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revision 1.14 by cnh, Thu Feb 28 19:32:20 2002 UTC revision 1.15 by heimbach, Wed Apr 24 11:01:46 2002 UTC
# Line 557  Because of the local character of the de Line 557  Because of the local character of the de
557  (a derivative is defined w.r.t. a point along the trajectory),  (a derivative is defined w.r.t. a point along the trajectory),
558  the intermediate results of the model trajectory  the intermediate results of the model trajectory
559  $\vec{v}^{(\lambda+1)}={\cal M}_{\lambda}(v^{(\lambda)})$  $\vec{v}^{(\lambda+1)}={\cal M}_{\lambda}(v^{(\lambda)})$
560  are needed to evaluate the intermediate Jacobian  may be required to evaluate the intermediate Jacobian
561  $M_{\lambda}|_{\vec{v}^{(\lambda)}} \, \delta \vec{v}^{(\lambda)} $.  $M_{\lambda}|_{\vec{v}^{(\lambda)}} \, \delta \vec{v}^{(\lambda)} $.
562    This is the case e.g. for nonlinear expressions
563    (momentum advection, nonlinear equation of state), state-dependent
564    conditional statements (parameterization schemes).
565  In the forward mode, the intermediate results are required  In the forward mode, the intermediate results are required
566  in the same order as computed by the full forward model ${\cal M}$,  in the same order as computed by the full forward model ${\cal M}$,
567  but in the reverse mode they are required in the reverse order.  but in the reverse mode they are required in the reverse order.
# Line 569  point of evaluation has to be recomputed Line 572  point of evaluation has to be recomputed
572    
573  A method to balance the amount of recomputations vs.  A method to balance the amount of recomputations vs.
574  storage requirements is called {\sf checkpointing}  storage requirements is called {\sf checkpointing}
575  (e.g. \cite{res-eta:98}).  (e.g. \cite{gri:92}, \cite{res-eta:98}).
576  It is depicted in \ref{fig:3levelcheck} for a 3-level checkpointing  It is depicted in \ref{fig:3levelcheck} for a 3-level checkpointing
577  [as an example, we give explicit numbers for a 3-day  [as an example, we give explicit numbers for a 3-day
578  integration with a 1-hourly timestep in square brackets].  integration with a 1-hourly timestep in square brackets].
# Line 580  In a first step, the model trajectory is Line 583  In a first step, the model trajectory is
583  $ {n}^{lev3} $ subsections [$ {n}^{lev3} $=3 1-day intervals],  $ {n}^{lev3} $ subsections [$ {n}^{lev3} $=3 1-day intervals],
584  with the label $lev3$ for this outermost loop.  with the label $lev3$ for this outermost loop.
585  The model is then integrated along the full trajectory,  The model is then integrated along the full trajectory,
586  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
587  [i.e. 3 times, at  [i.e. 3 times, at
588  $ i = 0,1,2 $ corresponding to $ k_{i}^{lev3} = 0, 24, 48 $].  $ i = 0,1,2 $ corresponding to $ k_{i}^{lev3} = 0, 24, 48 $].
589    In addition, the cost function is computed, if needed.
590  %  %
591  \item [$lev2$]  \item [$lev2$]
592  In a second step each subsection itself is divided into  In a second step each subsection itself is divided into
593  $ {n}^{lev2} $ sub-subsections  $ {n}^{lev2} $ subsections
594  [$ {n}^{lev2} $=4 6-hour intervals per subsection].  [$ {n}^{lev2} $=4 6-hour intervals per subsection].
595  The model picks up at the last outermost dumped state  The model picks up at the last outermost dumped state
596  $ v_{k_{n}^{lev3}} $ and is integrated forward in time along  $ v_{k_{n}^{lev3}} $ and is integrated forward in time along
597  the last subsection, with the label $lev2$ for this    the last subsection, with the label $lev2$ for this  
598  intermediate loop.  intermediate loop.
599  The model state is now stored at every $ k_{i}^{lev2} $-th  The model state is now stored to disk at every $ k_{i}^{lev2} $-th
600  timestep  timestep
601  [i.e. 4 times, at  [i.e. 4 times, at
602  $ i = 0,1,2,3 $ corresponding to $ k_{i}^{lev2} = 48, 54, 60, 66 $].  $ i = 0,1,2,3 $ corresponding to $ k_{i}^{lev2} = 48, 54, 60, 66 $].
# Line 600  $ i = 0,1,2,3 $ corresponding to $ k_{i} Line 604  $ i = 0,1,2,3 $ corresponding to $ k_{i}
604  \item [$lev1$]  \item [$lev1$]
605  Finally, the model picks up at the last intermediate dump state  Finally, the model picks up at the last intermediate dump state
606  $ v_{k_{n}^{lev2}} $ and is integrated forward in time along  $ v_{k_{n}^{lev2}} $ and is integrated forward in time along
607  the last sub-subsection, with the label $lev1$ for this    the last subsection, with the label $lev1$ for this  
608  intermediate loop.  intermediate loop.
609  Within this sub-subsection only, the model state is stored  Within this sub-subsection only, parts of the model state is stored
610  at every timestep  to memory at every timestep
611  [i.e. every hour $ i=0,...,5$ corresponding to  [i.e. every hour $ i=0,...,5$ corresponding to
612  $ k_{i}^{lev1} = 66, 67, \ldots, 71 $].  $ k_{i}^{lev1} = 66, 67, \ldots, 71 $].
613  Thus, the  final state $ v_n = v_{k_{n}^{lev1}} $ is reached  The  final state $ v_n = v_{k_{n}^{lev1}} $ is reached
614  and the model state of all proceeding timesteps along the last  and the model state of all preceding timesteps along the last
615  sub-subsections are available, enabling integration backwards  innermost subsection are available, enabling integration backwards
616  in time along the last sub-subsection.  in time along the last subsection.
617  Thus, the adjoint can be computed along this last  The adjoint can thus be computed along this last
618  sub-subsection $k_{n}^{lev2}$.  subsection $k_{n}^{lev2}$.
619  %  %
620  \end{itemize}  \end{itemize}
621  %  %
622  This procedure is repeated consecutively for each previous  This procedure is repeated consecutively for each previous
623  sub-subsection $k_{n-1}^{lev2}, \ldots, k_{1}^{lev2} $  subsection $k_{n-1}^{lev2}, \ldots, k_{1}^{lev2} $
624  carrying the adjoint computation to the initial time  carrying the adjoint computation to the initial time
625  of the subsection $k_{n}^{lev3}$.  of the subsection $k_{n}^{lev3}$.
626  Then, the procedure is repeated for the previous subsection  Then, the procedure is repeated for the previous subsection
# Line 627  $k_{1}^{lev3}$. Line 631  $k_{1}^{lev3}$.
631  For the full model trajectory of  For the full model trajectory of
632  $ n^{lev3} \cdot n^{lev2} \cdot n^{lev1} $ timesteps  $ n^{lev3} \cdot n^{lev2} \cdot n^{lev1} $ timesteps
633  the required storing of the model state was significantly reduced to  the required storing of the model state was significantly reduced to
634  $ n^{lev1} + n^{lev2} + n^{lev3} $  $ n^{lev2} + n^{lev3} $ to disk and roughly $ n^{lev1} $ to memory
635  [i.e. for the 3-day integration with a total oof 72 timesteps  [i.e. for the 3-day integration with a total oof 72 timesteps
636  the model state was stored 13 times].  the model state was stored 7 times to disk and roughly 6 times
637    to memory].
638  This saving in memory comes at a cost of a required  This saving in memory comes at a cost of a required
639  3 full forward integrations of the model (one for each  3 full forward integrations of the model (one for each
640  checkpointing level).  checkpointing level).
641  The balance of storage vs. recomputation certainly depends  The optimal balance of storage vs. recomputation certainly depends
642  on the computing resources available.  on the computing resources available and may be adjusted by
643    adjusting the partitioning among the
644    $ n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1} $.
645    
646  \begin{figure}[t!]  \begin{figure}[t!]
647  \begin{center}  \begin{center}
# Line 682  If the option {\tt ALLOW\_AUTODIFF\_TAMC Line 689  If the option {\tt ALLOW\_AUTODIFF\_TAMC
689  {\it the\_model\_main}, instead of calling {\it the\_main\_loop},  {\it the\_model\_main}, instead of calling {\it the\_main\_loop},
690  invokes the adjoint of this routine, {\it adthe\_main\_loop},  invokes the adjoint of this routine, {\it adthe\_main\_loop},
691  which is the toplevel routine in terms of reverse mode computation.  which is the toplevel routine in terms of reverse mode computation.
692  The routine {\it adthe\_main\_loop} has been generated using TAMC.  The routine {\it adthe\_main\_loop} has been generated by TAMC.
693  It contains both the forward integration of the full model,  It contains both the forward integration of the full model,
694  any additional storing that is required for efficient checkpointing,  any additional storing that is required for efficient checkpointing,
695  and the reverse integration of the adjoint model.  and the reverse integration of the adjoint model.
# Line 690  The structure of {\it adthe\_main\_loop} Line 697  The structure of {\it adthe\_main\_loop}
697  simplified for clarification; in particular, no checkpointing  simplified for clarification; in particular, no checkpointing
698  procedures are shown here.  procedures are shown here.
699  Prior to the call of {\it adthe\_main\_loop}, the routine  Prior to the call of {\it adthe\_main\_loop}, the routine
700  {\it ctrl\_unpack} is invoked to unpack the control vector,  {\it ctrl\_unpack} is invoked to unpack the control vector
701  and following that call, the routine {\it ctrl\_pack}  or initialise the control variables.
702    Following the call of {\it adthe\_main\_loop},
703    the routine {\it ctrl\_pack}
704  is invoked to pack the control vector  is invoked to pack the control vector
705  (cf. Section \ref{section_ctrl}).  (cf. Section \ref{section_ctrl}).
706  If gradient checks are to be performed, the option  If gradient checks are to be performed, the option
# Line 706  the gradient has been computed via the a Line 715  the gradient has been computed via the a
715  The cost function $ {\cal J} $ is referred to as the {\sf dependent variable}.  The cost function $ {\cal J} $ is referred to as the {\sf dependent variable}.
716  It is a function of the input variables $ \vec{u} $ via the composition  It is a function of the input variables $ \vec{u} $ via the composition
717  $ {\cal J}(\vec{u}) \, = \, {\cal J}(M(\vec{u})) $.  $ {\cal J}(\vec{u}) \, = \, {\cal J}(M(\vec{u})) $.
718  The input is referred to as the  The input are referred to as the
719  {\sf independent variables} or {\sf control variables}.  {\sf independent variables} or {\sf control variables}.
720  All aspects relevant to the treatment of the cost function $ {\cal J} $  All aspects relevant to the treatment of the cost function $ {\cal J} $
721  (parameter setting, initialization, accumulation,  (parameter setting, initialization, accumulation,
722  final evaluation), are controlled by the package {\it pkg/cost}.  final evaluation), are controlled by the package {\it pkg/cost}.
723    The aspects relevant to the treatment of the independent variables
724    are controlled by the package {\it pkg/ctrl} and will be treated
725    in the next section.
726    
727  \input{part5/doc_cost_flow}  \input{part5/doc_cost_flow}
728    
# Line 745  Call {\it genmake} with the option Line 757  Call {\it genmake} with the option
757  {\tt genmake -enable=cost}.  {\tt genmake -enable=cost}.
758  %  %
759  \end{enumerate}  \end{enumerate}
760    N.B.: In general the following packages ought to be enabled
761    simultaneously: {\it autodiff, cost, ctrl}.
762  The basic CPP option to enable the cost function is {\bf ALLOW\_COST}.  The basic CPP option to enable the cost function is {\bf ALLOW\_COST}.
763  Each specific cost function contribution has its own option.  Each specific cost function contribution has its own option.
764  For the present example the option is {\bf ALLOW\_COST\_TRACER}.  For the present example the option is {\bf ALLOW\_COST\_TRACER}.
765  All cost-specific options are set in {\it ECCO\_CPPOPTIONS.h}  All cost-specific options are set in {\it ECCO\_CPPOPTIONS.h}
766  Since the cost function is usually used in conjunction with  Since the cost function is usually used in conjunction with
767  automatic differentiation, the CPP option  automatic differentiation, the CPP option
768  {\bf ALLOW\_ADJOINT\_RUN} should be defined  {\bf ALLOW\_ADJOINT\_RUN} (file {\it CPP\_OPTIONS.h}) and
769  (file {\it CPP\_OPTIONS.h}).  {\bf ALLOW\_AUTODIFF\_TAMC} (file {\it ECCO\_CPPOPTIONS.h})
770    should be defined.
771    
772  \subsubsection{Initialization}  \subsubsection{Initialization}
773  %  %
774  The initialization of the {\it cost} package is readily enabled  The initialization of the {\it cost} package is readily enabled
775  as soon as the CPP option {\bf ALLOW\_ADJOINT\_RUN} is defined.  as soon as the CPP option {\bf ALLOW\_COST} is defined.
776  %  %
777  \begin{itemize}  \begin{itemize}
778  %  %
# Line 831  from each contribution and sums over all Line 846  from each contribution and sums over all
846  \begin{equation}  \begin{equation}
847  {\cal J} \, = \,  {\cal J} \, = \,
848  {\rm fc} \, = \,  {\rm fc} \, = \,
849  {\rm mult\_tracer} \sum_{bi,\,bj}^{nSx,\,nSy}  {\rm mult\_tracer} \sum_{\text{global sum}} \sum_{bi,\,bj}^{nSx,\,nSy}
850  {\rm objf\_tracer}(bi,bj) \, + \, ...  {\rm objf\_tracer}(bi,bj) \, + \, ...
851  \end{equation}  \end{equation}
852  %  %
# Line 879  are controlled by the package {\it pkg/c Line 894  are controlled by the package {\it pkg/c
894  %  %
895  To enable the directory to be included to the compile list,  To enable the directory to be included to the compile list,
896  {\bf ctrl} has to be added to the {\bf enable} list in  {\bf ctrl} has to be added to the {\bf enable} list in
897  {\it .genmakerc} (or {\it genmake} itself).  {\it .genmakerc} or in {\it genmake} itself (analogous to {\it cost}
898    package, cf. previous section).
899  Each control variable is enabled via its own CPP option  Each control variable is enabled via its own CPP option
900  in {\it ECCO\_CPPOPTIONS.h}.  in {\it ECCO\_CPPOPTIONS.h}.
901    
# Line 1023  in the code takes on the form Line 1039  in the code takes on the form
1039  %  %
1040  Note, that reading an active variable corresponds  Note, that reading an active variable corresponds
1041  to a variable assignment. Its derivative corresponds  to a variable assignment. Its derivative corresponds
1042  to a write statement of the adjoint variable.  to a write statement of the adjoint variable, followed by
1043    a reset.
1044  The 'active file' routines have been designed  The 'active file' routines have been designed
1045  to support active read and corresponding adjoint active write  to support active read and corresponding adjoint active write
1046  operations (and vice versa).  operations (and vice versa).
# Line 1140  at intermediate times can be written usi Line 1157  at intermediate times can be written usi
1157  {\it addummy\_in\_stepping}.  {\it addummy\_in\_stepping}.
1158  This routine is part of the adjoint support package  This routine is part of the adjoint support package
1159  {\it pkg/autodiff} (cf.f. below).  {\it pkg/autodiff} (cf.f. below).
1160    The procedure is enabled using via the CPP-option
1161    {\bf ALLOW\_AUTODIFF\_MONITOR} (file {\it ECCO\_CPPOPTIONS.h}).
1162  To be part of the adjoint code, the corresponding S/R  To be part of the adjoint code, the corresponding S/R
1163  {\it dummy\_in\_stepping} has to be called in the forward  {\it dummy\_in\_stepping} has to be called in the forward
1164  model (S/R {\it the\_main\_loop}) at the appropriate place.  model (S/R {\it the\_main\_loop}) at the appropriate place.
1165    The adjoint common blocks are extracted from the adjoint code
1166    via the header file {\it adcommon.h}.
1167    
1168  {\it dummy\_in\_stepping} is essentially empty,  {\it dummy\_in\_stepping} is essentially empty,
1169  the corresponding adjoint routine is hand-written rather  the corresponding adjoint routine is hand-written rather
# Line 1169  the common blocks Line 1190  the common blocks
1190  {\bf /adtr1\_r/}, {\bf /adffields/},  {\bf /adtr1\_r/}, {\bf /adffields/},
1191  which have been extracted from the adjoint code to enable  which have been extracted from the adjoint code to enable
1192  access to the adjoint variables.  access to the adjoint variables.
1193    
1194    {\bf WARNING:} If the structure of the common blocks
1195    {\bf /dynvars\_r/}, {\bf /dynvars\_cd/}, etc., changes
1196    similar changes will occur in the adjoint common blocks.
1197    Therefore, consistency between the TAMC-generated common blocks
1198    and those in {\it adcommon.h} have to be checked.
1199  %  %
1200  \end{itemize}  \end{itemize}
1201    
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