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% $Name$ |
% $Name$ |
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|
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Author: Patrick Heimbach |
Author: Patrick Heimbach |
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|
\label{ask_the_author:doc_ad_2} |
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{\sf Automatic differentiation} (AD), also referred to as algorithmic |
{\sf Automatic differentiation} (AD), also referred to as algorithmic |
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(or, more loosely, computational) differentiation, involves |
(or, more loosely, computational) differentiation, involves |
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under consideration, |
under consideration, |
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% |
% |
| 68 |
\begin{equation} |
\begin{equation} |
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\begin{split} |
\begin{aligned} |
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{\cal M} \, : & \, U \,\, \longrightarrow \, V \\ |
{\cal M} \, : & \, U \,\, \longrightarrow \, V \\ |
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~ & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \, |
~ & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \, |
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{\cal M}(\vec{u}) |
{\cal M}(\vec{u}) |
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\label{fulloperator} |
\label{fulloperator} |
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\end{split} |
\end{aligned} |
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\end{equation} |
\end{equation} |
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% |
% |
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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. |
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$\left\langle \,\, , \,\, \right\rangle $ |
$\left\langle \,\, , \,\, \right\rangle $ |
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% |
% |
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\begin{equation} |
\begin{equation} |
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\begin{split} |
\begin{aligned} |
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{\cal J} & = \, |
{\cal J} & = \, |
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{\cal J} |_{\vec{u}^{(0)}} \, + \, |
{\cal J} |_{\vec{u}^{(0)}} \, + \, |
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\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 |
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{\cal J} |_{\vec{v}^{(0)}} \, + \, |
{\cal J} |_{\vec{v}^{(0)}} \, + \, |
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\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 |
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\, + \, O(\delta \vec{v}^2) |
\, + \, O(\delta \vec{v}^2) |
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\end{split} |
\end{aligned} |
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\label{deljidentity} |
\label{deljidentity} |
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\end{equation} |
\end{equation} |
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% |
% |
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invoking the adjoint $ M^{\ast } $ of the tangent linear model $ M $ |
invoking the adjoint $ M^{\ast } $ of the tangent linear model $ M $ |
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% |
% |
| 203 |
\begin{equation} |
\begin{equation} |
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\begin{split} |
\begin{aligned} |
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\nabla _{u}{\cal J}^T |_{\vec{u}} & |
\nabla _{u}{\cal J}^T |_{\vec{u}} & |
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= \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}} \\ |
= \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}} \\ |
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~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\ |
~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\ |
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~ & = \, \delta \vec{u}^{\ast} |
~ & = \, \delta \vec{u}^{\ast} |
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\end{split} |
\end{aligned} |
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\label{adjoint} |
\label{adjoint} |
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\end{equation} |
\end{equation} |
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% |
% |
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= \nabla_v {\cal J} \cdot \delta \vec{v} $ ) |
= \nabla_v {\cal J} \cdot \delta \vec{v} $ ) |
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% |
% |
| 256 |
\begin{equation} |
\begin{equation} |
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\begin{split} |
\begin{aligned} |
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\nabla_v {\cal J} (M(\delta \vec{u})) & = \, |
\nabla_v {\cal J} (M(\delta \vec{u})) & = \, |
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\nabla_v {\cal J} \cdot M_{\Lambda} |
\nabla_v {\cal J} \cdot M_{\Lambda} |
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\cdot ...... \cdot M_{\lambda} \cdot ...... \cdot |
\cdot ...... \cdot M_{\lambda} \cdot ...... \cdot |
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M_{1} \cdot M_{0} \cdot \delta \vec{u} \\ |
M_{1} \cdot M_{0} \cdot \delta \vec{u} \\ |
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~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\ |
~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\ |
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\end{split} |
\end{aligned} |
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\label{forward} |
\label{forward} |
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\end{equation} |
\end{equation} |
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% |
% |
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% |
% |
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\begin{equation} |
\begin{equation} |
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\boxed{ |
\boxed{ |
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\begin{split} |
\begin{aligned} |
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M^T ( \nabla_v {\cal J}^T) & = \, |
M^T ( \nabla_v {\cal J}^T) & = \, |
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M_{0}^T \cdot M_{1}^T |
M_{0}^T \cdot M_{1}^T |
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\cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot |
\cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot |
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\cdot ...... \cdot |
\cdot ...... \cdot |
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\nabla_{v^{(\lambda)}} {\cal J}^T \\ |
\nabla_{v^{(\lambda)}} {\cal J}^T \\ |
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~ & = \, \nabla_u {\cal J}^T |
~ & = \, \nabla_u {\cal J}^T |
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\end{split} |
\end{aligned} |
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} |
} |
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\label{reverse} |
\label{reverse} |
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\end{equation} |
\end{equation} |
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% |
% |
| 297 |
\begin{equation} |
\begin{equation} |
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\boxed{ |
\boxed{ |
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\begin{split} |
\begin{aligned} |
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\nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}} |
\nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}} |
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& = \, |
& = \, |
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M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot |
M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot |
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M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\ |
M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\ |
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~ & = \, \delta \vec{v}^{(\lambda) \, \ast} |
~ & = \, \delta \vec{v}^{(\lambda) \, \ast} |
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\end{split} |
\end{aligned} |
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} |
} |
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\end{equation} |
\end{equation} |
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% |
% |
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$ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}} |
$ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}} |
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{\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $, |
{\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $, |
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for intermediate components, yielding |
for intermediate components, yielding |
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|
{\small |
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\begin{equation} |
\begin{equation} |
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\small |
\begin{aligned} |
|
\begin{split} |
|
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\left( |
\left( |
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\begin{array}{c} |
\begin{array}{c} |
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\delta v^{(\lambda) \, \ast}_1 \\ |
\delta v^{(\lambda) \, \ast}_1 \\ |
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\delta v^{\ast}_{n} \\ |
\delta v^{\ast}_{n} \\ |
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\end{array} |
\end{array} |
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\right) |
\right) |
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\end{split} |
\end{aligned} |
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\end{equation} |
\end{equation} |
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|
} |
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|
|
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Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in |
Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in |
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showing the advantage of the reverse over the forward mode |
showing the advantage of the reverse over the forward mode |
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Then, $ \nabla_v {\cal J} $ takes the form |
Then, $ \nabla_v {\cal J} $ takes the form |
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% |
% |
| 541 |
\begin{equation*} |
\begin{equation*} |
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\begin{split} |
\begin{aligned} |
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\nabla_v {\cal J}^T & = \, 2 \, \, H \cdot |
\nabla_v {\cal J}^T & = \, 2 \, \, H \cdot |
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\left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\ |
\left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\ |
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~ & = \, 2 \sum_{j} \left\{ \sum_k |
~ & = \, 2 \sum_{j} \left\{ \sum_k |
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\frac{\partial {\cal H}_k}{\partial v_{j}} |
\frac{\partial {\cal H}_k}{\partial v_{j}} |
| 547 |
\left( {\cal H}_k (\vec{v}) - d_k \right) |
\left( {\cal H}_k (\vec{v}) - d_k \right) |
| 548 |
\right\} \, {\vec{f}_{j}} \\ |
\right\} \, {\vec{f}_{j}} \\ |
| 549 |
\end{split} |
\end{aligned} |
| 550 |
\end{equation*} |
\end{equation*} |
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% |
% |
| 552 |
where $H_{kj} = \partial {\cal H}_k / \partial v_{j} $ is the |
where $H_{kj} = \partial {\cal H}_k / \partial v_{j} $ is the |
| 732 |
{\tt ALLOW\_GRADIENT\_CHECK} is defined. In this case |
{\tt ALLOW\_GRADIENT\_CHECK} is defined. In this case |
| 733 |
the driver routine {\it grdchk\_main} is called after |
the driver routine {\it grdchk\_main} is called after |
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the gradient has been computed via the adjoint |
the gradient has been computed via the adjoint |
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(cf. Section \ref{section_grdchk}). |
(cf. Section \ref{sec:ad_gradient_check}). |
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|
|
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%------------------------------------------------------------------ |
%------------------------------------------------------------------ |
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|
|
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In order to configure AD-related setups the following packages need |
In order to configure AD-related setups the following packages need |
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to be enabled: |
to be enabled: |
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{\it |
{\it |
| 745 |
\begin{table}[h!] |
\begin{table}[!ht] |
| 746 |
\begin{tabular}{l} |
\begin{tabular}{l} |
| 747 |
autodiff \\ |
autodiff \\ |
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ctrl \\ |
ctrl \\ |
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to generate, and on which AD tool is available (TAF or TAMC), |
to generate, and on which AD tool is available (TAF or TAMC), |
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the following {\tt make} targets are available: |
the following {\tt make} targets are available: |
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|
|
| 789 |
\begin{table}[h!] |
\begin{table}[!ht] |
| 790 |
{\footnotesize |
{\footnotesize |
| 791 |
\begin{tabular}{|ccll|} |
\begin{tabular}{|ccll|} |
| 792 |
\hline |
\hline |
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the chosen cost function contributions. |
the chosen cost function contributions. |
| 1107 |
In the present example ({\bf ALLOW\_COST\_TRACER}), |
In the present example ({\bf ALLOW\_COST\_TRACER}), |
| 1108 |
S/R {\it cost\_tracer} is called. |
S/R {\it cost\_tracer} is called. |
| 1109 |
It accumulates {\bf objf\_tracer} according to eqn. (\ref{???}). |
It accumulates {\bf objf\_tracer} according to eqn. (\ref{ask_the_author:doc_ad_2}). |
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% |
% |
| 1111 |
\subsubsection{Finalize all contributions} |
\subsubsection{Finalize all contributions} |
| 1112 |
% |
% |
| 1269 |
a perturbation anomaly is added to the field in S/R |
a perturbation anomaly is added to the field in S/R |
| 1270 |
{\it ctrl\_map\_ini} |
{\it ctrl\_map\_ini} |
| 1271 |
% |
% |
| 1272 |
|
%\begin{eqnarray} |
| 1273 |
\begin{equation} |
\begin{equation} |
| 1274 |
\begin{split} |
\begin{aligned} |
| 1275 |
u & = \, u_{[0]} \, + \, \Delta u \\ |
u & = \, u_{[0]} \, + \, \Delta u \\ |
| 1276 |
{\bf tr1}(...) & = \, {\bf tr1_{ini}}(...) \, + \, {\bf xx\_tr1}(...) |
{\bf tr1}(...) & = \, {\bf tr1_{ini}}(...) \, + \, {\bf xx\_tr1}(...) |
| 1277 |
\label{perturb} |
\label{perturb} |
| 1278 |
\end{split} |
\end{aligned} |
| 1279 |
\end{equation} |
\end{equation} |
| 1280 |
|
%\end{eqnarray} |
| 1281 |
% |
% |
| 1282 |
{\bf xx\_tr1} is a 3-dim. global array |
{\bf xx\_tr1} is a 3-dim. global array |
| 1283 |
holding the perturbation. In the case of a simple |
holding the perturbation. In the case of a simple |
| 1503 |
$ u_{[k+1]} $ then serves as input for a forward/adjoint run |
$ u_{[k+1]} $ then serves as input for a forward/adjoint run |
| 1504 |
to determine $ {\cal J} $ and $ \nabla _{u}{\cal J} $ at iteration step |
to determine $ {\cal J} $ and $ \nabla _{u}{\cal J} $ at iteration step |
| 1505 |
$ k+1 $. |
$ k+1 $. |
| 1506 |
Tab. \ref{???} sketches the flow between forward/adjoint model |
Tab. \ref{ask_the_author:doc_ad_2} sketches the flow between forward/adjoint model |
| 1507 |
and the minimization routine. |
and the minimization routine. |
| 1508 |
|
|
| 1509 |
|
{\scriptsize |
| 1510 |
\begin{eqnarray*} |
\begin{eqnarray*} |
|
\scriptsize |
|
| 1511 |
\begin{array}{ccccc} |
\begin{array}{ccccc} |
| 1512 |
u_{[0]} \,\, , \,\, \Delta u_{[k]} & ~ & ~ & ~ & ~ \\ |
u_{[0]} \,\, , \,\, \Delta u_{[k]} & ~ & ~ & ~ & ~ \\ |
| 1513 |
{\Big\downarrow} |
{\Big\downarrow} |
| 1558 |
~ & ~ & ~ & ~ & \Delta u_{[k+1]} \\ |
~ & ~ & ~ & ~ & \Delta u_{[k+1]} \\ |
| 1559 |
\end{array} |
\end{array} |
| 1560 |
\end{eqnarray*} |
\end{eqnarray*} |
| 1561 |
|
} |
| 1562 |
|
|
| 1563 |
The routines {\it ctrl\_unpack} and {\it ctrl\_pack} provide |
The routines {\it ctrl\_unpack} and {\it ctrl\_pack} provide |
| 1564 |
the link between the model and the minimization routine. |
the link between the model and the minimization routine. |
| 1565 |
As described in Section \ref{???} |
As described in Section \ref{ask_the_author:doc_ad_2} |
| 1566 |
the {\it unpack} and {\it pack} routines read and write |
the {\it unpack} and {\it pack} routines read and write |
| 1567 |
control and gradient {\it vectors} which are compressed |
control and gradient {\it vectors} which are compressed |
| 1568 |
to contain only wet points, in addition to the full |
to contain only wet points, in addition to the full |
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