadqX2wGcLVsc=">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</latexit> 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 17/30

18.

Exact gradient computation Forward models solved by a s-stage Runge—Kutta (RK) method n+1 <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> d dt = n +h s X bi dn,i , i=1 t = (r x f ) t <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 and adjoint model solved by another s-stage RK method where 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> provide exact gradient as 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 18/30

19.

Adjoint consistency in time Continuous Continuous Take adjoint Discretize Discretize Discrete Take adjoint … G1 Substitute into def. of G2 G0 (Almost impossible to compute) In general, G1 and G2 are different. Neither G1 nor G2 gives an exact gradient (G0) > = because > n n+1 n+1 <latexit sha1_base64="56Y7u9cyHM3ItBECtCsjHh8yGmo=">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</latexit> 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 n in both cases. ©Shin-ichi Ito 19/30

20.

Adjoint consistency in time Continuous Continuous Take adjoint Discretize via RK Discrete Discretize via Sanz-Serna (RK) Take adjoint G1 … Substitute into def. of (Almost impossible to compute) G2 G0 Sanz-Serna’s scheme guarantees that G1 and G2 are identical They give an exact gradient (G0) since >n+1 n+1 = >n n <latexit sha1_base64="56Y7u9cyHM3ItBECtCsjHh8yGmo=">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</latexit> Open problems(?) Other types of schemes Consistency in space, i.e., PDEs (e.g., advection eq.) 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 20/30

21.

Towards exact Hessian computation Exact gradient can be computed by using Sanz-Serna’s scheme. Exact Hessian ??? Forward model Tangent linear (TL) model <latexit sha1_base64="Yn4fesY/G2osg9+bcjgQv8fTPeE=">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</latexit> Adjoint model d dt t = (r x f ) t SOA model Sanz-Serna (2016) 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ? ©Shin-ichi Ito 20/30

22.
[beta]
Towards exact Hessian computation
Our idea: Equivalence between SOA model and adjoint model

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

De ne augmented vectors:

d
dt

t

= (r x f )

t

and

Set of adjoint and SOA models constitutes a large adjoint system
→ Many techniques for adjoint method can be used for SOA method.
e.g., Sanz-Serna’s idea, adjoint code generator
fi

地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」

©Shin-ichi Ito

21/30

23.

Exact gradient and Hessian computations The augmented models solved by Sanz-Serna (2016), i.e., the augmented forward model solved by a s-stage RK method and the augmented adjoint model solved by another s-stage RK method where 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> provide exact gradient and Hessian computations. Note that our method itself is not limited to RK. 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 22/30

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Numerical veri cations - Inhomogeneous wave equation fi ©Shin-ichi Ito 23/30

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Inhomogeneous wave equation Validate our method through numerical test of data assimilation based on a one-dimensional inhomogeneous wave equation ∂2U(z, t) ∂ ∂ = E(z) U(z, t) ] ∂t 2 ∂z [ ∂z (discretized in space by nite volume method) Time integral scheme: Heun method (2-stage RK) Assume initial states of and are known but is not. Cost function: where fi 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 : in the above movie ©Shin-ichi Ito 24/30

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[beta]
Discretization in space
∂2U(z, t)
∂
∂
=
E(z)
U(z,
t)
]
∂t 2
∂z [
∂z

d 1 2 …

d-1 d 1

z

Discretization via nite volume method (staggered lattice)

d
U i = Vi
dt
d
Vi =
dt

<latexit sha1_base64="L+2BqlzWn0/1JS6vSP7Z5e5cY5I=">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</latexit>

(i = 1, . . . , d)
8
E 32 (U2 U1 ) Ed+ 12 (U1 Ud )
<
1
1 (Ui+1
1 (Ui
E
U
)
E
Ui 1 )
i
i+
i
2
2
2
z :
Ed+ 12 (U1 Ud ) Ed 12 (Ud Ud 1 )

(i = 1)
(i = 2, . . . , d
(i = d)

1)

d
Ei+ 12 = 0 (i = 1, . . . , d),
dt

fi

地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」

©Shin-ichi Ito

25/30

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Time integral schemes Heun method Augmented forward model Augmented adjoint model Naively Our method Heun method Pn,2 = pn+1 , Pn,2 = pn+1 , ⌘ 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 , <latexit sha1_base64="+B1N3A2zEwHO1V2IZxk8UhQuGJg=">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</latexit> Pn,1 = pn+1 + hl¯n,2 . ⌘ h ⇣¯ pn = pn+1 + ln,1 + l¯n,2 , 2 l¯n,2 = rqG(qn+1 )> Pn,2 , l¯n,1 = rqG(qn )> Pn,1 , <latexit sha1_base64="CBreDbPCesB9QaOp1LDWarNY89c=">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</latexit> Pn,1 = pn+1 + hl¯n,2 . 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 26/30

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Symmetry of Hessian matrix : Degree of asymmetry, which is zero if Our method Heun method is symmetric. Discretization errors resulting from Heun method : Optimum sol. Only accumulation of machine errors (e.g., round-off error) Our method reproduces the symmetry of Hessian without any discretization errors even when using a large step size. 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 27/30

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Convergence to optimum solution Optimizer : Regularized Newton method where Our method Heun method Our method x2 Heun method Number of adjoint evaluations Our method contributes to rapid convergence. 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 28/30

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Large-scale uncertainty quanti cation Slow-slip motion model * Damped EOM [Hirahara et al.(2019)] dτ = dt G dV ∙ (Vplate − V ) + (Vplate − Vlock) − 2c dt * Rate-and-state dependent friction law τ = τ0 + A log V + B log θ * Aging law dθ Vθ =1− dt L Results A, B, and L are spatially dependent Ito et al., in prep. Time integrator: Runge-Kutta-Fehlberg method UQ for frictional parameter elds • Uncertainty fields High Accuracy Low −1 HL,L L ©Shin-ichi Ito fi 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 fi Large Unc. of Unc. of A − B 𝖪 𝒢 Uncertainty −1 Hθ,θ −1 HV,V Unc. of A Small • Accuracy of dy parts is higher th silent parts beca dynamic data ha 29/30

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Conclusions * We proposed a method to compute an exact Hessian-vector product. * The fact that the second-order adjoint system can be reformulated to a part of a large adjoint system is the key point to obtain the exact Hessian-vector product based on the Sanz-Serna scheme. * Our method is capable of being applied to various types of ODEs. * Seismic imaging, aerodynamic design, structural materials, … * Neural ODE * Thank you for your attention 地震研特定研究 (B) 「固体地球現象の理解と予測に向けたデータ同化法の開発」 ©Shin-ichi Ito 30/30

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End ©Shin-ichi Ito 30/30