| 65 |
under consideration, |
under consideration, |
| 66 |
% |
% |
| 67 |
\begin{equation} |
\begin{equation} |
| 68 |
\begin{split} |
\begin{aligned} |
| 69 |
{\cal M} \, : & \, U \,\, \longrightarrow \, V \\ |
{\cal M} \, : & \, U \,\, \longrightarrow \, V \\ |
| 70 |
~ & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \, |
~ & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \, |
| 71 |
{\cal M}(\vec{u}) |
{\cal M}(\vec{u}) |
| 72 |
\label{fulloperator} |
\label{fulloperator} |
| 73 |
\end{split} |
\end{aligned} |
| 74 |
\end{equation} |
\end{equation} |
| 75 |
% |
% |
| 76 |
The vectors $ \vec{u} \in U $ and $ v \in V $ may be represented w.r.t. |
The vectors $ \vec{u} \in U $ and $ v \in V $ may be represented w.r.t. |
| 150 |
$\left\langle \,\, , \,\, \right\rangle $ |
$\left\langle \,\, , \,\, \right\rangle $ |
| 151 |
% |
% |
| 152 |
\begin{equation} |
\begin{equation} |
| 153 |
\begin{split} |
\begin{aligned} |
| 154 |
{\cal J} & = \, |
{\cal J} & = \, |
| 155 |
{\cal J} |_{\vec{u}^{(0)}} \, + \, |
{\cal J} |_{\vec{u}^{(0)}} \, + \, |
| 156 |
\left\langle \, \nabla _{u}{\cal J}^T |_{\vec{u}^{(0)}} \, , \, \delta \vec{u} \, \right\rangle |
\left\langle \, \nabla _{u}{\cal J}^T |_{\vec{u}^{(0)}} \, , \, \delta \vec{u} \, \right\rangle |
| 159 |
{\cal J} |_{\vec{v}^{(0)}} \, + \, |
{\cal J} |_{\vec{v}^{(0)}} \, + \, |
| 160 |
\left\langle \, \nabla _{v}{\cal J}^T |_{\vec{v}^{(0)}} \, , \, \delta \vec{v} \, \right\rangle |
\left\langle \, \nabla _{v}{\cal J}^T |_{\vec{v}^{(0)}} \, , \, \delta \vec{v} \, \right\rangle |
| 161 |
\, + \, O(\delta \vec{v}^2) |
\, + \, O(\delta \vec{v}^2) |
| 162 |
\end{split} |
\end{aligned} |
| 163 |
\label{deljidentity} |
\label{deljidentity} |
| 164 |
\end{equation} |
\end{equation} |
| 165 |
% |
% |
| 200 |
invoking the adjoint $ M^{\ast } $ of the tangent linear model $ M $ |
invoking the adjoint $ M^{\ast } $ of the tangent linear model $ M $ |
| 201 |
% |
% |
| 202 |
\begin{equation} |
\begin{equation} |
| 203 |
\begin{split} |
\begin{aligned} |
| 204 |
\nabla _{u}{\cal J}^T |_{\vec{u}} & |
\nabla _{u}{\cal J}^T |_{\vec{u}} & |
| 205 |
= \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}} \\ |
= \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}} \\ |
| 206 |
~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\ |
~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\ |
| 207 |
~ & = \, \delta \vec{u}^{\ast} |
~ & = \, \delta \vec{u}^{\ast} |
| 208 |
\end{split} |
\end{aligned} |
| 209 |
\label{adjoint} |
\label{adjoint} |
| 210 |
\end{equation} |
\end{equation} |
| 211 |
% |
% |
| 253 |
= \nabla_v {\cal J} \cdot \delta \vec{v} $ ) |
= \nabla_v {\cal J} \cdot \delta \vec{v} $ ) |
| 254 |
% |
% |
| 255 |
\begin{equation} |
\begin{equation} |
| 256 |
\begin{split} |
\begin{aligned} |
| 257 |
\nabla_v {\cal J} (M(\delta \vec{u})) & = \, |
\nabla_v {\cal J} (M(\delta \vec{u})) & = \, |
| 258 |
\nabla_v {\cal J} \cdot M_{\Lambda} |
\nabla_v {\cal J} \cdot M_{\Lambda} |
| 259 |
\cdot ...... \cdot M_{\lambda} \cdot ...... \cdot |
\cdot ...... \cdot M_{\lambda} \cdot ...... \cdot |
| 260 |
M_{1} \cdot M_{0} \cdot \delta \vec{u} \\ |
M_{1} \cdot M_{0} \cdot \delta \vec{u} \\ |
| 261 |
~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\ |
~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\ |
| 262 |
\end{split} |
\end{aligned} |
| 263 |
\label{forward} |
\label{forward} |
| 264 |
\end{equation} |
\end{equation} |
| 265 |
% |
% |
| 267 |
% |
% |
| 268 |
\begin{equation} |
\begin{equation} |
| 269 |
\boxed{ |
\boxed{ |
| 270 |
\begin{split} |
\begin{aligned} |
| 271 |
M^T ( \nabla_v {\cal J}^T) & = \, |
M^T ( \nabla_v {\cal J}^T) & = \, |
| 272 |
M_{0}^T \cdot M_{1}^T |
M_{0}^T \cdot M_{1}^T |
| 273 |
\cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot |
\cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot |
| 276 |
\cdot ...... \cdot |
\cdot ...... \cdot |
| 277 |
\nabla_{v^{(\lambda)}} {\cal J}^T \\ |
\nabla_{v^{(\lambda)}} {\cal J}^T \\ |
| 278 |
~ & = \, \nabla_u {\cal J}^T |
~ & = \, \nabla_u {\cal J}^T |
| 279 |
\end{split} |
\end{aligned} |
| 280 |
} |
} |
| 281 |
\label{reverse} |
\label{reverse} |
| 282 |
\end{equation} |
\end{equation} |
| 295 |
% |
% |
| 296 |
\begin{equation} |
\begin{equation} |
| 297 |
\boxed{ |
\boxed{ |
| 298 |
\begin{split} |
\begin{aligned} |
| 299 |
\nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}} |
\nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}} |
| 300 |
& = \, |
& = \, |
| 301 |
M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot |
M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot |
| 302 |
M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\ |
M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\ |
| 303 |
~ & = \, \delta \vec{v}^{(\lambda) \, \ast} |
~ & = \, \delta \vec{v}^{(\lambda) \, \ast} |
| 304 |
\end{split} |
\end{aligned} |
| 305 |
} |
} |
| 306 |
\end{equation} |
\end{equation} |
| 307 |
% |
% |
| 418 |
$ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}} |
$ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}} |
| 419 |
{\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $, |
{\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $, |
| 420 |
for intermediate components, yielding |
for intermediate components, yielding |
| 421 |
|
{\small |
| 422 |
\begin{equation} |
\begin{equation} |
| 423 |
\small |
\begin{aligned} |
|
\begin{split} |
|
| 424 |
\left( |
\left( |
| 425 |
\begin{array}{c} |
\begin{array}{c} |
| 426 |
\delta v^{(\lambda) \, \ast}_1 \\ |
\delta v^{(\lambda) \, \ast}_1 \\ |
| 465 |
\delta v^{\ast}_{n} \\ |
\delta v^{\ast}_{n} \\ |
| 466 |
\end{array} |
\end{array} |
| 467 |
\right) |
\right) |
| 468 |
\end{split} |
\end{aligned} |
| 469 |
\end{equation} |
\end{equation} |
| 470 |
|
} |
| 471 |
|
|
| 472 |
Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in |
Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in |
| 473 |
showing the advantage of the reverse over the forward mode |
showing the advantage of the reverse over the forward mode |
| 538 |
Then, $ \nabla_v {\cal J} $ takes the form |
Then, $ \nabla_v {\cal J} $ takes the form |
| 539 |
% |
% |
| 540 |
\begin{equation*} |
\begin{equation*} |
| 541 |
\begin{split} |
\begin{aligned} |
| 542 |
\nabla_v {\cal J}^T & = \, 2 \, \, H \cdot |
\nabla_v {\cal J}^T & = \, 2 \, \, H \cdot |
| 543 |
\left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\ |
\left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\ |
| 544 |
~ & = \, 2 \sum_{j} \left\{ \sum_k |
~ & = \, 2 \sum_{j} \left\{ \sum_k |
| 545 |
\frac{\partial {\cal H}_k}{\partial v_{j}} |
\frac{\partial {\cal H}_k}{\partial v_{j}} |
| 546 |
\left( {\cal H}_k (\vec{v}) - d_k \right) |
\left( {\cal H}_k (\vec{v}) - d_k \right) |
| 547 |
\right\} \, {\vec{f}_{j}} \\ |
\right\} \, {\vec{f}_{j}} \\ |
| 548 |
\end{split} |
\end{aligned} |
| 549 |
\end{equation*} |
\end{equation*} |
| 550 |
% |
% |
| 551 |
where $H_{kj} = \partial {\cal H}_k / \partial v_{j} $ is the |
where $H_{kj} = \partial {\cal H}_k / \partial v_{j} $ is the |
| 731 |
{\tt ALLOW\_GRADIENT\_CHECK} is defined. In this case |
{\tt ALLOW\_GRADIENT\_CHECK} is defined. In this case |
| 732 |
the driver routine {\it grdchk\_main} is called after |
the driver routine {\it grdchk\_main} is called after |
| 733 |
the gradient has been computed via the adjoint |
the gradient has been computed via the adjoint |
| 734 |
(cf. Section \ref{section_grdchk}). |
(cf. Section \ref{sec:ad_gradient_check}). |
| 735 |
|
|
| 736 |
%------------------------------------------------------------------ |
%------------------------------------------------------------------ |
| 737 |
|
|
| 741 |
In order to configure AD-related setups the following packages need |
In order to configure AD-related setups the following packages need |
| 742 |
to be enabled: |
to be enabled: |
| 743 |
{\it |
{\it |
| 744 |
\begin{table}[h!] |
\begin{table}[!ht] |
| 745 |
\begin{tabular}{l} |
\begin{tabular}{l} |
| 746 |
autodiff \\ |
autodiff \\ |
| 747 |
ctrl \\ |
ctrl \\ |
| 785 |
to generate, and on which AD tool is available (TAF or TAMC), |
to generate, and on which AD tool is available (TAF or TAMC), |
| 786 |
the following {\tt make} targets are available: |
the following {\tt make} targets are available: |
| 787 |
|
|
| 788 |
\begin{table}[h!] |
\begin{table}[!ht] |
| 789 |
{\footnotesize |
{\footnotesize |
| 790 |
\begin{tabular}{|ccll|} |
\begin{tabular}{|ccll|} |
| 791 |
\hline |
\hline |
| 1105 |
the chosen cost function contributions. |
the chosen cost function contributions. |
| 1106 |
In the present example ({\bf ALLOW\_COST\_TRACER}), |
In the present example ({\bf ALLOW\_COST\_TRACER}), |
| 1107 |
S/R {\it cost\_tracer} is called. |
S/R {\it cost\_tracer} is called. |
| 1108 |
It accumulates {\bf objf\_tracer} according to eqn. (\ref{???}). |
It accumulates {\bf objf\_tracer} according to eqn. (ref:ask-the-author). |
| 1109 |
% |
% |
| 1110 |
\subsubsection{Finalize all contributions} |
\subsubsection{Finalize all contributions} |
| 1111 |
% |
% |
| 1268 |
a perturbation anomaly is added to the field in S/R |
a perturbation anomaly is added to the field in S/R |
| 1269 |
{\it ctrl\_map\_ini} |
{\it ctrl\_map\_ini} |
| 1270 |
% |
% |
| 1271 |
|
%\begin{eqnarray} |
| 1272 |
\begin{equation} |
\begin{equation} |
| 1273 |
\begin{split} |
\begin{aligned} |
| 1274 |
u & = \, u_{[0]} \, + \, \Delta u \\ |
u & = \, u_{[0]} \, + \, \Delta u \\ |
| 1275 |
{\bf tr1}(...) & = \, {\bf tr1_{ini}}(...) \, + \, {\bf xx\_tr1}(...) |
{\bf tr1}(...) & = \, {\bf tr1_{ini}}(...) \, + \, {\bf xx\_tr1}(...) |
| 1276 |
\label{perturb} |
\label{perturb} |
| 1277 |
\end{split} |
\end{aligned} |
| 1278 |
\end{equation} |
\end{equation} |
| 1279 |
|
%\end{eqnarray} |
| 1280 |
% |
% |
| 1281 |
{\bf xx\_tr1} is a 3-dim. global array |
{\bf xx\_tr1} is a 3-dim. global array |
| 1282 |
holding the perturbation. In the case of a simple |
holding the perturbation. In the case of a simple |
| 1502 |
$ u_{[k+1]} $ then serves as input for a forward/adjoint run |
$ u_{[k+1]} $ then serves as input for a forward/adjoint run |
| 1503 |
to determine $ {\cal J} $ and $ \nabla _{u}{\cal J} $ at iteration step |
to determine $ {\cal J} $ and $ \nabla _{u}{\cal J} $ at iteration step |
| 1504 |
$ k+1 $. |
$ k+1 $. |
| 1505 |
Tab. \ref{???} sketches the flow between forward/adjoint model |
Tab. ref:ask-the-author sketches the flow between forward/adjoint model |
| 1506 |
and the minimization routine. |
and the minimization routine. |
| 1507 |
|
|
| 1508 |
|
{\scriptsize |
| 1509 |
\begin{eqnarray*} |
\begin{eqnarray*} |
|
\scriptsize |
|
| 1510 |
\begin{array}{ccccc} |
\begin{array}{ccccc} |
| 1511 |
u_{[0]} \,\, , \,\, \Delta u_{[k]} & ~ & ~ & ~ & ~ \\ |
u_{[0]} \,\, , \,\, \Delta u_{[k]} & ~ & ~ & ~ & ~ \\ |
| 1512 |
{\Big\downarrow} |
{\Big\downarrow} |
| 1557 |
~ & ~ & ~ & ~ & \Delta u_{[k+1]} \\ |
~ & ~ & ~ & ~ & \Delta u_{[k+1]} \\ |
| 1558 |
\end{array} |
\end{array} |
| 1559 |
\end{eqnarray*} |
\end{eqnarray*} |
| 1560 |
|
} |
| 1561 |
|
|
| 1562 |
The routines {\it ctrl\_unpack} and {\it ctrl\_pack} provide |
The routines {\it ctrl\_unpack} and {\it ctrl\_pack} provide |
| 1563 |
the link between the model and the minimization routine. |
the link between the model and the minimization routine. |
| 1564 |
As described in Section \ref{???} |
As described in Section ref:ask-the-author |
| 1565 |
the {\it unpack} and {\it pack} routines read and write |
the {\it unpack} and {\it pack} routines read and write |
| 1566 |
control and gradient {\it vectors} which are compressed |
control and gradient {\it vectors} which are compressed |
| 1567 |
to contain only wet points, in addition to the full |
to contain only wet points, in addition to the full |
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