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revision 1.20 by edhill, Wed Apr 5 02:27:33 2006 UTC revision 1.23 by jmc, Mon Aug 30 23:09:19 2010 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  Author: Patrick Heimbach  Author: Patrick Heimbach
5    \label{ask_the_author:doc_ad_2}
6    
7  {\sf Automatic differentiation} (AD), also referred to as algorithmic  {\sf Automatic differentiation} (AD), also referred to as algorithmic
8  (or, more loosely, computational) differentiation, involves  (or, more loosely, computational) differentiation, involves
# Line 10  existing fully non-linear prognostic cod Line 11  existing fully non-linear prognostic cod
11  software tool is used that parses and transforms source files  software tool is used that parses and transforms source files
12  according to a set of linguistic and mathematical rules.  AD tools are  according to a set of linguistic and mathematical rules.  AD tools are
13  like source-to-source translators in that they parse a program code as  like source-to-source translators in that they parse a program code as
14  input and produce a new program code as output.  However, unlike a  input and produce a new program code as output
15    (we restrict our discussion to source-to-source tools, ignoring
16    operator-overloading tools).  However, unlike a
17  pure source-to-source translation, the output program represents a new  pure source-to-source translation, the output program represents a new
18  algorithm, such as the evaluation of the Jacobian, the Hessian, or  algorithm, such as the evaluation of the Jacobian, the Hessian, or
19  higher derivative operators.  In principle, a variety of derived  higher derivative operators.  In principle, a variety of derived
# Line 21  Model Compiler (TAMC) and its successor Line 24  Model Compiler (TAMC) and its successor
24  Algorithms in Fortran), developed by Ralf Giering (\cite{gie-kam:98},  Algorithms in Fortran), developed by Ralf Giering (\cite{gie-kam:98},
25  \cite{gie:99,gie:00}).  The first application of the adjoint of MITgcm  \cite{gie:99,gie:00}).  The first application of the adjoint of MITgcm
26  for sensitivity studies has been published by \cite{maro-eta:99}.  for sensitivity studies has been published by \cite{maro-eta:99}.
27  \cite{sta-eta:97,sta-eta:01} use MITgcm and its adjoint for ocean  \cite{stam-etal:97,stam-etal:02} use MITgcm and its adjoint for ocean
28  state estimation studies.  In the following we shall refer to TAMC and  state estimation studies.  In the following we shall refer to TAMC and
29  TAF synonymously, except were explicitly stated otherwise.  TAF synonymously, except were explicitly stated otherwise.
30    
31  TAMC exploits the chain rule for computing the first derivative of a  As of mid-2007 we are also able to generate fairly efficient
32    adjoint code of the MITgcm using a new, open-source AD tool,
33    called OpenAD (see \cite{naum-etal:06,utke-etal:08}.
34    This enables us for the first time to compare adjoint models
35    generated from different AD tools, providing an additional
36    accuracy check, complementary to finite-difference gradient checks.
37    OpenAD and its application to  MITgcm is described in detail
38    in section \ref{sec_ad_openad}.
39    
40    The AD tool exploits the chain rule for computing the first derivative of a
41  function with respect to a set of input variables.  Treating a given  function with respect to a set of input variables.  Treating a given
42  forward code as a composition of operations -- each line representing  forward code as a composition of operations -- each line representing
43  a compositional element, the chain rule is rigorously applied to the  a compositional element, the chain rule is rigorously applied to the
# Line 54  model output variable $\vec{v}=(v_1,\ldo Line 66  model output variable $\vec{v}=(v_1,\ldo
66  under consideration,  under consideration,
67  %  %
68  \begin{equation}  \begin{equation}
69  \begin{split}  \begin{aligned}
70  {\cal M} \, : & \, U \,\, \longrightarrow \, V \\  {\cal M} \, : & \, U \,\, \longrightarrow \, V \\
71  ~      & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \,  ~      & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \,
72  {\cal M}(\vec{u})  {\cal M}(\vec{u})
73  \label{fulloperator}  \label{fulloperator}
74  \end{split}  \end{aligned}
75  \end{equation}  \end{equation}
76  %  %
77  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.
# Line 139  w.r.t. their corresponding inner product Line 151  w.r.t. their corresponding inner product
151  $\left\langle \,\, , \,\, \right\rangle $  $\left\langle \,\, , \,\, \right\rangle $
152  %  %
153  \begin{equation}  \begin{equation}
154  \begin{split}  \begin{aligned}
155  {\cal J} & = \,  {\cal J} & = \,
156  {\cal J} |_{\vec{u}^{(0)}} \, + \,  {\cal J} |_{\vec{u}^{(0)}} \, + \,
157  \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
# Line 148  $\left\langle \,\, , \,\, \right\rangle Line 160  $\left\langle \,\, , \,\, \right\rangle
160  {\cal J} |_{\vec{v}^{(0)}} \, + \,  {\cal J} |_{\vec{v}^{(0)}} \, + \,
161  \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
162  \, + \, O(\delta \vec{v}^2)  \, + \, O(\delta \vec{v}^2)
163  \end{split}  \end{aligned}
164  \label{deljidentity}  \label{deljidentity}
165  \end{equation}  \end{equation}
166  %  %
# Line 189  the gradient $ \nabla _{u}{\cal J} $ can Line 201  the gradient $ \nabla _{u}{\cal J} $ can
201  invoking the adjoint $ M^{\ast } $ of the tangent linear model $ M $  invoking the adjoint $ M^{\ast } $ of the tangent linear model $ M $
202  %  %
203  \begin{equation}  \begin{equation}
204  \begin{split}  \begin{aligned}
205  \nabla _{u}{\cal J}^T |_{\vec{u}} &  \nabla _{u}{\cal J}^T |_{\vec{u}} &
206  = \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}}  \\  = \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}}  \\
207  ~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\  ~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\
208  ~ & = \, \delta \vec{u}^{\ast}  ~ & = \, \delta \vec{u}^{\ast}
209  \end{split}  \end{aligned}
210  \label{adjoint}  \label{adjoint}
211  \end{equation}  \end{equation}
212  %  %
# Line 242  $ \langle \, \nabla _{v}{\cal J}^T \, , Line 254  $ \langle \, \nabla _{v}{\cal J}^T \, ,
254  = \nabla_v {\cal J} \cdot \delta \vec{v} $ )  = \nabla_v {\cal J} \cdot \delta \vec{v} $ )
255  %  %
256  \begin{equation}  \begin{equation}
257  \begin{split}  \begin{aligned}
258  \nabla_v {\cal J} (M(\delta \vec{u})) & = \,  \nabla_v {\cal J} (M(\delta \vec{u})) & = \,
259  \nabla_v {\cal J} \cdot M_{\Lambda}  \nabla_v {\cal J} \cdot M_{\Lambda}
260  \cdot ...... \cdot M_{\lambda} \cdot ...... \cdot  \cdot ...... \cdot M_{\lambda} \cdot ...... \cdot
261  M_{1} \cdot M_{0} \cdot \delta \vec{u} \\  M_{1} \cdot M_{0} \cdot \delta \vec{u} \\
262  ~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\  ~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\
263  \end{split}  \end{aligned}
264  \label{forward}  \label{forward}
265  \end{equation}  \end{equation}
266  %  %
# Line 256  whereas in reverse mode we have Line 268  whereas in reverse mode we have
268  %  %
269  \begin{equation}  \begin{equation}
270  \boxed{  \boxed{
271  \begin{split}  \begin{aligned}
272  M^T ( \nabla_v {\cal J}^T) & = \,  M^T ( \nabla_v {\cal J}^T) & = \,
273  M_{0}^T \cdot M_{1}^T  M_{0}^T \cdot M_{1}^T
274  \cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot  \cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot
# Line 265  M_{\Lambda}^T \cdot \nabla_v {\cal J}^T Line 277  M_{\Lambda}^T \cdot \nabla_v {\cal J}^T
277  \cdot ...... \cdot  \cdot ...... \cdot
278  \nabla_{v^{(\lambda)}} {\cal J}^T \\  \nabla_{v^{(\lambda)}} {\cal J}^T \\
279  ~ & = \, \nabla_u {\cal J}^T  ~ & = \, \nabla_u {\cal J}^T
280  \end{split}  \end{aligned}
281  }  }
282  \label{reverse}  \label{reverse}
283  \end{equation}  \end{equation}
# Line 284  $ \vec{v}^{(\lambda)} $ at each intermed Line 296  $ \vec{v}^{(\lambda)} $ at each intermed
296  %  %
297  \begin{equation}  \begin{equation}
298  \boxed{  \boxed{
299  \begin{split}  \begin{aligned}
300  \nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}}  \nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}}
301  & = \,  & = \,
302  M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot  M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot
303  M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\  M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\
304  ~ & = \, \delta \vec{v}^{(\lambda) \, \ast}  ~ & = \, \delta \vec{v}^{(\lambda) \, \ast}
305  \end{split}  \end{aligned}
306  }  }
307  \end{equation}  \end{equation}
308  %  %
# Line 407  and the shorthand notation for the adjoi Line 419  and the shorthand notation for the adjoi
419  $ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}}  $ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}}
420  {\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $,  {\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $,
421  for intermediate components, yielding  for intermediate components, yielding
422    {\small
423  \begin{equation}  \begin{equation}
424  \small  \begin{aligned}
 \begin{split}  
425  \left(  \left(
426  \begin{array}{c}  \begin{array}{c}
427  \delta v^{(\lambda) \, \ast}_1 \\  \delta v^{(\lambda) \, \ast}_1 \\
# Line 454  for intermediate components, yielding Line 466  for intermediate components, yielding
466  \delta v^{\ast}_{n} \\  \delta v^{\ast}_{n} \\
467  \end{array}  \end{array}
468  \right)  \right)
469  \end{split}  \end{aligned}
470  \end{equation}  \end{equation}
471    }
472    
473  Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in  Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in
474  showing the advantage of the reverse over the forward mode  showing the advantage of the reverse over the forward mode
# Line 526  operator which maps the model state spac Line 539  operator which maps the model state spac
539  Then, $ \nabla_v {\cal J} $ takes the form  Then, $ \nabla_v {\cal J} $ takes the form
540  %  %
541  \begin{equation*}  \begin{equation*}
542  \begin{split}  \begin{aligned}
543  \nabla_v {\cal J}^T & = \, 2 \, \, H \cdot  \nabla_v {\cal J}^T & = \, 2 \, \, H \cdot
544  \left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\  \left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\
545  ~          & = \, 2 \sum_{j} \left\{ \sum_k  ~          & = \, 2 \sum_{j} \left\{ \sum_k
546  \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*}
551  %  %
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
# Line 652  $ n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1} Line 665  $ n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1}
665  %\psfrag{v_kn^lev2}{\mathinfigure{v_{k_{n}^{lev2}}}}  %\psfrag{v_kn^lev2}{\mathinfigure{v_{k_{n}^{lev2}}}}
666  %\psfrag{v_k1^lev1}{\mathinfigure{v_{k_{1}^{lev1}}}}  %\psfrag{v_k1^lev1}{\mathinfigure{v_{k_{1}^{lev1}}}}
667  %\psfrag{v_kn^lev1}{\mathinfigure{v_{k_{n}^{lev1}}}}  %\psfrag{v_kn^lev1}{\mathinfigure{v_{k_{n}^{lev1}}}}
668  %\mbox{\epsfig{file=part5/checkpointing.eps, width=0.8\textwidth}}  %\mbox{\epsfig{file=s_autodiff/figs/checkpointing.eps, width=0.8\textwidth}}
669  \resizebox{5.5in}{!}{\includegraphics{part5/checkpointing.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_autodiff/figs/checkpointing.eps}}
670  %\psfull  %\psfull
671  \end{center}  \end{center}
672  \caption{  \caption{
# Line 681  Schematic view of intermediate dump and Line 694  Schematic view of intermediate dump and
694    
695  In this section we describe in a general fashion  In this section we describe in a general fashion
696  the parts of the code that are relevant for automatic  the parts of the code that are relevant for automatic
697  differentiation using the software tool TAF.  differentiation using the software tool TAF.
698    Modifications to use OpenAD are described in \ref{sec_ad_openad}.
699    
700  \input{part5/doc_ad_the_model}  \input{s_autodiff/text/doc_ad_the_model}
701    
702  The basic flow is depicted in \ref{fig:adthemodel}.  The basic flow is depicted in \ref{fig:adthemodel}.
703  If CPP option \texttt{ALLOW\_AUTODIFF\_TAMC} is defined,  If CPP option \texttt{ALLOW\_AUTODIFF\_TAMC} is defined,
# Line 718  If gradient checks are to be performed, Line 732  If gradient checks are to be performed,
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
734  the gradient has been computed via the adjoint  the gradient has been computed via the adjoint
735  (cf. Section \ref{section_grdchk}).  (cf. Section \ref{sec:ad_gradient_check}).
736    
737  %------------------------------------------------------------------  %------------------------------------------------------------------
738    
# Line 728  the gradient has been computed via the a Line 742  the gradient has been computed via the a
742  In order to configure AD-related setups the following packages need  In order to configure AD-related setups the following packages need
743  to be enabled:  to be enabled:
744  {\it  {\it
745  \begin{table}[h!]  \begin{table}[!ht]
746  \begin{tabular}{l}  \begin{tabular}{l}
747  autodiff \\  autodiff \\
748  ctrl \\  ctrl \\
# Line 764  w.r.t. the 3-level checkpointing (see se Line 778  w.r.t. the 3-level checkpointing (see se
778    
779  %------------------------------------------------------------------  %------------------------------------------------------------------
780    
781  \subsection{Building the AD code  \subsection{Building the AD code using TAF
782  \label{section_ad_build}}  \label{section_ad_build}}
783    
784  The build process of an AD code is very similar to building  The build process of an AD code is very similar to building
# Line 772  the forward model. However, depending on Line 786  the forward model. However, depending on
786  to generate, and on which AD tool is available (TAF or TAMC),  to generate, and on which AD tool is available (TAF or TAMC),
787  the following {\tt make} targets are available:  the following {\tt make} targets are available:
788    
789  \begin{table}[h!]  \begin{table}[!ht]
790  {\footnotesize  {\footnotesize
791  \begin{tabular}{ccll}  \begin{tabular}{|ccll|}
792    \hline
793  ~ & {\it AD-target} & {\it output} & {\it description} \\  ~ & {\it AD-target} & {\it output} & {\it description} \\
794  \hline  \hline
795  \hline  \hline
# Line 793  generates code for $<$MODE$>$ using $<$T Line 808  generates code for $<$MODE$>$ using $<$T
808  ~ & ~ & ~ & and compiles all code \\  ~ & ~ & ~ & and compiles all code \\
809  ~ & ~ & ~ & (use of TAF is set as default) \\  ~ & ~ & ~ & (use of TAF is set as default) \\
810  \hline  \hline
 \hline  
811  \end{tabular}  \end{tabular}
812  }  }
813  \end{table}  \end{table}
# Line 802  Here, the following placeholders are use Line 816  Here, the following placeholders are use
816  %  %
817  \begin{itemize}  \begin{itemize}
818  %  %
819  \item [$<$TOOL$>$]  \item $<$TOOL$>$
820  %  %
821  \begin{itemize}  \begin{itemize}
822  %  %
# Line 811  Here, the following placeholders are use Line 825  Here, the following placeholders are use
825  %  %
826  \end{itemize}  \end{itemize}
827  %  %
828  \item [$<$MODE$>$]  \item $<$MODE$>$
829  %  %
830  \begin{itemize}  \begin{itemize}
831  %  %
# Line 853  The {\tt make <MODE>all} target consists Line 867  The {\tt make <MODE>all} target consists
867  \item  \item
868  A header file {\tt AD\_CONFIG.h} is generated which contains a CPP option  A header file {\tt AD\_CONFIG.h} is generated which contains a CPP option
869  on which code ought to be generated. Depending on the {\tt make} target,  on which code ought to be generated. Depending on the {\tt make} target,
870  the contents is  the contents is one of the following:
871  \begin{itemize}  \begin{itemize}
872  \item  \item
873  {\tt \#define ALLOW\_ADJOINT\_RUN}  {\tt \#define ALLOW\_ADJOINT\_RUN}
# Line 869  consisting of all {\tt .f} files that ar Line 883  consisting of all {\tt .f} files that ar
883  and all {\tt .flow} files that are part of the list {\bf AD\_FLOW\_FILES}.  and all {\tt .flow} files that are part of the list {\bf AD\_FLOW\_FILES}.
884  %  %
885  \item  \item
886  The AD tool is invoked with the {\bf <MODE>\_<TOOL>\_FLAGS}.  The AD tool is invoked with the {\tt <MODE>\_<TOOL>\_FLAGS}.
887  The default AD tool flags in {\tt genmake2} can be overrwritten by  The default AD tool flags in {\tt genmake2} can be overrwritten by
888  an {\tt adjoint\_options} file (similar to the platform-specific  an {\tt adjoint\_options} file (similar to the platform-specific
889  {\tt build\_options}, see Section ???.  {\tt build\_options}, see Section ???.
# Line 1002  The aspects relevant to the treatment of Line 1016  The aspects relevant to the treatment of
1016  are controlled by the package {\it pkg/ctrl} and will be treated  are controlled by the package {\it pkg/ctrl} and will be treated
1017  in the next section.  in the next section.
1018    
1019  \input{part5/doc_cost_flow}  \input{s_autodiff/text/doc_cost_flow}
1020    
1021  \subsubsection{Enabling the package}  \subsubsection{Enabling the package}
1022    
# Line 1092  Within this 'driver' routine, S/R are ca Line 1106  Within this 'driver' routine, S/R are ca
1106  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}).
1110  %  %
1111  \subsubsection{Finalize all contributions}  \subsubsection{Finalize all contributions}
1112  %  %
# Line 1117  from each contribution and sums over all Line 1131  from each contribution and sums over all
1131  \end{equation}  \end{equation}
1132  %  %
1133  The total cost function {\bf fc} will be the  The total cost function {\bf fc} will be the
1134  'dependent' variable in the argument list for TAMC, i.e.  'dependent' variable in the argument list for TAF, i.e.
1135  \begin{verbatim}  \begin{verbatim}
1136  tamc -output 'fc' ...  taf -output 'fc' ...
1137  \end{verbatim}  \end{verbatim}
1138    
1139  %%%% \end{document}  %%%% \end{document}
1140    
1141  \input{part5/doc_ad_the_main}  \input{s_autodiff/text/doc_ad_the_main}
1142    
1143  \subsection{The control variables (independent variables)  \subsection{The control variables (independent variables)
1144  \label{section_ctrl}}  \label{section_ctrl}}
# Line 1144  All aspects relevant to the treatment of Line 1158  All aspects relevant to the treatment of
1158  (parameter setting, initialization, perturbation)  (parameter setting, initialization, perturbation)
1159  are controlled by the package {\it pkg/ctrl}.  are controlled by the package {\it pkg/ctrl}.
1160    
1161  \input{part5/doc_ctrl_flow}  \input{s_autodiff/text/doc_ctrl_flow}
1162    
1163  \subsubsection{genmake and CPP options}  \subsubsection{genmake and CPP options}
1164  %  %
# Line 1234  and gradient are generated and initialis Line 1248  and gradient are generated and initialis
1248  %  %
1249  The dependency flow for differentiation w.r.t. the controls  The dependency flow for differentiation w.r.t. the controls
1250  starts with adding a perturbation onto the input variable,  starts with adding a perturbation onto the input variable,
1251  thus defining the independent or control variables for TAMC.  thus defining the independent or control variables for TAF.
1252  Three types of controls may be considered:  Three types of controls may be considered:
1253  %  %
1254  \begin{itemize}  \begin{itemize}
# Line 1255  temperature and salinity are initialised Line 1269  temperature and salinity are initialised
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
1284  sensitivity study this array is identical to zero.  sensitivity study this array is identical to zero.
1285  However, it's specification is essential in the context  However, it's specification is essential in the context
1286  of automatic differentiation since TAMC  of automatic differentiation since TAF
1287  treats the corresponding line in the code symbolically  treats the corresponding line in the code symbolically
1288  when determining the differentiation chain and its origin.  when determining the differentiation chain and its origin.
1289  Thus, the variable names are part of the argument list  Thus, the variable names are part of the argument list
1290  when calling TAMC:  when calling TAF:
1291  %  %
1292  \begin{verbatim}  \begin{verbatim}
1293  tamc -input 'xx_tr1 ...' ...  taf -input 'xx_tr1 ...' ...
1294  \end{verbatim}  \end{verbatim}
1295  %  %
1296  Now, as mentioned above, MITgcm avoids maintaining  Now, as mentioned above, MITgcm avoids maintaining
1297  an array for each control variable by reading the  an array for each control variable by reading the
1298  perturbation to a temporary array from file.  perturbation to a temporary array from file.
1299  To ensure the symbolic link to be recognized by TAMC, a scalar  To ensure the symbolic link to be recognized by TAF, a scalar
1300  dummy variable {\bf xx\_tr1\_dummy} is introduced  dummy variable {\bf xx\_tr1\_dummy} is introduced
1301  and an 'active read' routine of the adjoint support  and an 'active read' routine of the adjoint support
1302  package {\it pkg/autodiff} is invoked.  package {\it pkg/autodiff} is invoked.
1303  The read-procedure is tagged with the variable  The read-procedure is tagged with the variable
1304  {\bf xx\_tr1\_dummy} enabling TAMC to recognize the  {\bf xx\_tr1\_dummy} enabling TAF to recognize the
1305  initialization of the perturbation.  initialization of the perturbation.
1306  The modified call of TAMC thus reads  The modified call of TAF thus reads
1307  %  %
1308  \begin{verbatim}  \begin{verbatim}
1309  tamc -input 'xx_tr1_dummy ...' ...  taf -input 'xx_tr1_dummy ...' ...
1310  \end{verbatim}  \end{verbatim}
1311  %  %
1312  and the modified operation to (\ref{perturb})  and the modified operation to (\ref{perturb})
# Line 1487  u_{[k+1]} \, = \,  u_{[0]} \, + \, \Delt Line 1503  u_{[k+1]} \, = \,  u_{[0]} \, + \, \Delt
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}
# Line 1542  ad \, v_{[k]} (\delta {\cal J}) = Line 1558  ad \, v_{[k]} (\delta {\cal J}) =
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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