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revision 1.18 by heimbach, Mon Aug 1 22:31:36 2005 UTC revision 1.24 by jmc, Tue Aug 31 20:56:21 2010 UTC
# Line 4  Line 4 
4  Author: Patrick Heimbach  Author: Patrick Heimbach
5    
6  {\sf Automatic differentiation} (AD), also referred to as algorithmic  {\sf Automatic differentiation} (AD), also referred to as algorithmic
7  (or, more loosely, computational) differentiation, involves  (or, more loosely, computational) differentiation, involves
8  automatically deriving code to calculate  automatically deriving code to calculate partial derivatives from an
9  partial derivatives from an existing fully non-linear prognostic code.  existing fully non-linear prognostic code.  (see \cite{gri:00}).  A
10  (see \cite{gri:00}).  software tool is used that parses and transforms source files
11  A software tool is used that parses and transforms source files  according to a set of linguistic and mathematical rules.  AD tools are
12  according to a set of linguistic and mathematical rules.  like source-to-source translators in that they parse a program code as
13  AD tools are like source-to-source translators in that  input and produce a new program code as output
14  they parse a program code as input and produce a new program code  (we restrict our discussion to source-to-source tools, ignoring
15  as output.  operator-overloading tools).  However, unlike a
16  However, unlike a pure source-to-source translation, the output program  pure source-to-source translation, the output program represents a new
17  represents a new algorithm, such as the evaluation of the  algorithm, such as the evaluation of the Jacobian, the Hessian, or
18  Jacobian, the Hessian, or higher derivative operators.  higher derivative operators.  In principle, a variety of derived
19  In principle, a variety of derived algorithms  algorithms can be generated automatically in this way.
20  can be generated automatically in this way.  
21    MITgcm has been adapted for use with the Tangent linear and Adjoint
22  The MITGCM has been adapted for use with the  Model Compiler (TAMC) and its successor TAF (Transformation of
23  Tangent linear and Adjoint Model Compiler (TAMC) and its successor TAF  Algorithms in Fortran), developed by Ralf Giering (\cite{gie-kam:98},
24  (Transformation of Algorithms in Fortran), developed  \cite{gie:99,gie:00}).  The first application of the adjoint of MITgcm
25  by Ralf Giering (\cite{gie-kam:98}, \cite{gie:99,gie:00}).  for sensitivity studies has been published by \cite{maro-eta:99}.
26  The first application of the adjoint of the MITGCM for sensitivity  \cite{stam-etal:97,stam-etal:02} use MITgcm and its adjoint for ocean
27  studies has been published by \cite{maro-eta:99}.  state estimation studies.  In the following we shall refer to TAMC and
28  \cite{sta-eta:97,sta-eta:01} use the MITGCM and its adjoint  TAF synonymously, except were explicitly stated otherwise.
29  for ocean state estimation studies.  
30  In the following we shall refer to TAMC and TAF synonymously,  As of mid-2007 we are also able to generate fairly efficient
31  except were explicitly stated otherwise.  adjoint code of the MITgcm using a new, open-source AD tool,
32    called OpenAD (see \cite{naum-etal:06,utke-etal:08}.
33  TAMC exploits the chain rule for computing the first  This enables us for the first time to compare adjoint models
34  derivative of a function with  generated from different AD tools, providing an additional
35  respect to a set of input variables.  accuracy check, complementary to finite-difference gradient checks.
36  Treating a given forward code as a composition of operations --  OpenAD and its application to  MITgcm is described in detail
37  each line representing a compositional element, the chain rule is  in section \ref{sec_ad_openad}.
38  rigorously applied to the code, line by line. The resulting  
39  tangent linear or adjoint code,  The AD tool exploits the chain rule for computing the first derivative of a
40  then, may be thought of as the composition in  function with respect to a set of input variables.  Treating a given
41  forward or reverse order, respectively, of the  forward code as a composition of operations -- each line representing
42  Jacobian matrices of the forward code's compositional elements.  a compositional element, the chain rule is rigorously applied to the
43    code, line by line. The resulting tangent linear or adjoint code,
44    then, may be thought of as the composition in forward or reverse
45    order, respectively, of the Jacobian matrices of the forward code's
46    compositional elements.
47    
48  %**********************************************************************  %**********************************************************************
49  \section{Some basic algebra}  \section{Some basic algebra}
# Line 61  model output variable $\vec{v}=(v_1,\ldo Line 65  model output variable $\vec{v}=(v_1,\ldo
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.
# Line 146  w.r.t. their corresponding inner product Line 150  w.r.t. their corresponding inner product
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
# Line 155  $\left\langle \,\, , \,\, \right\rangle Line 159  $\left\langle \,\, , \,\, \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  %  %
# Line 196  the gradient $ \nabla _{u}{\cal J} $ can Line 200  the gradient $ \nabla _{u}{\cal J} $ can
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  %  %
# Line 249  $ \langle \, \nabla _{v}{\cal J}^T \, , Line 253  $ \langle \, \nabla _{v}{\cal J}^T \, ,
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  %  %
# Line 263  whereas in reverse mode we have Line 267  whereas in reverse mode we have
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
# Line 272  M_{\Lambda}^T \cdot \nabla_v {\cal J}^T Line 276  M_{\Lambda}^T \cdot \nabla_v {\cal J}^T
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}
# Line 291  $ \vec{v}^{(\lambda)} $ at each intermed Line 295  $ \vec{v}^{(\lambda)} $ at each intermed
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  %  %
# Line 414  and the shorthand notation for the adjoi Line 418  and the shorthand notation for the adjoi
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 \\
# Line 461  for intermediate components, yielding Line 465  for intermediate components, yielding
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
# Line 533  operator which maps the model state spac Line 538  operator which maps the model state spac
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
# Line 659  $ n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1} Line 664  $ n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1}
664  %\psfrag{v_kn^lev2}{\mathinfigure{v_{k_{n}^{lev2}}}}  %\psfrag{v_kn^lev2}{\mathinfigure{v_{k_{n}^{lev2}}}}
665  %\psfrag{v_k1^lev1}{\mathinfigure{v_{k_{1}^{lev1}}}}  %\psfrag{v_k1^lev1}{\mathinfigure{v_{k_{1}^{lev1}}}}
666  %\psfrag{v_kn^lev1}{\mathinfigure{v_{k_{n}^{lev1}}}}  %\psfrag{v_kn^lev1}{\mathinfigure{v_{k_{n}^{lev1}}}}
667  %\mbox{\epsfig{file=part5/checkpointing.eps, width=0.8\textwidth}}  %\mbox{\epsfig{file=s_autodiff/figs/checkpointing.eps, width=0.8\textwidth}}
668  \resizebox{5.5in}{!}{\includegraphics{part5/checkpointing.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_autodiff/figs/checkpointing.eps}}
669  %\psfull  %\psfull
670  \end{center}  \end{center}
671  \caption{  \caption{
# Line 688  Schematic view of intermediate dump and Line 693  Schematic view of intermediate dump and
693    
694  In this section we describe in a general fashion  In this section we describe in a general fashion
695  the parts of the code that are relevant for automatic  the parts of the code that are relevant for automatic
696  differentiation using the software tool TAF.  differentiation using the software tool TAF.
697    Modifications to use OpenAD are described in \ref{sec_ad_openad}.
698    
699  \input{part5/doc_ad_the_model}  \input{s_autodiff/text/doc_ad_the_model}
700    
701  The basic flow is depicted in \ref{fig:adthemodel}.  The basic flow is depicted in \ref{fig:adthemodel}.
702  If CPP option {\tt ALLOW\_AUTODIFF\_TAMC} is defined, the driver routine  If CPP option \texttt{ALLOW\_AUTODIFF\_TAMC} is defined,
703    the driver routine
704  {\it the\_model\_main}, instead of calling {\it the\_main\_loop},  {\it the\_model\_main}, instead of calling {\it the\_main\_loop},
705  invokes the adjoint of this routine, {\it adthe\_main\_loop},  invokes the adjoint of this routine, {\it adthe\_main\_loop}
706  which is the toplevel routine in terms of automatic differentiation.  (case \texttt{\#define ALLOW\_ADJOINT\_RUN}), or
707  The routine {\it adthe\_main\_loop} has been generated by TAF.  the tangent linear of this routine {\it g\_the\_main\_loop}
708    (case \texttt{\#define ALLOW\_TANGENTLINEAR\_RUN}),
709    which are the toplevel routines in terms of automatic differentiation.
710    The routines {\it adthe\_main\_loop} or {\it g\_the\_main\_loop}
711    are generated by TAF.
712  It contains both the forward integration of the full model, the  It contains both the forward integration of the full model, the
713  cost function calculation,  cost function calculation,
714  any additional storing that is required for efficient checkpointing,  any additional storing that is required for efficient checkpointing,
# Line 720  If gradient checks are to be performed, Line 731  If gradient checks are to be performed,
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    
# Line 730  the gradient has been computed via the a Line 741  the gradient has been computed via the a
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 \\
# Line 766  w.r.t. the 3-level checkpointing (see se Line 777  w.r.t. the 3-level checkpointing (see se
777    
778  %------------------------------------------------------------------  %------------------------------------------------------------------
779    
780  \subsection{Building the AD code  \subsection{Building the AD code using TAF
781  \label{section_ad_build}}  \label{section_ad_build}}
782    
783  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 774  the forward model. However, depending on Line 785  the forward model. However, depending on
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
792  ~ & {\it AD-target} & {\it output} & {\it description} \\  ~ & {\it AD-target} & {\it output} & {\it description} \\
793  \hline  \hline
794  \hline  \hline
# Line 795  generates code for $<$MODE$>$ using $<$T Line 807  generates code for $<$MODE$>$ using $<$T
807  ~ & ~ & ~ & and compiles all code \\  ~ & ~ & ~ & and compiles all code \\
808  ~ & ~ & ~ & (use of TAF is set as default) \\  ~ & ~ & ~ & (use of TAF is set as default) \\
809  \hline  \hline
 \hline  
810  \end{tabular}  \end{tabular}
811  }  }
812  \end{table}  \end{table}
# Line 804  Here, the following placeholders are use Line 815  Here, the following placeholders are use
815  %  %
816  \begin{itemize}  \begin{itemize}
817  %  %
818  \item [$<$TOOL$>$]  \item $<$TOOL$>$
819  %  %
820  \begin{itemize}  \begin{itemize}
821  %  %
# Line 813  Here, the following placeholders are use Line 824  Here, the following placeholders are use
824  %  %
825  \end{itemize}  \end{itemize}
826  %  %
827  \item [$<$MODE$>$]  \item $<$MODE$>$
828  %  %
829  \begin{itemize}  \begin{itemize}
830  %  %
# Line 855  The {\tt make <MODE>all} target consists Line 866  The {\tt make <MODE>all} target consists
866  \item  \item
867  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
868  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,
869  the contents is  the contents is one of the following:
870  \begin{itemize}  \begin{itemize}
871  \item  \item
872  {\tt \#define ALLOW\_ADJOINT\_RUN}  {\tt \#define ALLOW\_ADJOINT\_RUN}
# Line 871  consisting of all {\tt .f} files that ar Line 882  consisting of all {\tt .f} files that ar
882  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}.
883  %  %
884  \item  \item
885  The AD tool is invoked with the {\bf <MODE>\_<TOOL>\_FLAGS}.  The AD tool is invoked with the {\tt <MODE>\_<TOOL>\_FLAGS}.
886  The default AD tool flags in {\tt genmake2} can be overrwritten by  The default AD tool flags in {\tt genmake2} can be overrwritten by
887  an {\tt adjoint\_options} file (similar to the platform-specific  an {\tt adjoint\_options} file (similar to the platform-specific
888  {\tt build\_options}, see Section ???.  {\tt build\_options}, see Section ???.
# Line 950  The flow directives for the core MITgcm Line 961  The flow directives for the core MITgcm
961  {\tt eesupp/src/} and {\tt model/src/}  {\tt eesupp/src/} and {\tt model/src/}
962  reside in {\tt pkg/autodiff/}.  reside in {\tt pkg/autodiff/}.
963  This directory also contains hand-written adjoint code  This directory also contains hand-written adjoint code
964  for the MITgcm WRAPPER (see Section ???).  for the MITgcm WRAPPER (section \ref{chap:sarch}).
965    
966  Flow directives for package-specific routines are contained in  Flow directives for package-specific routines are contained in
967  the corresponding package directories in the file  the corresponding package directories in the file
# Line 1004  The aspects relevant to the treatment of Line 1015  The aspects relevant to the treatment of
1015  are controlled by the package {\it pkg/ctrl} and will be treated  are controlled by the package {\it pkg/ctrl} and will be treated
1016  in the next section.  in the next section.
1017    
1018  \input{part5/doc_cost_flow}  \input{s_autodiff/text/doc_cost_flow}
1019    
1020  \subsubsection{Enabling the package}  \subsubsection{Enabling the package}
1021    
# Line 1094  Within this 'driver' routine, S/R are ca Line 1105  Within this 'driver' routine, S/R are ca
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  %  %
# Line 1119  from each contribution and sums over all Line 1130  from each contribution and sums over all
1130  \end{equation}  \end{equation}
1131  %  %
1132  The total cost function {\bf fc} will be the  The total cost function {\bf fc} will be the
1133  'dependent' variable in the argument list for TAMC, i.e.  'dependent' variable in the argument list for TAF, i.e.
1134  \begin{verbatim}  \begin{verbatim}
1135  tamc -output 'fc' ...  taf -output 'fc' ...
1136  \end{verbatim}  \end{verbatim}
1137    
1138  %%%% \end{document}  %%%% \end{document}
1139    
1140  \input{part5/doc_ad_the_main}  \input{s_autodiff/text/doc_ad_the_main}
1141    
1142  \subsection{The control variables (independent variables)  \subsection{The control variables (independent variables)
1143  \label{section_ctrl}}  \label{section_ctrl}}
# Line 1146  All aspects relevant to the treatment of Line 1157  All aspects relevant to the treatment of
1157  (parameter setting, initialization, perturbation)  (parameter setting, initialization, perturbation)
1158  are controlled by the package {\it pkg/ctrl}.  are controlled by the package {\it pkg/ctrl}.
1159    
1160  \input{part5/doc_ctrl_flow}  \input{s_autodiff/text/doc_ctrl_flow}
1161    
1162  \subsubsection{genmake and CPP options}  \subsubsection{genmake and CPP options}
1163  %  %
# Line 1204  and their gradients: {\it ctrl\_unpack} Line 1215  and their gradients: {\it ctrl\_unpack}
1215  \\  \\
1216  %  %
1217  Two important issues related to the handling of the control  Two important issues related to the handling of the control
1218  variables in the MITGCM need to be addressed.  variables in MITgcm need to be addressed.
1219  First, in order to save memory, the control variable arrays  First, in order to save memory, the control variable arrays
1220  are not kept in memory, but rather read from file and added  are not kept in memory, but rather read from file and added
1221  to the initial fields during the model initialization phase.  to the initial fields during the model initialization phase.
# Line 1236  and gradient are generated and initialis Line 1247  and gradient are generated and initialis
1247  %  %
1248  The dependency flow for differentiation w.r.t. the controls  The dependency flow for differentiation w.r.t. the controls
1249  starts with adding a perturbation onto the input variable,  starts with adding a perturbation onto the input variable,
1250  thus defining the independent or control variables for TAMC.  thus defining the independent or control variables for TAF.
1251  Three types of controls may be considered:  Three types of controls may be considered:
1252  %  %
1253  \begin{itemize}  \begin{itemize}
# Line 1257  temperature and salinity are initialised Line 1268  temperature and salinity are initialised
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
1283  sensitivity study this array is identical to zero.  sensitivity study this array is identical to zero.
1284  However, it's specification is essential in the context  However, it's specification is essential in the context
1285  of automatic differentiation since TAMC  of automatic differentiation since TAF
1286  treats the corresponding line in the code symbolically  treats the corresponding line in the code symbolically
1287  when determining the differentiation chain and its origin.  when determining the differentiation chain and its origin.
1288  Thus, the variable names are part of the argument list  Thus, the variable names are part of the argument list
1289  when calling TAMC:  when calling TAF:
1290  %  %
1291  \begin{verbatim}  \begin{verbatim}
1292  tamc -input 'xx_tr1 ...' ...  taf -input 'xx_tr1 ...' ...
1293  \end{verbatim}  \end{verbatim}
1294  %  %
1295  Now, as mentioned above, the MITGCM avoids maintaining  Now, as mentioned above, MITgcm avoids maintaining
1296  an array for each control variable by reading the  an array for each control variable by reading the
1297  perturbation to a temporary array from file.  perturbation to a temporary array from file.
1298  To ensure the symbolic link to be recognized by TAMC, a scalar  To ensure the symbolic link to be recognized by TAF, a scalar
1299  dummy variable {\bf xx\_tr1\_dummy} is introduced  dummy variable {\bf xx\_tr1\_dummy} is introduced
1300  and an 'active read' routine of the adjoint support  and an 'active read' routine of the adjoint support
1301  package {\it pkg/autodiff} is invoked.  package {\it pkg/autodiff} is invoked.
1302  The read-procedure is tagged with the variable  The read-procedure is tagged with the variable
1303  {\bf xx\_tr1\_dummy} enabling TAMC to recognize the  {\bf xx\_tr1\_dummy} enabling TAF to recognize the
1304  initialization of the perturbation.  initialization of the perturbation.
1305  The modified call of TAMC thus reads  The modified call of TAF thus reads
1306  %  %
1307  \begin{verbatim}  \begin{verbatim}
1308  tamc -input 'xx_tr1_dummy ...' ...  taf -input 'xx_tr1_dummy ...' ...
1309  \end{verbatim}  \end{verbatim}
1310  %  %
1311  and the modified operation to (\ref{perturb})  and the modified operation to (\ref{perturb})
# Line 1489  u_{[k+1]} \, = \,  u_{[0]} \, + \, \Delt Line 1502  u_{[k+1]} \, = \,  u_{[0]} \, + \, \Delt
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}
# Line 1544  ad \, v_{[k]} (\delta {\cal J}) = Line 1557  ad \, v_{[k]} (\delta {\cal J}) =
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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