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-revision 1.1 by adcroft,
Wed Aug  8 16:16:26 2001 UTC
+revision 1.19 by heimbach,
Tue Aug  2 22:26:58 2005 UTC
 Line 1
  % $Header$
  % $Name$
+ Author: Patrick Heimbach
  {\sf Automatic differentiation} (AD), also referred to as algorithmic
  (or, more loosely, computational) differentiation, involves
  automatically deriving code to calculate
-Line 18 
 In principle, a variety of derived algor
+Line 20 
 In principle, a variety of derived algor
  can be generated automatically in this way.
  The MITGCM has been adapted for use with the
- Tangent linear and Adjoint Model Compiler (TAMC) and its succssor TAF
+ Tangent linear and Adjoint Model Compiler (TAMC) and its successor TAF
  (Transformation of Algorithms in Fortran), developed
  by Ralf Giering (\cite{gie-kam:98}, \cite{gie:99,gie:00}).
- The first application of the adjoint of the MITGCM for senistivity
+ The first application of the adjoint of the MITGCM for sensitivity
  studies has been published by \cite{maro-eta:99}.
  \cite{sta-eta:97,sta-eta:01} use the MITGCM and its adjoint
  for ocean state estimation studies.
+ In the following we shall refer to TAMC and TAF synonymously,
+ except were explicitly stated otherwise.
  TAMC exploits the chain rule for computing the first
  derivative of a function with
  respect to a set of input variables.
  Treating a given forward code as a composition of operations --
- each line representing a compositional element -- the chain rule is
+ each line representing a compositional element, the chain rule is
  rigorously applied to the code, line by line. The resulting
  tangent linear or adjoint code,
  then, may be thought of as the composition in
  forward or reverse order, respectively, of the
- Jacobian matrices of the forward code compositional elements.
+ Jacobian matrices of the forward code's compositional elements.
  %**********************************************************************
  \section{Some basic algebra}
  \label{sec_ad_algebra}
+ \begin{rawhtml}
+ <!-- CMIREDIR:sec_ad_algebra: -->
+ \end{rawhtml}
  %**********************************************************************
  Let $ \cal{M} $ be a general nonlinear, model, i.e. a
-Line 50 
 $\vec{u}=(u_1,\ldots,u_m)$
+Line 57 
 $\vec{u}=(u_1,\ldots,u_m)$
  such as forcing functions) to the $n$-dimensional space
  $V \subset I\!\!R^n$ of
  model output variable $\vec{v}=(v_1,\ldots,v_n)$
- (model state, model diagnostcs, objective function, ...)
+ (model state, model diagnostics, objective function, ...)
  under consideration,
  %
  \begin{equation}
-Line 105 
 In contrast to the full nonlinear model
+Line 112 
 In contrast to the full nonlinear model
  $ M $ is just a matrix
  which can readily be used to find the forward sensitivity of $\vec{v}$ to
  perturbations in  $u$,
- but if there are very many input variables $(>>O(10^{6})$ for
+ but if there are very many input variables $(\gg O(10^{6})$ for
  large-scale oceanographic application), it quickly becomes
  prohibitive to proceed directly as in (\ref{tangent_linear}),
  if the impact of each component $ {\bf e_{i}} $ is to be assessed.
-Line 130 
 or a measure of some model-to-data misfi
+Line 137 
 or a measure of some model-to-data misfi
  \label{compo}
  \end{eqnarray}
  %
- The linear approximation of $ {\cal J} $,
+ The perturbation of $ {\cal J} $ around a fixed point $ {\cal J}_0 $,
  \[
- {\cal J} \, \approx \, {\cal J}_0 \, + \, \delta {\cal J}
+ {\cal J} \, = \, {\cal J}_0 \, + \, \delta {\cal J}
  \]
  can be expressed in both bases of $ \vec{u} $ and $ \vec{v} $
  w.r.t. their corresponding inner product
-Line 152 
 $\left\langle \,\, , \,\, \right\rangle
+Line 159 
 $\left\langle \,\, , \,\, \right\rangle
  \label{deljidentity}
  \end{equation}
  %
- (note, that the gradient $ \nabla f $ is a pseudo-vector, therefore
+ (note, that the gradient $ \nabla f $ is a co-vector, therefore
  its transpose is required in the above inner product).
  Then, using the representation of
  $ \delta {\cal J} =
-Line 168 
 transpose of $ A $,
+Line 175 
 transpose of $ A $,
  \[
  A^{\ast} \, = \, A^T
  \]
- and from eq. (\ref{tangent_linear}), we note that
+ and from eq. (\ref{tangent_linear}), (\ref{deljidentity}),
+ we note that
  (omitting $|$'s):
  %
  \begin{equation}
-Line 204 
 the adjoint variable of the model state
+Line 212 
 the adjoint variable of the model state
  $ \delta \vec{u}^{\ast} $ the adjoint variable of the control variable $ \vec{u} $.
  The {\sf reverse} nature of the adjoint calculation can be readily
- seen as follows. Let us decompose ${\cal J}(u)$, thus:
+ seen as follows.
+ Consider a model integration which consists of $ \Lambda $
+ consecutive operations
+ $ {\cal M}_{\Lambda} (  {\cal M}_{\Lambda-1} (
+ ...... ( {\cal M}_{\lambda} (
+ ......
+ ( {\cal M}_{1} ( {\cal M}_{0}(\vec{u}) )))) $,
+ where the ${\cal M}$'s could be the elementary steps, i.e. single lines
+ in the code of the model, or successive time steps of the
+ model integration,
+ starting at step 0 and moving up to step $\Lambda$, with intermediate
+ ${\cal M}_{\lambda} (\vec{u}) = \vec{v}^{(\lambda+1)}$ and final
+ ${\cal M}_{\Lambda} (\vec{u}) = \vec{v}^{(\Lambda+1)} = \vec{v}$.
+ Let ${\cal J}$ be a cost function which explicitly depends on the
+ final state $\vec{v}$ only
+ (this restriction is for clarity reasons only).
+ %
+ ${\cal J}(u)$ may be decomposed according to:
  %
  \begin{equation}
  {\cal J}({\cal M}(\vec{u})) \, = \,
-Line 215 
 seen as follows. Let us decompose ${\cal
+Line 240 
 seen as follows. Let us decompose ${\cal
  \label{compos}
  \end{equation}
  %
- where the ${\cal M}$'s could be the elementary steps, i.e. single lines
+ Then, according to the chain rule, the forward calculation reads,
- in the code of the model,
+ in terms of the Jacobi matrices
- starting at step 0 and moving up to step $\Lambda$, with intermediate
- ${\cal M}_{\lambda} (\vec{u}) = \vec{v}^{(\lambda+1)}$ and final
- ${\cal M}_{\Lambda} (\vec{u}) = \vec{v}^{(\Lambda+1)} = \vec{v}$
- Then, according to the chain rule the forward calculation reads in
- terms of the Jacobi matrices
  (we've omitted the $ | $'s which, nevertheless are important
  to the aspect of {\it tangent} linearity;
- note also that per definition
+ note also that by definition
  $ \langle \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{v} \, \rangle
  = \nabla_v {\cal J} \cdot \delta \vec{v} $ )
  %
-Line 259 
 M_{\Lambda}^T \cdot \nabla_v {\cal J}^T
+Line 279 
 M_{\Lambda}^T \cdot \nabla_v {\cal J}^T
  %
  clearly expressing the reverse nature of the calculation.
  Eq. (\ref{reverse}) is at the heart of automatic adjoint compilers.
- The intermediate steps $\lambda$ in
+ If the intermediate steps $\lambda$ in
  eqn. (\ref{compos}) -- (\ref{reverse})
- could represent the model state (forward or adjoint) at each
+ represent the model state (forward or adjoint) at each
- intermediate time step in which case
+ intermediate time step as noted above, then correspondingly,
- $ {\cal M}(\vec{v}^{(\lambda)}) = \vec{v}^{(\lambda+1)} $, and correspondingly,
+ $ M^T (\delta \vec{v}^{(\lambda) \, \ast}) =
- $ M^T (\delta \vec{v}^{(\lambda) \, \ast}) = \delta \vec{v}^{(\lambda-1) \, \ast} $,
+ \delta \vec{v}^{(\lambda-1) \, \ast} $ for the adjoint variables.
- but they can also be viewed more generally as
+ It thus becomes evident that the adjoint calculation also
- single lines of code in the numerical algorithm.
+ yields the adjoint of each model state component
- In both cases it becomes evident that the adjoint calculation
+ $ \vec{v}^{(\lambda)} $ at each intermediate step $ \lambda $, namely
- yields at the same time the adjoint of each model state component
- $ \vec{v}^{(\lambda)} $ at each intermediate step $ l $, namely
  %
  \begin{equation}
  \boxed{
-Line 285 
 M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \c
+Line 303 
 M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \c
  %
  in close analogy to eq. (\ref{adjoint})
  We note in passing that that the $\delta \vec{v}^{(\lambda) \, \ast}$
- are the Lagrange multipliers of the model state $ \vec{v}^{(\lambda)}$.
+ are the Lagrange multipliers of the model equations which determine
+ $ \vec{v}^{(\lambda)}$.
- In coponents, eq. (\ref{adjoint}) reads as follows.
+ In components, eq. (\ref{adjoint}) reads as follows.
  Let
  \[
  \begin{array}{rclcrcl}
-Line 308 
 Let
+Line 327 
 Let
  \end{array}
  \]
  denote the perturbations in $\vec{u}$ and $\vec{v}$, respectively,
- and their adjoint varaiables;
+ and their adjoint variables;
  further
  \[
  M \, = \, \left(
-Line 395 
 and the shorthand notation for the adjoi
+Line 414 
 and the shorthand notation for the adjoi
  $ \delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}}
  {\cal J}^T $, $ j = 1, \ldots , n_{\lambda} $,
  for intermediate components, yielding
- \[
+ \begin{equation}
- \footnotesize
+ \small
+ \begin{split}
  \left(
  \begin{array}{c}
  \delta v^{(\lambda) \, \ast}_1 \\
-Line 404 
 for intermediate components, yielding
+Line 424 
 for intermediate components, yielding
  \delta v^{(\lambda) \, \ast}_{n_{\lambda}} \\
  \end{array}
  \right)
- \, = \,
+ \, = &
  \left(
  \begin{array}{ccc}
  \frac{\partial ({\cal M}_{\lambda})_1}{\partial v^{(\lambda)}_1}
- & \ldots &
+ & \ldots \,\, \ldots &
  \frac{\partial ({\cal M}_{\lambda})_{n_{\lambda+1}}}{\partial v^{(\lambda)}_1} \\
  \vdots & ~ & \vdots \\
  \frac{\partial ({\cal M}_{\lambda})_1}{\partial v^{(\lambda)}_{n_{\lambda}}}
- & \ldots  &
+ & \ldots \,\, \ldots  &
  \frac{\partial ({\cal M}_{\lambda})_{n_{\lambda+1}}}{\partial v^{(\lambda)}_{n_{\lambda}}} \\
  \end{array}
  \right)
- %
  \cdot
  %
+ \\ ~ & ~
+ \\ ~ &
+ %
  \left(
  \begin{array}{ccc}
  \frac{\partial ({\cal M}_{\lambda+1})_1}{\partial v^{(\lambda+1)}_1}
-Line 431 
 for intermediate components, yielding
+Line 453 
 for intermediate components, yielding
  \frac{\partial ({\cal M}_{\lambda+1})_{n_{\lambda+2}}}{\partial v^{(\lambda+1)}_{n_{\lambda+1}}} \\
  \end{array}
  \right)
- \cdot \ldots \ldots \cdot
+ \cdot \, \ldots \, \cdot
  \left(
  \begin{array}{c}
  \delta v^{\ast}_1 \\
-Line 439 
 for intermediate components, yielding
+Line 461 
 for intermediate components, yielding
  \delta v^{\ast}_{n} \\
  \end{array}
  \right)
- \]
+ \end{split}
+ \end{equation}
  Eq. (\ref{forward}) and (\ref{reverse}) are perhaps clearest in
  showing the advantage of the reverse over the forward mode
-Line 450 
 variables $u$
+Line 473 
 variables $u$
  {\it all} intermediate states $ \vec{v}^{(\lambda)} $) are sought.
  In order to be able to solve for each component of the gradient
  $ \partial {\cal J} / \partial u_{i} $ in (\ref{forward})
- a forward calulation has to be performed for each component seperately,
+ a forward calculation has to be performed for each component separately,
  i.e. $ \delta \vec{u} = \delta u_{i} {\vec{e}_{i}} $
  for  the $i$-th forward calculation.
  Then, (\ref{forward}) represents the
-Line 460 
 In contrast, eq. (\ref{reverse}) yields
+Line 483 
 In contrast, eq. (\ref{reverse}) yields
  gradient $\nabla _{u}{\cal J}$ (and all intermediate gradients
  $\nabla _{v^{(\lambda)}}{\cal J}$) within a single reverse calculation.
- Note, that in case $ {\cal J} $ is a vector-valued function
+ Note, that if $ {\cal J} $ is a vector-valued function
  of dimension $ l > 1 $,
  eq. (\ref{reverse}) has to be modified according to
  \[
-Line 468 
 M^T \left( \nabla_v {\cal J}^T \left(\de
+Line 491 
 M^T \left( \nabla_v {\cal J}^T \left(\de
  \, = \,
  \nabla_u {\cal J}^T \cdot \delta \vec{J}
  \]
- where now $ \delta \vec{J} \in I\!\!R $ is a vector of dimenison $ l $.
+ where now $ \delta \vec{J} \in I\!\!R^l $ is a vector of
+ dimension $ l $.
  In this case $ l $ reverse simulations have to be performed
  for each $ \delta J_{k}, \,\, k = 1, \ldots, l $.
  Then, the reverse mode is more efficient as long as
  $ l < n $, otherwise the forward mode is preferable.
- Stricly, the reverse mode is called adjoint mode only for
+ Strictly, the reverse mode is called adjoint mode only for
  $ l = 1 $.
  A detailed analysis of the underlying numerical operations
-Line 503 
 operator onto the $j$-th component ${\bf
+Line 527 
 operator onto the $j$-th component ${\bf
  \paragraph{Example 2:
  $ {\cal J} = \langle \, {\cal H}(\vec{v}) - \vec{d} \, ,
   \, {\cal H}(\vec{v}) - \vec{d} \, \rangle $} ~ \\
- The cost function represents the quadratic model vs.data misfit.
+ The cost function represents the quadratic model vs. data misfit.
  Here, $ \vec{d} $ is the data vector and $ {\cal H} $ represents the
  operator which maps the model state space onto the data space.
  Then, $ \nabla_v {\cal J} $ takes the form
-Line 534 
 H \cdot \left( {\cal H}(\vec{v}) - \vec{
+Line 558 
 H \cdot \left( {\cal H}(\vec{v}) - \vec{
  We note an important aspect of the forward vs. reverse
  mode calculation.
- Because of the locality of the derivative,
+ Because of the local character of the derivative
+ (a derivative is defined w.r.t. a point along the trajectory),
  the intermediate results of the model trajectory
  $\vec{v}^{(\lambda+1)}={\cal M}_{\lambda}(v^{(\lambda)})$
- are needed to evaluate the intermediate Jacobian
+ may be required to evaluate the intermediate Jacobian
  $M_{\lambda}|_{\vec{v}^{(\lambda)}} \, \delta \vec{v}^{(\lambda)} $.
+ This is the case e.g. for nonlinear expressions
+ (momentum advection, nonlinear equation of state), state-dependent
+ conditional statements (parameterization schemes).
  In the forward mode, the intermediate results are required
  in the same order as computed by the full forward model ${\cal M}$,
- in the reverse mode they are required in the reverse order.
+ but in the reverse mode they are required in the reverse order.
  Thus, in the reverse mode the trajectory of the forward model
  integration ${\cal M}$ has to be stored to be available in the reverse
- calculation. Alternatively, the model state would have to be
+ calculation. Alternatively, the complete model state up to the
- recomputed whenever its value is required.
+ point of evaluation has to be recomputed whenever its value is required.
  A method to balance the amount of recomputations vs.
  storage requirements is called {\sf checkpointing}
- (e.g. \cite{res-eta:98}).
+ (e.g. \cite{gri:92}, \cite{res-eta:98}).
- It is depicted in Fig. ... for a 3-level checkpointing
+ It is depicted in \ref{fig:3levelcheck} for a 3-level checkpointing
- [as concrete example, we give explicit numbers for a 3-day
+ [as an example, we give explicit numbers for a 3-day
  integration with a 1-hourly timestep in square brackets].
  \begin{itemize}
  %
-Line 559 
 integration with a 1-hourly timestep in
+Line 587 
 integration with a 1-hourly timestep in
  In a first step, the model trajectory is subdivided into
  $ {n}^{lev3} $ subsections [$ {n}^{lev3} $=3 1-day intervals],
  with the label $lev3$ for this outermost loop.
- The model is then integrated over the full trajectory,
+ The model is then integrated along the full trajectory,
- and the model state stored only at every $ k_{i}^{lev3} $-th timestep
+ and the model state stored to disk only at every $ k_{i}^{lev3} $-th timestep
  [i.e. 3 times, at
  $ i = 0,1,2 $ corresponding to $ k_{i}^{lev3} = 0, 24, 48 $].
+ In addition, the cost function is computed, if needed.
  %
  \item [$lev2$]
- In a second step each subsection is itself divided into
+ In a second step each subsection itself is divided into
- $ {n}^{lev2} $ subsubsections
+ $ {n}^{lev2} $ subsections
  [$ {n}^{lev2} $=4 6-hour intervals per subsection].
  The model picks up at the last outermost dumped state
- $ v_{k_{n}^{lev3}} $ and is integrated forward in time over
+ $ v_{k_{n}^{lev3}} $ and is integrated forward in time along
  the last subsection, with the label $lev2$ for this
  intermediate loop.
- The model state is now stored only at every $ k_{i}^{lev2} $-th
+ The model state is now stored to disk at every $ k_{i}^{lev2} $-th
  timestep
  [i.e. 4 times, at
  $ i = 0,1,2,3 $ corresponding to $ k_{i}^{lev2} = 48, 54, 60, 66 $].
  %
  \item [$lev1$]
- Finally, the mode picks up at the last intermediate dump state
+ Finally, the model picks up at the last intermediate dump state
- $ v_{k_{n}^{lev2}} $ and is integrated forward in time over
+ $ v_{k_{n}^{lev2}} $ and is integrated forward in time along
- the last subsubsection, with the label $lev1$ for this
+ the last subsection, with the label $lev1$ for this
  intermediate loop.
- Within this subsubsection only, the model state is stored
+ Within this sub-subsection only, parts of the model state is stored
- at every timestep
+ to memory at every timestep
  [i.e. every hour $ i=0,...,5$ corresponding to
  $ k_{i}^{lev1} = 66, 67, \ldots, 71 $].
- Thus, the  final state $ v_n = v_{k_{n}^{lev1}} $ is reached
+ The  final state $ v_n = v_{k_{n}^{lev1}} $ is reached
- and the model state of all peceeding timesteps over the last
+ and the model state of all preceding timesteps along the last
- subsubsections are available, enabling integration backwards
+ innermost subsection are available, enabling integration backwards
- in time over the last subsubsection.
+ in time along the last subsection.
- Thus, the adjoint can be computed over this last
+ The adjoint can thus be computed along this last
- subsubsection $k_{n}^{lev2}$.
+ subsection $k_{n}^{lev2}$.
  %
  \end{itemize}
  %
  This procedure is repeated consecutively for each previous
- subsubsection $k_{n-1}^{lev2}, \ldots, k_{1}^{lev2} $
+ subsection $k_{n-1}^{lev2}, \ldots, k_{1}^{lev2} $
  carrying the adjoint computation to the initial time
  of the subsection $k_{n}^{lev3}$.
  Then, the procedure is repeated for the previous subsection
-Line 607 
 $k_{1}^{lev3}$.
+Line 636 
 $k_{1}^{lev3}$.
  For the full model trajectory of
  $ n^{lev3} \cdot n^{lev2} \cdot n^{lev1} $ timesteps
  the required storing of the model state was significantly reduced to
- $ n^{lev1} + n^{lev2} + n^{lev3} $
+ $ n^{lev2} + n^{lev3} $ to disk and roughly $ n^{lev1} $ to memory
  [i.e. for the 3-day integration with a total oof 72 timesteps
- the model state was stored 13 times].
+ the model state was stored 7 times to disk and roughly 6 times
+ to memory].
  This saving in memory comes at a cost of a required
 full forward integrations of the model (one for each
  checkpointing level).
- The balance of storage vs. recomputation certainly depends
+ The optimal balance of storage vs. recomputation certainly depends
- on the computing resources available.
+ on the computing resources available and may be adjusted by
+ adjusting the partitioning among the
+ $ n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1} $.
  \begin{figure}[t!]
- \centering
+ \begin{center}
  %\psdraft
- \psfrag{v_k1^lev3}{\mathinfigure{v_{k_{1}^{lev3}}}}
+ %\psfrag{v_k1^lev3}{\mathinfigure{v_{k_{1}^{lev3}}}}
- \psfrag{v_kn-1^lev3}{\mathinfigure{v_{k_{n-1}^{lev3}}}}
+ %\psfrag{v_kn-1^lev3}{\mathinfigure{v_{k_{n-1}^{lev3}}}}
- \psfrag{v_kn^lev3}{\mathinfigure{v_{k_{n}^{lev3}}}}
+ %\psfrag{v_kn^lev3}{\mathinfigure{v_{k_{n}^{lev3}}}}
- \psfrag{v_k1^lev2}{\mathinfigure{v_{k_{1}^{lev2}}}}
+ %\psfrag{v_k1^lev2}{\mathinfigure{v_{k_{1}^{lev2}}}}
- \psfrag{v_kn-1^lev2}{\mathinfigure{v_{k_{n-1}^{lev2}}}}
+ %\psfrag{v_kn-1^lev2}{\mathinfigure{v_{k_{n-1}^{lev2}}}}
- \psfrag{v_kn^lev2}{\mathinfigure{v_{k_{n}^{lev2}}}}
+ %\psfrag{v_kn^lev2}{\mathinfigure{v_{k_{n}^{lev2}}}}
- \psfrag{v_k1^lev1}{\mathinfigure{v_{k_{1}^{lev1}}}}
+ %\psfrag{v_k1^lev1}{\mathinfigure{v_{k_{1}^{lev1}}}}
- \psfrag{v_kn^lev1}{\mathinfigure{v_{k_{n}^{lev1}}}}
+ %\psfrag{v_kn^lev1}{\mathinfigure{v_{k_{n}^{lev1}}}}
- \mbox{\epsfig{file=part5/checkpointing.eps, width=0.8\textwidth}}
+ %\mbox{\epsfig{file=part5/checkpointing.eps, width=0.8\textwidth}}
+ \resizebox{5.5in}{!}{\includegraphics{part5/checkpointing.eps}}
  %\psfull
- \caption
+ \end{center}
- {Schematic view of intermediate dump and restart for
+ \caption{
+ Schematic view of intermediate dump and restart for
 -level checkpointing.}
- \label{fig:erswns}
+ \label{fig:3levelcheck}
  \end{figure}
- \subsection{Optimal perturbations}
+ % \subsection{Optimal perturbations}
- \label{optpert}
+ % \label{sec_optpert}
- \subsection{Error covariance estimate and Hessian matrix}
+ % \subsection{Error covariance estimate and Hessian matrix}
- \label{sec_hessian}
+ % \label{sec_hessian}
  \newpage
  %**********************************************************************
- \section{AD-specific setup by example: sensitivity of carbon sequestration}
+ \section{TLM and ADM generation in general}
- \label{sec_ad_setup_ex}
+ \label{sec_ad_setup_gen}
+ \begin{rawhtml}
+ <!-- CMIREDIR:sec_ad_setup_gen: -->
+ \end{rawhtml}
  %**********************************************************************
- The MITGCM has been adapted to enable AD using TAMC or TAF
+ In this section we describe in a general fashion
- (we'll refer to TAMC and TAF interchangeably, except where
+ the parts of the code that are relevant for automatic
- distinctions are explicitly mentioned).
+ differentiation using the software tool TAF.
- The present description, therefore, is specific to the
- use of TAMC as AD tool.
+ \input{part5/doc_ad_the_model}
- The following sections describe the steps which are necessary to
- generate a tangent linear or adjoint model of the MITGCM.
+ The basic flow is depicted in \ref{fig:adthemodel}.
- We take as an example the sensitivity of carbon sequestration
+ If CPP option \texttt{ALLOW\_AUTODIFF\_TAMC} is defined,
- in the ocean.
+ the driver routine
- The AD-relevant hooks in the code are sketched in
+ {\it the\_model\_main}, instead of calling {\it the\_main\_loop},
- \reffig{adthemodel}, \reffig{adthemain}.
+ invokes the adjoint of this routine, {\it adthe\_main\_loop}
+ (case \texttt{\#define ALLOW\_ADJOINT\_RUN}), or
- \subsection{Overview of the experiment}
+ the tangent linear of this routine {\it g\_the\_main\_loop}
+ (case \texttt{\#define ALLOW\_TANGENTLINEAR\_RUN}),
- We describe an adjoint sensitivity analysis of outgassing from
+ which are the toplevel routines in terms of automatic differentiation.
- the ocean into the atmosphere of a carbon like tracer injected
+ The routines {\it adthe\_main\_loop} or {\it g\_the\_main\_loop}
- into the ocean interior (see \cite{hil-eta:01}).
+ are generated by TAF.
+ It contains both the forward integration of the full model, the
- \subsubsection{Passive tracer equation}
+ cost function calculation,
+ any additional storing that is required for efficient checkpointing,
- For this work the MITGCM was augmented with a thermodynamically
+ and the reverse integration of the adjoint model.
- inactive tracer, $C$. Tracer residing in the ocean
- model surface layer is outgassed according to a relaxation time scale,
+ [DESCRIBE IN A SEPARATE SECTION THE WORKING OF THE TLM]
- $\mu$. Within the ocean interior, the tracer is passively advected
- by the ocean model currents. The full equation for the time evolution
+ In Fig. \ref{fig:adthemodel}
- %
+ the structure of {\it adthe\_main\_loop} has been strongly
- \begin{equation}
+ simplified to focus on the essentials; in particular, no checkpointing
- \label{carbon_ddt}
+ procedures are shown here.
- \frac{\partial C}{\partial t} \, = \,
+ Prior to the call of {\it adthe\_main\_loop}, the routine
- -U\cdot \nabla C \, - \, \mu C \, + \, \Gamma(C) \,+ \, S
+ {\it ctrl\_unpack} is invoked to unpack the control vector
- \end{equation}
+ or initialise the control variables.
- %
+ Following the call of {\it adthe\_main\_loop},
- also includes a source term $S$. This term
+ the routine {\it ctrl\_pack}
- represents interior sources of $C$ such as would arise due to
+ is invoked to pack the control vector
- direct injection.
+ (cf. Section \ref{section_ctrl}).
- The velocity term, $U$, is the sum of the
+ If gradient checks are to be performed, the option
- model Eulerian circulation and an eddy-induced velocity, the latter
+ {\tt ALLOW\_GRADIENT\_CHECK} is defined. In this case
- parameterized according to Gent/McWilliams (\cite{gen:90, dan:95}).
+ the driver routine {\it grdchk\_main} is called after
- The convection function, $\Gamma$, mixes $C$ vertically wherever the
+ the gradient has been computed via the adjoint
- fluid is locally statically unstable.
+ (cf. Section \ref{section_grdchk}).
- The outgassing time scale, $\mu$, in eqn. (\ref{carbon_ddt})
+ %------------------------------------------------------------------
- is set so that \( 1/\mu \sim 1 \ \mathrm{year} \) for the surface
- ocean and $\mu=0$ elsewhere. With this value, eqn. (\ref{carbon_ddt})
+ \subsection{General setup
- is valid as a prognostic equation for small perturbations in oceanic
+ \label{section_ad_setup}}
- carbon concentrations. This configuration provides a
- powerful tool for examining the impact of large-scale ocean circulation
+ In order to configure AD-related setups the following packages need
- on $ CO_2 $ outgassing due to interior injections.
+ to be enabled:
- As source we choose a constant in time injection of
+ {\it
- $ S = 1 \,\, {\rm mol / s}$.
+ \begin{table}[h!]
+ \begin{tabular}{l}
- \subsubsection{Model configuration}
+ autodiff \\
+ ctrl \\
- The model configuration employed has a constant
+ cost \\
- $4^\circ \times 4^\circ$ resolution horizontal grid and realistic
+ grdchk \\
- geography and bathymetry. Twenty vertical layers are used with
+ \end{tabular}
- vertical spacing ranging
+ \end{table}
- from 50 m near the surface to 815 m at depth.
+ }
- Driven to steady-state by climatalogical wind-stress, heat and
+ The packages are enabled by adding them to your experiment-specific
- fresh-water forcing the model reproduces well known large-scale
+ configuration file
- features of the ocean general circulation.
+ {\it packages.conf} (see Section ???).
- \subsubsection{Outgassing cost function}
+ The following AD-specific CPP option files need to be customized:
- To quantify and understand outgassing due to injections of $C$
- in eqn. (\ref{carbon_ddt}),
- we define a cost function $ {\cal J} $ that measures the total amount of
- tracer outgassed at each timestep:
- %
- \begin{equation}
- \label{cost_tracer}
- {\cal J}(t=T)=\int_{t=0}^{t=T}\int_{A} \mu C \, dA \, dt
- \end{equation}
- %
- Equation(\ref{cost_tracer}) integrates the outgassing term, $\mu C$,
- from (\ref{carbon_ddt})
- over the entire ocean surface area, $A$, and accumulates it
- up to time $T$.
- Physically, ${\cal J}$ can be thought of as representing the amount of
- $CO_2$ that our model predicts would be outgassed following an
- injection at rate $S$.
- The sensitivity of ${\cal J}$ to the spatial location of $S$,
- $\frac{\partial {\cal J}}{\partial S}$,
- can be used to identify regions from which circulation
- would cause $CO_2$ to rapidly outgas following injection
- and regions in which $CO_2$ injections would remain effectively
- sequesterd within the ocean.
- \subsection{Code configuration}
- The model configuration for this experiment resides under the
- directory {\it verification/carbon/}.
- The code customisation routines are in {\it verification/carbon/code/}:
  %
  \begin{itemize}
  %
- \item {\it .genmakerc}
+ \item {\it ECCO\_CPPOPTIONS.h} \\
- %
+ This header file collects CPP options for the packages
- \item {\it COST\_CPPOPTIONS.h}
+ {\it autodiff, cost, ctrl} as well as AD-unrelated options for
- %
+ the external forcing package {\it exf}.
- \item {\it CPP\_EEOPTIONS.h}
+ \footnote{NOTE: These options are not set in their package-specific
- %
+ headers such as {\it COST\_CPPOPTIONS.h}, but are instead collected
- \item {\it CPP\_OPTIONS.h}
+ in the single header file {\it ECCO\_CPPOPTIONS.h}.
- %
+ The package-specific header files serve as simple
- \item {\it CTRL\_OPTIONS.h}
+ placeholders at this point.}
  %
- \item {\it ECCO\_OPTIONS.h}
+ \item {\it tamc.h} \\
- %
+ This header configures the splitting of the time stepping loop
- \item {\it SIZE.h}
+ w.r.t. the 3-level checkpointing (see section ???).
- %
- \item {\it adcommon.h}
- %
- \item {\it tamc.h}
  %
  \end{itemize}
+ %------------------------------------------------------------------
+ \subsection{Building the AD code
+ \label{section_ad_build}}
+ The build process of an AD code is very similar to building
+ the forward model. However, depending on which AD code one wishes
+ to generate, and on which AD tool is available (TAF or TAMC),
+ the following {\tt make} targets are available:
+ \begin{table}[h!]
+ {\footnotesize
+ \begin{tabular}{ccll}
+ ~ & {\it AD-target} & {\it output} & {\it description} \\
+ \hline
+ \hline
+ (1) & {\tt <MODE><TOOL>only} & {\tt <MODE>\_<TOOL>\_output.f}  &
+ generates code for $<$MODE$>$ using $<$TOOL$>$ \\
+ ~ & ~ & ~ & no {\tt make} dependencies on {\tt .F .h} \\
+ ~ & ~ & ~ & useful for compiling on remote platforms \\
+ \hline
+ (2) & {\tt <MODE><TOOL>} & {\tt <MODE>\_<TOOL>\_output.f}  &
+ generates code for $<$MODE$>$ using $<$TOOL$>$ \\
+ ~ & ~ & ~ & includes {\tt make} dependencies on {\tt .F .h} \\
+ ~ & ~ & ~ & i.e. input for $<$TOOL$>$ may be re-generated \\
+ \hline
+ (3) & {\tt <MODE>all} & {\tt mitgcmuv\_<MODE>}  &
+ generates code for $<$MODE$>$ using $<$TOOL$>$ \\
+ ~ & ~ & ~ & and compiles all code \\
+ ~ & ~ & ~ & (use of TAF is set as default) \\
+ \hline
+ \hline
+ \end{tabular}
+ }
+ \end{table}
  %
- The runtime flag and parameters settings are contained in
+ Here, the following placeholders are used
- {\it verification/carbon/input/},
- together with the forcing fields and and restart files:
  %
  \begin{itemize}
  %
- \item {\it data}
+ \item [$<$TOOL$>$]
- %
- \item {\it data.cost}
- %
- \item {\it data.ctrl}
- %
- \item {\it data.pkg}
- %
- \item {\it eedata}
  %
- \item {\it topog.bin}
+ \begin{itemize}
- %
- \item {\it windx.bin, windy.bin}
- %
- \item {\it salt.bin, theta.bin}
- %
- \item {\it SSS.bin, SST.bin}
  %
- \item {\it pickup*}
+ \item {\tt TAF}
+ \item {\tt TAMC}
  %
  \end{itemize}
  %
- Finally, the file to generate the adjoint code resides in
+ \item [$<$MODE$>$]
- $ adjoint/ $:
  %
  \begin{itemize}
  %
- \item {\it makefile}
+ \item {\tt ad} generates the adjoint model (ADM)
+ \item {\tt ftl} generates the tangent linear model (TLM)
+ \item {\tt svd} generates both ADM and TLM for \\
+ singular value decomposition (SVD) type calculations
  %
  \end{itemize}
  %
+ \end{itemize}
- Below we describe the customisations of this files which are
+ For example, to generate the adjoint model using TAF after routines ({\tt .F})
- specific to this experiment.
+ or headers ({\tt .h}) have been modified, but without compilation,
+ type {\tt make adtaf};
- \subsubsection{File {\it .genmakerc}}
+ or, to generate the tangent linear model using TAMC without
- This file overwites default settings of {\it genmake}.
+ re-generating the input code, type {\tt make ftltamconly}.
- In the present example it is used to switch on the following
- packages which are related to automatic differentiation
- and are disabled by default: \\
- \hspace*{4ex} {\tt set ENABLE=( autodiff cost ctrl ecco )}  \\
- Other packages which are not needed are switched off: \\
- \hspace*{4ex} {\tt set DISABLE=( aim obcs zonal\_filt shap\_filt cal exf )}
- \subsubsection{File {\it COST\_CPPOPTIONS.h,  CTRL\_OPTIONS.h}}
- These files used to contain package-specific CPP-options
- (see Section \ref{???}).
- For technical reasons those options have been grouped together
- in the file {\it ECCO\_OPTIONS.h}.
- To retain the modularity, the files have been kept and contain
- the standard include of the {\it CPP\_OPTIONS.h} file.
- \subsubsection{File {\it CPP\_EEOPTIONS.h}}
- This file contains 'wrapper'-specific CPP options.
- It only needs to be changed if the code is to be run
- in  parallel environment (see Section \ref{???}).
- \subsubsection{File {\it CPP\_OPTIONS.h}}
- This file contains model-specific CPP options
- (see Section \ref{???}).
- Most options are related to the forward model setup.
- They are identical to the global steady circulation setup of
- {\it verification/exp2/}.
- The option specific to this experiment is \\
- \hspace*{4ex} {\tt \#define ALLOW\_MIT\_ADJOINT\_RUN} \\
- This flag enables the inclusion of some AD-related fields
- concerning initialisation, link between control variables
- and forward model variables, and the call to the top-level
- forward/adjoint subroutine {\it adthe\_main\_loop}
- instead of {\it the\_main\_loop}.
- \subsubsection{File {\it ECCO\_OPTIONS.h}}
- The CPP options of several AD-related packages are grouped
- in this file:
- %
- \begin{itemize}
- %
- \item
- Adjoint support package: {\it pkg/autodiff/} \\
- This package contains hand-written adjoint code such as
- active file handling, flow directives for files which must not
- be differentiated, and TAMC-specific header files. \\
- \hspace*{4ex} {\tt \#define ALLOW\_AUTODIFF\_TAMC} \\
- defines TAMC-related features in the code. \\
- \hspace*{4ex} {\tt \#define ALLOW\_TAMC\_CHECKPOINTING} \\
- enables the checkpointing feature of TAMC
- (see Section \ref{???}).
- In the present example a 3-level checkpointing is implemented.
- The code contains the relevant store directives, common block
- and tape initialisations, storing key computation,
- and loop index handling.
- The checkpointing length at each level is defined in
- file {\it tamc.h}, cf. below.
- %
- \item Cost function package: {\it pkg/cost/} \\
- This package contains all relevant routines for
- initialising, accumulating and finalizing the cost function
- (see Section \ref{???}). \\
- \hspace*{4ex} {\tt \#define ALLOW\_COST} \\
- enables all general aspects of the cost function handling,
- in particular the hooks in the foorward code for
- initialising, accumulating and finalizing the cost function. \\
- \hspace*{4ex} {\tt \#define ALLOW\_COST\_TRACER} \\
- includes the subroutine with the cost function for this
- particular experiment, eqn. (\ref{cost_tracer}).
- %
- \item Control variable package: {\it pkg/ctrl/} \\
- This package contains all relevant routines for
- the handling of the control vector.
- Each control variable can be enabled/disabled with its own flag: \\
- \begin{tabular}{ll}
- \hspace*{2ex} {\tt \#define ALLOW\_THETA0\_CONTROL} &
- initial temperature \\
- \hspace*{2ex} {\tt \#define ALLOW\_SALT0\_CONTROL} &
- initial salinity \\
- \hspace*{2ex} {\tt \#define ALLOW\_TR0\_CONTROL} &
- initial passive tracer concentration \\
- \hspace*{2ex} {\tt \#define ALLOW\_TAUU0\_CONTROL} &
- zonal wind stress \\
- \hspace*{2ex} {\tt \#define ALLOW\_TAUV0\_CONTROL} &
- meridional wind stress \\
- \hspace*{2ex} {\tt \#define ALLOW\_SFLUX0\_CONTROL} &
- freshwater flux \\
- \hspace*{2ex} {\tt \#define ALLOW\_HFLUX0\_CONTROL} &
- heat flux \\
- \hspace*{2ex} {\tt \#undef ALLOW\_DIFFKR\_CONTROL} &
- diapycnal diffusivity \\
- \hspace*{2ex} {\tt \#undef ALLOW\_KAPPAGM\_CONTROL} &
- isopycnal diffusivity \\
- \end{tabular}
- %
- \end{itemize}
- \subsubsection{File {\it SIZE.h}}
+ A typical full build process to generate the ADM via TAF would
+ look like follows:
+ \begin{verbatim}
+ % mkdir build
+ % cd build
+ % ../../../tools/genmake2 -mods=../code_ad
+ % make depend
+ % make adall
+ \end{verbatim}
- The file contains the grid point dimensions of the forward
+ %------------------------------------------------------------------
- model. It is identical to the {\it verification/exp2/}: \\
- \hspace*{4ex} {\tt sNx = 90} \\
- \hspace*{4ex} {\tt sNy = 40} \\
- \hspace*{4ex} {\tt Nr = 20} \\
- It correpsponds to a single-tile/single-processor setup:
- {\tt nSx = nSy = 1, nPx = nPy = 1},
- with standard overlap dimensioning
- {\tt OLx = OLy = 3}.
- \subsubsection{File {\it adcommon.h}}
- This file contains common blocks of some adjoint variables
- that are generated by TAMC.
- The common blocks are used by the adjoint support routine
- {\it addummy\_in\_stepping} which needs to access those variables:
- \begin{tabular}{ll}
- \hspace*{4ex} {\tt common /addynvars\_r/} &
- \hspace*{4ex} is related to {\it DYNVARS.h} \\
- \hspace*{4ex} {\tt common /addynvars\_cd/} &
- \hspace*{4ex} is related to {\it DYNVARS.h} \\
- \hspace*{4ex} {\tt common /adtr1\_r/} &
- \hspace*{4ex} is related to {\it TR1.h} \\
- \hspace*{4ex} {\tt common /adffields/} &
- \hspace*{4ex} is related to {\it FFIELDS.h}\\
- \end{tabular}
- Note that if the structure of the common block changes in the
+ \subsection{The AD build process in detail
- above header files of the forward code, the structure
+ \label{section_ad_build_detail}}
- of the adjoint common blocks will change accordingly.
- Thus, it has to be made sure that the structure of the
- adjoint common block in the hand-written file {\it adcommon.h}
- complies with the automatically generated adjoint common blocks
- in {\it adjoint\_model.F}.
- \subsubsection{File {\it tamc.h}}
+ The {\tt make <MODE>all} target consists of the following procedures:
- This routine contains the dimensions for TAMC checkpointing.
+ \begin{enumerate}
  %
+ \item
+ A header file {\tt AD\_CONFIG.h} is generated which contains a CPP option
+ on which code ought to be generated. Depending on the {\tt make} target,
+ the contents is
  \begin{itemize}
+ \item
+ {\tt \#define ALLOW\_ADJOINT\_RUN}
+ \item
+ {\tt \#define ALLOW\_TANGENTLINEAR\_RUN}
+ \item
+ {\tt \#define ALLOW\_ECCO\_OPTIMIZATION}
+ \end{itemize}
  %
- \item {\tt \#ifdef ALLOW\_TAMC\_CHECKPOINTING} \\
+ \item
--level checkpointing is enabled, i.e. the timestepping
+ A single file {\tt <MODE>\_input\_code.f} is concatenated
- is divided into three different levels (see Section \ref{???}).
+ consisting of all {\tt .f} files that are part of the list {\bf AD\_FILES}
- The model state of the outermost ({\tt nchklev\_3}) and the
+ and all {\tt .flow} files that are part of the list {\bf AD\_FLOW\_FILES}.
- itermediate ({\tt nchklev\_2}) timestepping loop are stored to file
+ %
- (handled in {\it the\_main\_loop}).
+ \item
- The innermost loop ({\tt nchklev\_1})
+ The AD tool is invoked with the {\bf <MODE>\_<TOOL>\_FLAGS}.
- avoids I/O by storing all required variables
+ The default AD tool flags in {\tt genmake2} can be overrwritten by
- to common blocks. This storing may also be necessary if
+ an {\tt adjoint\_options} file (similar to the platform-specific
- no checkpointing is chosen
+ {\tt build\_options}, see Section ???.
- (nonlinear functions, if-statements, iterative loops, ...).
+ The AD tool writes the resulting AD code into the file
- In the present example the dimensions are chosen as follows: \\
+ {\tt <MODE>\_input\_code\_ad.f}
- \hspace*{4ex} {\tt nchklev\_1      =  36 } \\
+ %
- \hspace*{4ex} {\tt nchklev\_2      =  30 } \\
+ \item
- \hspace*{4ex} {\tt nchklev\_3      =  60 } \\
+ A short sed script {\tt adjoint\_sed} is applied to
- To guarantee that the checkpointing intervals span the entire
+ {\tt <MODE>\_input\_code\_ad.f}
- integration period the relation \\
+ to reinstate {\bf myThid} into the CALL argument list of active file I/O.
- \hspace*{4ex} {\tt nchklev\_1*nchklev\_2*nchklev\_3 $ \ge $ nTimeSteps} \\
+ The result is written to file {\tt <MODE>\_<TOOL>\_output.f}.
- where {\tt nTimeSteps} is either specified in {\it data}
+ %
- or computed via \\
+ \item
- \hspace*{4ex} {\tt nTimeSteps = (endTime-startTime)/deltaTClock }.
+ All routines are compiled and an executable is generated
- %
+ (see Table ???).
- \item {\tt \#undef ALLOW\_TAMC\_CHECKPOINTING} \\
- No checkpointing is enabled.
- In this case the relevant counter is {\tt nchklev\_0}.
- Similar to above, the following relation has to be satisfied \\
- \hspace*{4ex} {\tt nchklev\_0 $ \ge $ nTimeSteps}.
  %
- \end{itemize}
+ \end{enumerate}
- \subsubsection{File {\it makefile}}
+ \subsubsection{The list AD\_FILES and {\tt .list} files}
- This file contains all relevant paramter flags and
+ Not all routines are presented to the AD tool.
- lists to run TAMC.
+ Routines typically hidden are diagnostics routines which
- It is assumed that TAMC is available to you, either locally,
+ do not influence the cost function, but may create
- being installed on your network, or remotely through the 'TAMC Utility'.
+ artificial flow dependencies such as I/O of active variables.
- TAMC is called with the command {\tt tamc} followed by a
- number of options. They are described in detail in the
+ {\tt genmake2} generates a list (or variable) {\bf AD\_FILES}
- TAMC manual \cite{gie:99}.
+ which contains all routines that are shown to the AD tool.
- Here we briefly discuss the main flags used in the {\it makefile}
+ This list is put together from all files with suffix {\tt .list}
+ that {\tt genmake2} finds in its search directories.
+ The list file for the core MITgcm routines is in {\tt model/src/}
+ is called {\tt model\_ad\_diff.list}.
+ Note that no wrapper routine is shown to TAF. These are either
+ not visible at all to the AD code, or hand-written AD code
+ is available (see next section).
+ Each package directory contains its package-specific
+ list file {\tt <PKG>\_ad\_diff.list}. For example,
+ {\tt pkg/ptracers/} contains the file {\tt ptracers\_ad\_diff.list}.
+ Thus, enabling a package will automatically extend the
+ {\bf AD\_FILES} list of {\tt genmake2} to incorporate the
+ package-specific routines.
+ Note that you will need to regenerate the {\tt Makefile} if
+ you enable a package (e.g. by adding it to {\tt packages.conf})
+ and a {\tt Makefile} already exists.
+ \subsubsection{The list AD\_FLOW\_FILES and {\tt .flow} files}
+ TAMC and TAF can evaluate user-specified directives
+ that start with a specific syntax ({\tt CADJ}, {\tt C\$TAF}, {\tt !\$TAF}).
+ The main categories of directives are STORE directives and
+ FLOW directives. Here, we are concerned with flow directives,
+ store directives are treated elsewhere.
+ Flow directives enable the AD tool to evaluate how it should treat
+ routines that are 'hidden' by the user, i.e. routines which are
+ not contained in the {\bf AD\_FILES} list (see previous section),
+ but which are called in part of the code that the AD tool does see.
+ The flow directive tell the AD tool
  %
  \begin{itemize}
- \item [{\tt tamc}] {\tt
- -input <variable names>
- -output <variable name> ... \\
- -toplevel <S/R name> -reverse <file names>
- }
- \end{itemize}
  %
- \begin{itemize}
+ \item which subroutine arguments are input/output
- %
+ \item which subroutine arguments are active
- \item {\tt -toplevel <S/R name>} \\
+ \item which subroutine arguments are required to compute the cost
- Name of the toplevel routine, with respect to which the
+ \item which subroutine arguments are dependent
- control flow analysis is performed.
- %
- \item {\tt -input <variable names>} \\
- List of independent variables $ u $ with respect to which the
- dependent variable $ J $ is differentiated.
- %
- \item {\tt -output <variable name>} \\
- Dependent variable $ J $  which is to be differentiated.
- %
- \item {\tt -reverse <file names>} \\
- Adjoint code is generated to compute the sensitivity of an
- independent variable w.r.t.  many dependent variables.
- The generated adjoint top-level routine computes the product
- of the transposed Jacobian matrix $ M^T $ times
- the gradient vector $ \nabla_v J $.
- \\
- {\tt <file names>} refers to the list of files {\it .f} which are to be
- analyzed by TAMC. This list is generally smaller than the full list
- of code to be compiled. The files not contained are either
- above the top-level routine (some initialisations), or are
- deliberately hidden from TAMC, either because hand-written
- adjoint routines exist, or the routines must not (or don't have to)
- be differentiated. For each routine which is part of the flow tree
- of the top-level routine, but deliberately hidden from TAMC,
- a corresponding file {\it .flow} exists containing flow directives
- for TAMC.
  %
  \end{itemize}
+ %
+ The syntax for the flow directives can be found in the
+ AD tool manuals.
+ {\tt genmake2} generates a list (or variable) {\bf AD\_FLOW\_FILES}
+ which contains all files with suffix{\tt .flow} that it finds
+ in its search directories.
+ The flow directives for the core MITgcm routines of
+ {\tt eesupp/src/} and {\tt model/src/}
+ reside in {\tt pkg/autodiff/}.
+ This directory also contains hand-written adjoint code
+ for the MITgcm WRAPPER (see Section ???).
+ Flow directives for package-specific routines are contained in
+ the corresponding package directories in the file
+ {\tt <PKG>\_ad.flow}, e.g. ptracers-specific directives are in
+ {\tt ptracers\_ad.flow}.
+ \subsubsection{Store directives for 3-level checkpointing}
+ The storing that is required at each period of the
+-level checkpointing is controled by three
+ top-level headers.
- \subsubsection{File {\it data}}
+ \begin{verbatim}
+ do ilev_3 = 1, nchklev_3
- \subsubsection{File {\it data.cost}}
+ #  include ``checkpoint_lev3.h''
+    do ilev_2 = 1, nchklev_2
- \subsubsection{File {\it data.ctrl}}
+ #     include ``checkpoint_lev2.h''
+       do ilev_1 = 1, nchklev_1
- \subsubsection{File {\it data.pkg}}
+ #        include ``checkpoint_lev1.h''
- \subsubsection{File {\it eedata}}
+ ...
- \subsubsection{File {\it topog.bin}}
+       end do
+    end do
- \subsubsection{File {\it windx.bin, windy.bin}}
+ end do
+ \end{verbatim}
- \subsubsection{File {\it salt.bin, theta.bin}}
- \subsubsection{File {\it SSS.bin, SST.bin}}
+ All files {\tt checkpoint\_lev?.h} are contained in directory
+ {\tt pkg/autodiff/}.
- \subsubsection{File {\it pickup*}}
- \subsection{Compiling the model and its adjoint}
+ \subsubsection{Changing the default AD tool flags: ad\_options files}
- \newpage
- %**********************************************************************
+ \subsubsection{Hand-written adjoint code}
- \section{TLM and ADM code generation in general}
- \label{sec_ad_setup_gen}
- %**********************************************************************
- In this section we describe in a general fashion
+ %------------------------------------------------------------------
- the parts of the code that are relevant for automatic
- differentiation using the software tool TAMC.
- \subsection{The cost function (dependent variable)}
+ \subsection{The cost function (dependent variable)
+ \label{section_cost}}
  The cost function $ {\cal J} $ is referred to as the {\sf dependent variable}.
  It is a function of the input variables $ \vec{u} $ via the composition
  $ {\cal J}(\vec{u}) \, = \, {\cal J}(M(\vec{u})) $.
- The input is referred to as the
+ The input are referred to as the
  {\sf independent variables} or {\sf control variables}.
  All aspects relevant to the treatment of the cost function $ {\cal J} $
- (parameter setting, initialisation, incrementation,
+ (parameter setting, initialization, accumulation,
- final evaluation), are controled by the package {\it pkg/cost}.
+ final evaluation), are controlled by the package {\it pkg/cost}.
+ The aspects relevant to the treatment of the independent variables
+ are controlled by the package {\it pkg/ctrl} and will be treated
+ in the next section.
+ \input{part5/doc_cost_flow}
+ \subsubsection{Enabling the package}
- \subsubsection{genmake and CPP options}
- %
- \begin{itemize}
- %
- \item
  \fbox{
  \begin{minipage}{12cm}
- {\it genmake}, {\it CPP\_OPTIONS.h}, {\it ECCO\_CPPOPTIONS.h}
+ {\it packages.conf}, {\it ECCO\_CPPOPTIONS.h}
  \end{minipage}
  }
- \end{itemize}
+ \begin{itemize}
- %
- The directory {\it pkg/cost} can be included to the
- compile list in 3 different ways (cf. Section \ref{???}):
  %
- \begin{enumerate}
+ \item
+ The package is enabled by adding {\it cost} to your file {\it packages.conf}
+ (see Section ???)
  %
- \item {\it genmake}: \\
+ \item
- Change the default settngs in the file {\it genmake} by adding
- {\bf cost} to the {\bf enable} list (not recommended).
- %
+ \end{itemize}
- \item {\it .genmakerc}: \\
- Customize the settings of {\bf enable}, {\bf disable} which are
- appropriate for your experiment in the file {\it .genmakerc}
- and add the file to your compile directory.
- %
- \item genmake-options: \\
- Call {\it genmake} with the option
- {\tt genmake -enable=cost}.
  %
- \end{enumerate}
- Since the cost function is usually used in conjunction with
+ N.B.: In general the following packages ought to be enabled
- automatic differentiation, the CPP option
+ simultaneously: {\it autodiff, cost, ctrl}.
- {\bf ALLOW\_ADJOINT\_RUN} should be defined
- (file {\it CPP\_OPTIONS.h}).
  The basic CPP option to enable the cost function is {\bf ALLOW\_COST}.
  Each specific cost function contribution has its own option.
  For the present example the option is {\bf ALLOW\_COST\_TRACER}.
  All cost-specific options are set in {\it ECCO\_CPPOPTIONS.h}
+ Since the cost function is usually used in conjunction with
+ automatic differentiation, the CPP option
+ {\bf ALLOW\_ADJOINT\_RUN} (file {\it CPP\_OPTIONS.h}) and
+ {\bf ALLOW\_AUTODIFF\_TAMC} (file {\it ECCO\_CPPOPTIONS.h})
+ should be defined.
- \subsubsection{Initialisation}
+ \subsubsection{Initialization}
  %
- The initialisation of the {\it cost} package is readily enabled
+ The initialization of the {\it cost} package is readily enabled
- as soon as the CPP option {\bf ALLOW\_ADJOINT\_RUN} is defined.
+ as soon as the CPP option {\bf ALLOW\_COST} is defined.
  %
  \begin{itemize}
  %
-Line 1152 
 Variables: {\it cost\_init}
+Line 1075 
 Variables: {\it cost\_init}
  }
  \\
  This S/R
- initialises the different cost function contributions.
+ initializes the different cost function contributions.
- The contribtion for the present example is {\bf objf\_tracer}
+ The contribution for the present example is {\bf objf\_tracer}
  which is defined on each tile (bi,bj).
  %
  \end{itemize}
  %
- \subsubsection{Incrementation}
+ \subsubsection{Accumulation}
  %
  \begin{itemize}
  %
-Line 1196 
 from each contribution and sums over all
+Line 1119 
 from each contribution and sums over all
  \begin{equation}
  {\cal J} \, = \,
  {\rm fc} \, = \,
- {\rm mult\_tracer} \sum_{bi,\,bj}^{nSx,\,nSy}
+ {\rm mult\_tracer} \sum_{\text{global sum}} \sum_{bi,\,bj}^{nSx,\,nSy}
  {\rm objf\_tracer}(bi,bj) \, + \, ...
  \end{equation}
  %
-Line 1206 
 The total cost function {\bf fc} will be
+Line 1129 
 The total cost function {\bf fc} will be
  tamc -output 'fc' ...
  \end{verbatim}
- \begin{figure}[t!]
+ %%%% \end{document}
- \input{part5/doc_ad_the_model}
- \label{fig:adthemodel}
- \caption{~}
- \end{figure}
- \begin{figure}
  \input{part5/doc_ad_the_main}
- \label{fig:adthemain}
- \caption{~}
- \end{figure}
- \subsection{The control variables (independent variables)}
+ \subsection{The control variables (independent variables)
+ \label{section_ctrl}}
  The control variables are a subset of the model input
  (initial conditions, boundary conditions, model parameters).
  Here we identify them with the variable $ \vec{u} $.
  All intermediate variables whose derivative w.r.t. control
- variables don't vanish are called {\sf active variables}.
+ variables do not vanish are called {\sf active variables}.
  All subroutines whose derivative w.r.t. the control variables
  don't vanish are called {\sf active routines}.
  Read and write operations from and to file can be viewed
-Line 1232 
 as variable assignments. Therefore, file
+Line 1148 
 as variable assignments. Therefore, file
  active variables are written and from which active variables
  are read are called {\sf active files}.
  All aspects relevant to the treatment of the control variables
- (parameter setting, initialisation, perturbation)
+ (parameter setting, initialization, perturbation)
- are controled by the package {\it pkg/ctrl}.
+ are controlled by the package {\it pkg/ctrl}.
+ \input{part5/doc_ctrl_flow}
  \subsubsection{genmake and CPP options}
  %
-Line 1249 
 are controled by the package {\it pkg/ct
+Line 1167 
 are controled by the package {\it pkg/ct
  %
  To enable the directory to be included to the compile list,
  {\bf ctrl} has to be added to the {\bf enable} list in
- {\it .genmakerc} (or {\it genmake} itself).
+ {\it .genmakerc} or in {\it genmake} itself (analogous to {\it cost}
+ package, cf. previous section).
  Each control variable is enabled via its own CPP option
  in {\it ECCO\_CPPOPTIONS.h}.
- \subsubsection{Initialisation}
+ \subsubsection{Initialization}
  %
  \begin{itemize}
  %
-Line 1293 
 Two important issues related to the hand
+Line 1212 
 Two important issues related to the hand
  variables in the MITGCM need to be addressed.
  First, in order to save memory, the control variable arrays
  are not kept in memory, but rather read from file and added
- to the initial (or first guess) fields.
+ to the initial fields during the model initialization phase.
  Similarly, the corresponding adjoint fields which represent
  the gradient of the cost function w.r.t. the control variables
- are written to to file.
+ are written to file at the end of the adjoint integration.
  Second, in addition to the files holding the 2-dim. and 3-dim.
- control variables and the gradient, a 1-dim. {\sf control vector}
+ control variables and the corresponding cost gradients,
+ a 1-dim. {\sf control vector}
  and {\sf gradient vector} are written to file. They contain
  only the wet points of the control variables and the corresponding
  gradient.
  This leads to a significant data compression.
- Furthermore, the control and the gradient vector can be passed to a
+ Furthermore, an option is available
+ ({\tt ALLOW\_NONDIMENSIONAL\_CONTROL\_IO}) to
+ non-dimensionalise the control and gradient vector,
+ which otherwise would contain different pieces of different
+ magnitudes and units.
+ Finally, the control and gradient vector can be passed to a
  minimization routine if an update of the control variables
  is sought as part of a minimization exercise.
-Line 1314 
 and gradient are generated and initialis
+Line 1239 
 and gradient are generated and initialis
  \subsubsection{Perturbation of the independent variables}
  %
- The dependency chain for differentiation starts
+ The dependency flow for differentiation w.r.t. the controls
- with adding a perturbation onto the the input variable,
+ starts with adding a perturbation onto the input variable,
  thus defining the independent or control variables for TAMC.
- Three classes of controls may be considered:
+ Three types of controls may be considered:
  %
  \begin{itemize}
  %
-Line 1332 
 Three classes of controls may be conside
+Line 1257 
 Three classes of controls may be conside
  Consider as an example the initial tracer distribution
  {\bf tr1} as control variable.
  After {\bf tr1} has been initialised in
- {\it ini\_tr1} (dynamical variables including
+ {\it ini\_tr1} (dynamical variables such as
  temperature and salinity are initialised in {\it ini\_fields}),
  a perturbation anomaly is added to the field in S/R
  {\it ctrl\_map\_ini}
-Line 1345 
 u         & = \, u_{[0]} \, + \, \Delta
+Line 1270 
 u         & = \, u_{[0]} \, + \, \Delta
  \end{split}
  \end{equation}
  %
- In principle {\bf xx\_tr1} is a 3-dim. global array
+ {\bf xx\_tr1} is a 3-dim. global array
  holding the perturbation. In the case of a simple
  sensitivity study this array is identical to zero.
- However, it's specification is essential since TAMC
+ However, it's specification is essential in the context
+ of automatic differentiation since TAMC
  treats the corresponding line in the code symbolically
  when determining the differentiation chain and its origin.
  Thus, the variable names are part of the argument list
-Line 1366 
 dummy variable {\bf xx\_tr1\_dummy} is i
+Line 1292 
 dummy variable {\bf xx\_tr1\_dummy} is i
  and an 'active read' routine of the adjoint support
  package {\it pkg/autodiff} is invoked.
  The read-procedure is tagged with the variable
- {\bf xx\_tr1\_dummy} enabbling TAMC to recognize the
+ {\bf xx\_tr1\_dummy} enabling TAMC to recognize the
- initialisation of the perturbation.
+ initialization of the perturbation.
  The modified call of TAMC thus reads
  %
  \begin{verbatim}
-Line 1386 
 in the code takes on the form
+Line 1312 
 in the code takes on the form
  %
  Note, that reading an active variable corresponds
  to a variable assignment. Its derivative corresponds
- to a write statement of the adjoint variable.
+ to a write statement of the adjoint variable, followed by
+ a reset.
  The 'active file' routines have been designed
- to support active read and corresponding active write
+ to support active read and corresponding adjoint active write
- operations.
+ operations (and vice versa).
  %
  \item
  \fbox{
-Line 1406 
 with the symbolic perturbation taking pl
+Line 1333 
 with the symbolic perturbation taking pl
  Note however an important difference:
  Since the boundary values are time dependent with a new
  forcing field applied at each time steps,
- the general problem may be be thought of as
+ the general problem may be thought of as
- a new control variable at each time step, i.e.
+ a new control variable at each time step
+ (or, if the perturbation is averaged over a certain period,
+ at each $ N $ timesteps), i.e.
  \[
  u_{\rm forcing} \, = \,
  \{ \, u_{\rm forcing} ( t_n ) \, \}_{
-Line 1432 
 calendar ({\it cal}~) and external forci
+Line 1361 
 calendar ({\it cal}~) and external forci
  %
  This routine is not yet implemented, but would proceed
  proceed along the same lines as the initial value sensitivity.
+ The mixing parameters {\bf diffkr} and {\bf kapgm}
+ are currently added as controls in {\it ctrl\_map\_ini.F}.
  %
  \end{itemize}
  %
  \subsubsection{Output of adjoint variables and gradient}
  %
- Two ways exist to generate output of adjoint fields.
+ Several ways exist to generate output of adjoint fields.
  %
  \begin{itemize}
  %
  \item
  \fbox{
  \begin{minipage}{12cm}
- {\it ctrl\_pack}:
+ {\it ctrl\_map\_ini, ctrl\_map\_forcing}:
  \end{minipage}
  }
  \\
- At the end of the forward/adjoint integration, the S/R
- {\it ctrl\_pack} is called which mirrors S/R {\it ctrl\_unpack}.
- It writes the following files:
- %
  \begin{itemize}
  %
- \item {\bf xx\_...}: the control variable fields
+ \item {\bf xx\_...}: the control variable fields \\
+ Before the forward integration, the control
+ variables are read from file {\bf xx\_ ...} and added to
+ the model field.
  %
  \item {\bf adxx\_...}: the adjoint variable fields, i.e. the gradient
- $ \nabla _{u}{\cal J} $ for each control variable,
+ $ \nabla _{u}{\cal J} $ for each control variable \\
+ After the adjoint integration the corresponding adjoint
+ variables are written to {\bf adxx\_ ...}.
+ %
+ \end{itemize}
  %
- \item {\bf vector\_ctrl}: the control vector
+ \item
+ \fbox{
+ \begin{minipage}{12cm}
+ {\it ctrl\_unpack, ctrl\_pack}:
+ \end{minipage}
+ }
+ \\
+ %
+ \begin{itemize}
  %
- \item {\bf vector\_grad}: the gradient vector
+ \item {\bf vector\_ctrl}: the control vector \\
+ At the very beginning of the model initialization,
+ the updated compressed control vector is read (or initialised)
+ and distributed to 2-dim. and 3-dim. control variable fields.
+ %
+ \item {\bf vector\_grad}: the gradient vector \\
+ At the very end of the adjoint integration,
+ the 2-dim. and 3-dim. adjoint variables are read,
+ compressed to a single vector and written to file.
  %
  \end{itemize}
  %
-Line 1474 
 $ \nabla _{u}{\cal J} $ for each control
+Line 1424 
 $ \nabla _{u}{\cal J} $ for each control
  }
  \\
  In addition to writing the gradient at the end of the
- forward/adjoint integration, many more adjoint variables,
+ forward/adjoint integration, many more adjoint variables
- representing the Lagrange multipliers of the model state
+ of the model state
- w.r.t. the model state
+ at intermediate times can be written using S/R
- at different times can be written using S/R
  {\it addummy\_in\_stepping}.
  This routine is part of the adjoint support package
  {\it pkg/autodiff} (cf.f. below).
+ The procedure is enabled using via the CPP-option
+ {\bf ALLOW\_AUTODIFF\_MONITOR} (file {\it ECCO\_CPPOPTIONS.h}).
  To be part of the adjoint code, the corresponding S/R
  {\it dummy\_in\_stepping} has to be called in the forward
  model (S/R {\it the\_main\_loop}) at the appropriate place.
+ The adjoint common blocks are extracted from the adjoint code
+ via the header file {\it adcommon.h}.
  {\it dummy\_in\_stepping} is essentially empty,
  the corresponding adjoint routine is hand-written rather
-Line 1491 
 than generated automatically.
+Line 1444 
 than generated automatically.
  Appropriate flow directives ({\it dummy\_in\_stepping.flow})
  ensure that TAMC does not automatically
  generate {\it addummy\_in\_stepping} by trying to differentiate
- {\it dummy\_in\_stepping}, but rather takes the hand-written routine.
+ {\it dummy\_in\_stepping}, but instead refers to
+ the hand-written routine.
  {\it dummy\_in\_stepping} is called in the forward code
  at the beginning of each
-Line 1501 
 each timestep in the adjoint calculation
+Line 1455 
 each timestep in the adjoint calculation
  {\it addynamics}.
  {\it addummy\_in\_stepping} includes the header files
- {\it adffields.h, addynamics.h, adtr1.h}.
+ {\it adcommon.h}.
- These header files are also hand-written. They contain
+ This header file is also hand-written. It contains
- the common blocks {\bf /addynvars\_r/}, {\bf /addynvars\_cd/},
+ the common blocks
+ {\bf /addynvars\_r/}, {\bf /addynvars\_cd/},
+ {\bf /addynvars\_diffkr/}, {\bf /addynvars\_kapgm/},
  {\bf /adtr1\_r/}, {\bf /adffields/},
  which have been extracted from the adjoint code to enable
  access to the adjoint variables.
+ {\bf WARNING:} If the structure of the common blocks
+ {\bf /dynvars\_r/}, {\bf /dynvars\_cd/}, etc., changes
+ similar changes will occur in the adjoint common blocks.
+ Therefore, consistency between the TAMC-generated common blocks
+ and those in {\it adcommon.h} have to be checked.
  %
  \end{itemize}
-Line 1521 
 The gradient $ \nabla _{u}{\cal J} |_{u_
+Line 1483 
 The gradient $ \nabla _{u}{\cal J} |_{u_
  with the value of the cost function itself $ {\cal J}(u_{[k]}) $
  at iteration step $ k $ serve
  as input to a minimization routine (e.g. quasi-Newton method,
- conjugate gradient, ...) to compute an update in the
+ conjugate gradient, ... \cite{gil-lem:89})
+ to compute an update in the
  control variable for iteration step $k+1$
  \[
  u_{[k+1]} \, = \,  u_{[0]} \, + \, \Delta u_{[k+1]}
-Line 1535 
 Tab. \ref{???} sketches the flow between
+Line 1498 
 Tab. \ref{???} sketches the flow between
  and the minimization routine.
  \begin{eqnarray*}
- \footnotesize
+ \scriptsize
  \begin{array}{ccccc}
  u_{[0]} \,\, ,  \,\, \Delta u_{[k]}    & ~ & ~ & ~ & ~ \\
  {\Big\downarrow}
-Line 1552 
 v_{[k]} = M \left( u_{[k]} \right) &
+Line 1515 
 v_{[k]} = M \left( u_{[k]} \right) &
  {\cal J}_{[k]} = {\cal J} \left( M \left( u_{[k]} \right) \right)} \\
  \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{~} \\
  \hline
+ \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{~}  \\
+ \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{{\Big\downarrow}} \\
+ \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{~}  \\
  \hline
  \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{~} \\
  \multicolumn{1}{|c}{
  \nabla_u {\cal J}_{[k]} (\delta {\cal J}) =
- T\!\!^{\ast} \cdot \nabla_v {\cal J} |_{v_{[k]}} (\delta {\cal J})} &
+ T^{\ast} \cdot \nabla_v {\cal J} |_{v_{[k]}} (\delta {\cal J})} &
  \stackrel{\bf adjoint}{\mathbf \longleftarrow} &
  ad \, v_{[k]} (\delta {\cal J}) =
  \nabla_v {\cal J} |_{v_{[k]}} (\delta {\cal J}) &
-Line 1565 
 ad \, v_{[k]} (\delta {\cal J}) =
+Line 1531 
 ad \, v_{[k]} (\delta {\cal J}) =
  \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{~} \\
  \hline
   ~ & ~ & ~ & ~ & ~ \\
- ~ & ~ &
+ \hspace*{15ex}{\Bigg\downarrow}
- {\cal J}_{[k]} \qquad {\Bigg\downarrow}  \qquad \nabla_u {\cal J}_{[k]}
+ \quad {\cal J}_{[k]}, \quad \nabla_u {\cal J}_{[k]}
-  & ~ & ~ \\
+  & ~ & ~ & ~ & ~ \\
   ~ & ~ & ~ & ~ & ~ \\
  \hline
  \multicolumn{1}{|c}{~} & ~ & ~ & ~ & \multicolumn{1}{c|}{~} \\
-Line 1595 
 The corresponding I/O flow looks as foll
+Line 1561 
 The corresponding I/O flow looks as foll
  \vspace*{0.5cm}
+ {\scriptsize
  \begin{tabular}{ccccc}
  {\bf vector\_ctrl\_$<$k$>$ } & ~ & ~ & ~ & ~ \\
  {\big\downarrow}  & ~ & ~ & ~ & ~ \\
-Line 1605 
 The corresponding I/O flow looks as foll
+Line 1572 
 The corresponding I/O flow looks as foll
  \cline{3-3}
  \multicolumn{1}{l}{\bf xx\_theta0...$<$k$>$} & ~ &
  \multicolumn{1}{|c|}{~} & ~ & ~ \\
- \multicolumn{1}{l}{\bf xx\_salt0...$<$k$>$} & $\longrightarrow$ &
+ \multicolumn{1}{l}{\bf xx\_salt0...$<$k$>$} &
+ $\stackrel{\mbox{read}}{\longrightarrow}$ &
  \multicolumn{1}{|c|}{forward integration} & ~ & ~ \\
  \multicolumn{1}{l}{\bf \vdots} & ~ & \multicolumn{1}{|c|}{~}
  & ~ & ~ \\
  \cline{3-3}
- ~ & ~ & ~ & ~ & ~ \\
+ ~ & ~ & $\downarrow$ & ~ & ~ \\
  \cline{3-3}
  ~ & ~ &
  \multicolumn{1}{|c|}{~} & ~ &
  \multicolumn{1}{l}{\bf adxx\_theta0...$<$k$>$}  \\
  ~ & ~ & \multicolumn{1}{|c|}{adjoint integration} &
- $\longrightarrow$ &
+ $\stackrel{\mbox{write}}{\longrightarrow}$ &
  \multicolumn{1}{l}{\bf adxx\_salt0...$<$k$>$} \\
  ~ & ~ & \multicolumn{1}{|c|}{~}
  & ~ & \multicolumn{1}{l}{\bf \vdots} \\
-Line 1628 
 $\longrightarrow$ &
+Line 1596 
 $\longrightarrow$ &
  ~ & ~ & ~ & ~ &  {\big\downarrow} \\
  ~ & ~ & ~ & ~ &  {\bf vector\_grad\_$<$k$>$ } \\
  \end{tabular}
+ }
  \vspace*{0.5cm}
- {\it ctrl\_unpack} reads in the updated control vector
+ {\it ctrl\_unpack} reads the updated control vector
  {\bf vector\_ctrl\_$<$k$>$}.
  It distributes the different control variables to
 -dim. and 3-dim. files {\it xx\_...$<$k$>$}.
- During the forward integration the control variables
+ At the start of the forward integration the control variables
- are read from {\it xx\_...$<$k$>$}.
+ are read from {\it xx\_...$<$k$>$} and added to the
- Correspondingly, the adjoint fields are written
+ field.
+ Correspondingly, at the end of the adjoint integration
+ the adjoint fields are written
  to {\it adxx\_...$<$k$>$}, again via the active file routines.
- Finally, {\it ctrl\_pack} collects all adjoint field files
+ Finally, {\it ctrl\_pack} collects all adjoint files
  and writes them to the compressed vector file
  {\bf vector\_grad\_$<$k$>$}.
- \subsection{TLM and ADM generation via TAMC}
- \subsection{Flow directives and adjoint support routines}
- \subsection{Store directives and checkpointing}
- \subsection{Gradient checks}
- \subsection{Second derivative generation via TAMC}
- \section{Example of adjoint code}

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