7. Automatic Differentiation¶
Author: Patrick Heimbach
Automatic differentiation (AD), also referred to as algorithmic (or, more loosely, computational) differentiation, involves automatically deriving code to calculate partial derivatives from an existing fully nonlinear prognostic code (see Griewank and Walther, 2008 [griewank:08]). A software tool is used that parses and transforms source files according to a set of linguistic and mathematical rules. AD tools are like sourcetosource translators in that they parse a program code as input and produce a new program code as output (we restrict our discussion to sourcetosource tools, ignoring operatoroverloading tools). However, unlike a pure sourcetosource translation, the output program represents a new algorithm, such as the evaluation of the Jacobian, the Hessian, or higher derivative operators. In principle, a variety of derived algorithms can be generated automatically in this way.
MITgcm has been adapted for use with the Tangent linear and Adjoint Model Compiler (TAMC) and its successor TAF (Transformation of Algorithms in Fortran), developed by Ralf Giering (Giering and Kaminski, 1998 [giering:98], Giering, 2000 [giering:00]). The first application of the adjoint of MITgcm for sensitivity studies was published by Marotzke et al. (1999) [maroeta:99]. Stammer et al. (1997, 2002) [stammer:97] [stammer:02] use 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.
As of mid2007 we are also able to generate fairly efficient adjoint code of the MITgcm using a new, opensource AD tool, called OpenAD (see Naumann, 2006 [naumann:06] and Utke et al., 2008 [utke:08]). This enables us for the first time to compare adjoint models generated from different AD tools, providing an additional accuracy check, complementary to finitedifference gradient checks. OpenAD and its application to MITgcm is described in detail in Section 7.5.
The AD tool 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 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’s compositional elements.
7.1. Some basic algebra¶
Let \(\cal{M}\) be a general nonlinear, model, i.e., a mapping from the \(m\)dimensional space \(U \subset \mathbb{R}^m\) of input variables \(\vec{u}=(u_1,\ldots,u_m)\) (model parameters, initial conditions, boundary conditions such as forcing functions) to the \(n\)dimensional space \(V \subset \mathbb{R}^n\) of model output variable \(\vec{v}=(v_1,\ldots,v_n)\) (model state, model diagnostics, objective function, …) under consideration:
The vectors \(\vec{u} \in U\) and \(\vec{v} \in V\) may be represented with respect to some given basis vectors \({\rm span} (U) = \{ {\vec{e}_i} \}_{i = 1, \ldots , m}\) and \({\rm span} (V) = \{ {\vec{f}_j} \}_{j = 1, \ldots , n}\) as
Two routes may be followed to determine the sensitivity of the output variable \(\vec{v}\) to its input \(\vec{u}\).
7.1.1. Forward or direct sensitivity¶
Consider a perturbation to the input variables \(\delta \vec{u}\) (typically a single component \(\delta \vec{u} = \delta u_{i} \, {\vec{e}_{i}}\)). Their effect on the output may be obtained via the linear approximation of the model \({\cal M}\) in terms of its Jacobian matrix \(M\), evaluated in the point \(u^{(0)}\) according to
with resulting output perturbation \(\delta \vec{v}\). In components \(M_{j i} \, = \, \partial {\cal M}_{j} / \partial u_{i}\), it reads
(7.2) is the tangent linear model (TLM). In contrast to the full nonlinear model \({\cal M}\), the operator \(M\) 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 \((\gg O(10^{6})\) for largescale oceanographic application), it quickly becomes prohibitive to proceed directly as in (7.2), if the impact of each component \({\bf e_{i}}\) is to be assessed.
7.1.2. Reverse or adjoint sensitivity¶
Let us consider the special case of a scalar objective function \({\cal J}(\vec{v})\) of the model output (e.g., the total meridional heat transport, the total uptake of CO_{2} in the Southern Ocean over a time interval, or a measure of some modeltodata misfit)
The perturbation of \({\cal J}\) around a fixed point \({\cal J}_0\),
can be expressed in both bases of \(\vec{u}\) and \(\vec{v}\) with respect to their corresponding inner product \(\left\langle \,\, , \,\, \right\rangle\)
(note, that the gradient \(\nabla f\) is a covector, therefore its transpose is required in the above inner product). Then, using the representation of \(\delta {\cal J} = \left\langle \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{v} \, \right\rangle\), the definition of an adjoint operator \(A^{\ast}\) of a given operator \(A\),
which for finitedimensional vector spaces is just the transpose of \(A\),
and from (7.2), (7.5), we note that (omitting \(\)’s):
With the identity (7.5), we then find that the gradient \(\nabla _{u}{\cal J}\) can be readily inferred by invoking the adjoint \(M^{\ast }\) of the tangent linear model \(M\)
(7.7) is the adjoint model (ADM), in which \(M^T\) is the adjoint (here, the transpose) of the tangent linear operator \(M\), \(\,\delta \vec{v}^{\ast}\) the adjoint variable of the model state \(\vec{v}\), and \(\delta \vec{u}^{\ast}\) the adjoint variable of the control variable \(\vec{u}\).
The reverse nature of the adjoint calculation can be readily seen as follows. Consider a model integration which consists of \(\Lambda\) consecutive operations \({\cal M}_{\Lambda} ( {\cal M}_{\Lambda1} ( ...... ( {\cal M}_{\lambda} (...... ( {\cal M}_{1} ( {\cal M}_{0}(\vec{u}) ))))\), 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:
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 tangent linearity; note also that by definition \(\langle \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{v} \, \rangle = \nabla_v {\cal J} \cdot \delta \vec{v}\) )
whereas in reverse mode we have
clearly expressing the reverse nature of the calculation. (7.10) is at the heart of automatic adjoint compilers. If the intermediate steps \(\lambda\) in (7.8) – (7.10) represent the model state (forward or adjoint) at each intermediate time step as noted above, then correspondingly, \(M^T (\delta \vec{v}^{(\lambda) \, \ast}) = \delta \vec{v}^{(\lambda1) \, \ast}\) for the adjoint variables. It thus becomes evident that the adjoint calculation also yields the adjoint of each model state component \(\vec{v}^{(\lambda)}\) at each intermediate step \(\lambda\), namely
in close analogy to (7.7) we note in passing that the \(\delta \vec{v}^{(\lambda) \, \ast}\) are the Lagrange multipliers of the model equations which determine \(\vec{v}^{(\lambda)}\).
In components, (7.7) reads as follows. Let
denote the perturbations in \(\vec{u}\) and \(\vec{v}\), respectively, and their adjoint variables; further
is the Jacobi matrix of \({\cal M}\) (an \(n \times m\) matrix) such that \(\delta \vec{v} = M \cdot \delta \vec{u}\), or
Then (7.7) takes the form
or
Furthermore, the adjoint \(\delta v^{(\lambda) \, \ast}\) of any intermediate state \(v^{(\lambda)}\) may be obtained, using the intermediate Jacobian (an \(n_{\lambda+1} \times n_{\lambda}\) matrix)
and the shorthand notation for the adjoint variables \(\delta v^{(\lambda) \, \ast}_{j} = \frac{\partial}{\partial v^{(\lambda)}_{j}} {\cal J}^T\), \(j = 1, \ldots , n_{\lambda}\), for intermediate components, yielding
(7.9) and (7.10) are perhaps clearest in showing the advantage of the reverse over the forward mode if the gradient \(\nabla _{u}{\cal J}\), i.e., the sensitivity of the cost function \({\cal J}\) with respect to all input variables \(u\) (or the sensitivity of the cost function with respect to 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 (7.9) 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, (7.9) represents the projection of \(\nabla_u {\cal J}\) onto the \(i\)th component. The full gradient is retrieved from the \(m\) forward calculations. In contrast, (7.10) yields the full gradient \(\nabla _{u}{\cal J}\) (and all intermediate gradients \(\nabla _{v^{(\lambda)}}{\cal J}\)) within a single reverse calculation.
Note, that if \({\cal J}\) is a vectorvalued function of dimension \(l > 1\), (7.10) has to be modified according to
where now \(\delta \vec{J} \in \mathbb{R}^l\) 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. Strictly, the reverse mode is called adjoint mode only for \(l = 1\).
A detailed analysis of the underlying numerical operations shows that the computation of \(\nabla _{u}{\cal J}\) in this way requires about two to five times the computation of the cost function. Alternatively, the gradient vector could be approximated by finite differences, requiring \(m\) computations of the perturbed cost function.
To conclude, we give two examples of commonly used types of cost functions:
7.1.2.1. Example 1: \({\cal J} = v_{j} (T)\)¶
The cost function consists of the \(j\)th component of the model state \(\vec{v}\) at time \(T\). Then \(\nabla_v {\cal J}^T = {\vec{f}_{j}}\) is just the \(j\)th unit vector. The \(\nabla_u {\cal J}^T\) is the projection of the adjoint operator onto the \(j\)th component \({\bf f_{j}}\),
7.1.2.2. 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. 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
where \(H_{kj} = \partial {\cal H}_k / \partial v_{j}\) is the Jacobi matrix of the data projection operator. Thus, the gradient \(\nabla_u {\cal J}\) is given by the adjoint operator, driven by the model vs. data misfit:
7.1.3. Storing vs. recomputation in reverse mode¶
We note an important aspect of the forward vs. reverse mode calculation. Because of the local character of the derivative (a derivative is defined with respect to a point along the trajectory), the intermediate results of the model trajectory \(\vec{v}^{(\lambda+1)}={\cal M}_{\lambda}(v^{(\lambda)})\) may be required to evaluate the intermediate Jacobian \(M_{\lambda}_{\vec{v}^{(\lambda)}} \, \delta \vec{v}^{(\lambda)}\). This is the case for example for nonlinear expressions (momentum advection, nonlinear equation of state), and statedependent conditional statements (parameterization schemes). In the forward mode, the intermediate results are required in the same order as computed by the full forward model \({\cal M}\), 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 complete model state up to the 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 checkpointing (e.g., Griewank, 1992 [griewank:92], Restrepo et al., 1998 [restrepo:98]). It is depicted in Figure 7.1 for a 3level checkpointing (as an example, we give explicit numbers for a 3day integration with a 1hourly timestep in square brackets).
 In a first step, the model trajectory is subdivided into \({n}^{lev3}\) subsections [\({n}^{lev3}\)=3 1day intervals], with the label \(lev3\) for this outermost loop. The model is then integrated along the full trajectory, 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.
 In a second step each subsection itself is divided into \({n}^{lev2}\) subsections [\({n}^{lev2}\)=4 6hour intervals per subsection]. The model picks up at the last outermost dumped state \(v_{k_{n}^{lev3}}\) and is integrated forward in time along the last subsection, with the label \(lev2\) for this intermediate loop. The model state is now stored to disk at every \(k_{i}^{lev2}\)th timestep [i.e. 4 times, at \(i = 0,1,2,3\) corresponding to \(k_{i}^{lev2} = 48, 54, 60, 66\)].
 Finally, the model picks up at the last intermediate dump state \(v_{k_{n}^{lev2}}\) and is integrated forward in time along the last subsection, with the label \(lev1\) for this intermediate loop. Within this subsubsection only, parts of the model state are stored to memory at every timestep [i.e. every hour \(i=0,...,5\) corresponding to \(k_{i}^{lev1} = 66, 67, \ldots, 71\)]. The final state \(v_n = v_{k_{n}^{lev1}}\) is reached and the model state of all preceding timesteps along the last innermost subsection are available, enabling integration backwards in time along the last subsection. The adjoint can thus be computed along this last subsection \(k_{n}^{lev2}\).
This procedure is repeated consecutively for each previous subsection \(k_{n1}^{lev2}, \ldots, k_{1}^{lev2}\) carrying the adjoint computation to the initial time of the subsection \(k_{n}^{lev3}\). Then, the procedure is repeated for the previous subsection \(k_{n1}^{lev3}\) carrying the adjoint computation to the initial time \(k_{1}^{lev3}\).
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 \(n^{lev2} + n^{lev3}\) to disk and roughly \(n^{lev1}\) to memory (i.e., for the 3day integration with a total of 72 timesteps 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 3 full forward integrations of the model (one for each checkpointing level). The optimal balance of storage vs. recomputation certainly depends on the computing resources available and may be adjusted by adjusting the partitioning among the \(n^{lev3}, \,\, n^{lev2}, \,\, n^{lev1}\).
7.2. TLM and ADM generation in general¶
In this section we describe in a general fashion the parts of the code that are relevant for automatic differentiation using the software tool TAF. Modifications to use OpenAD are described in Section 7.5.
The basic flow is as follows:
the_model_main

 initialise_fixed

 #ifdef ALLOW_ADJOINT_RUN
 
  ctrl_unpack
 
  adthe_main_loop
  
   initialise_varia
   ctrl_map_forcing
   do iloop = 1, nTimeSteps
    forward_step
    cost_tile
   end do
   cost_final
  
   adcost_final
   do iloop = nTimeSteps, 1, 1
    adcost_tile
    adforward_step
   end do
   adctrl_map_forcing
   adinitialise_varia
  o
 
  ctrl_pack
 
 #else
 
  the_main_loop
 
 #endif

 #ifdef ALLOW_GRADIENT_CHECK
 
  grdchk_main
 o
 #endif
o
If CPP option
ALLOW_AUTODIFF_TAMC is defined, the driver routine
the_model_main.F,
instead of calling the_model_loop.F, invokes the
adjoint of this routine, adthe_main_loop.F
(case
#define ALLOW_ADJOINT_RUN, or the tangent linear of this routine
g_the_main_loop.F
(case #define ALLOW_TANGENTLINEAR_RUN), which
are the toplevel routines in terms of automatic differentiation. The
routines adthe_main_loop.F
or g_the_main_loop.F
are generated by
TAF. It contains both the forward integration of the full model, the
cost function calculation, any additional storing that is required for
efficient checkpointing, and the reverse integration of the adjoint
model.
[DESCRIBE IN A SEPARATE SECTION THE WORKING OF THE TLM]
The above structure of adthe_main_loop.F
has been
strongly simplified to focus on the essentials; in particular, no
checkpointing procedures are shown here. Prior to the call of
adthe_main_loop.F
, the routine ctrl_unpack.F
is invoked to unpack the
control vector or initialize the control variables. Following the call
of adthe_main_loop.F
, the routine ctrl_pack.F
is invoked to pack the
control vector (cf. Section 7.2.5). If gradient checks are to
be performed, the option #define ALLOW_GRDCHK is chosen. In this case
the driver routine grdchk_main.F
is called after the gradient has been
computed via the adjoint (cf. Section 7.3).
7.2.1. General setup¶
In order to configure ADrelated setups the following packages need to be enabled:
The packages are enabled by adding them to your experimentspecific
configuration file packages.conf
(see Section ???).
The following ADspecific CPP option files need to be customized:
 ECCO_CPPOPTIONS.h This header file collects CPP options for pkg/autodiff, pkg/cost, pkg/ctrl as well as ADunrelated options for the external forcing package pkg/exf. (NOTE: These options are not set in their packagespecific headers such as COST_OPTIONS.h, but are instead collected in the single header file ECCO_CPPOPTIONS.h. The packagespecific header files serve as simple placeholders at this point.)
 tamc.h This header configures the splitting of the time stepping loop with respect to the 3level checkpointing (see section ???).
7.2.2. Building the AD code using TAF¶
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 make targets are available:
ADtarget  output  description 

«MODE»«TOOL»only  «MODE»_«TOOL»_output.f  generates code for «MODE» using «TOOL» 
no make dependencies on .F .h  
useful for compiling on remote platforms  
«MODE»«TOOL»  «MODE»_«TOOL»_output.f  generates code for «MODE» using «TOOL» 
includes make dependencies on .F .h  
i.e. input for «TOOL» may be regenerated  
«MODE»all  mitgcmuv_«MODE»  generates code for «MODE» using «TOOL» 
and compiles all code  
(use of TAF is set as default) 
Here, the following placeholders are used:
 «TOOL»
 TAF
 TAMC
 «MODE»
 ad generates the adjoint model (ADM)
 ftl generates the tangent linear model (TLM)
 svd generates both ADM and TLM for singular value decomposition (SVD) type calculations
For example, to generate the adjoint model using TAF after routines (.F
)
or headers (.h
) have been modified, but without compilation,
type make adtaf
; or, to generate the tangent linear model using TAMC without
regenerating the input code, type make ftltamconly
.
A typical full build process to generate the ADM via TAF would look like follows:
% mkdir build
% cd build
% ../../../tools/genmake2 mods=../code_ad
% make depend
% make adall
7.2.3. The AD build process in detail¶
The make «MODE»all
target consists of the following procedures:
 A header file
AD_CONFIG.h
is generated which contains a CPP option on which code ought to be generated. Depending on themake
target, the contents is one of the following: #define ALLOW_ADJOINT_RUN
 #define ALLOW_TANGENTLINEAR_RUN
 #define ALLOW_ECCO_OPTIMIZATION
 A single file
«MODE»_input_code.f
is concatenated consisting of all.f
files that are part of the listAD_FILES
and all.flow
files that are part of the listAD_FLOW_FILES
.  The AD tool is invoked with the
«MODE»_«TOOL»_FLAGS
. The default AD tool flags in genmake2 can be overwritten by a tools/adjoint_options file (similar to the platformspecific tools/build_options, see Section 3.5.2.2). The AD tool writes the resulting AD code into the file«MODE»_input_code_ad.f
.  A short sed script tools/adjoint_sed is applied to
«MODE»_input_code_ad.f
to reinstate myThid into the CALL argument list of active file I/O. The result is written to file«MODE»_«TOOL»_output.f
.  All routines are compiled and an executable is generated.
7.2.3.1. The list AD_FILES
and .list
files¶
Not all routines are presented to the AD tool. Routines typically hidden are diagnostics routines which do not influence the cost function, but may create artificial flow dependencies such as I/O of active variables.
genmake2 generates a list (or variable) AD_FILES
which contains all
routines that are shown to the AD tool. This list is put together from
all files with suffix .list
that genmake2 finds in its search
directories. The list file for the core MITgcm routines is model/src/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 handwritten
AD code is available (see next section).
Each package directory contains its packagespecific list file
«PKG»_ad_diff.list
. For example, pkg/ptracers contains the file
ptracers_ad_diff.list.
Thus, enabling a package will automatically
extend the AD_FILES
list of genmake2 to incorporate the
packagespecific routines. Note that you will need to regenerate the
makefile if you enable a package (e.g., by adding it to packages.conf
)
and a Makefile
already exists.
7.2.3.2. The list AD_FLOW_FILES
and .flow
files¶
TAMC and TAF can evaluate userspecified directives that start with a
specific syntax (CADJ
, C$TAF
, !$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 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:
 which subroutine arguments are input/output
 which subroutine arguments are active
 which subroutine arguments are required to compute the cost
 which subroutine arguments are dependent
The syntax for the flow directives can be found in the AD tool manuals.
genmake2 generates a list (or variable) AD_FLOW_FILES
which
contains all files with suffix.flow
that it finds in its search
directories. The flow directives for the core MITgcm routines of
eesupp/src/ and model/src/ reside in pkg/autodiff/. This directory also
contains handwritten adjoint code for the MITgcm WRAPPER (Section 6.2).
Flow directives for packagespecific routines are contained in the
corresponding package directories in the file «PKG»_ad.flow
, e.g.,
ptracersspecific directives are in ptracers_ad.flow.
7.2.3.3. Store directives for 3level checkpointing¶
The storing that is required at each period of the 3level checkpointing is controlled by three toplevel headers.
do ilev_3 = 1, nchklev_3
# include ``checkpoint_lev3.h''
do ilev_2 = 1, nchklev_2
# include ``checkpoint_lev2.h''
do ilev_1 = 1, nchklev_1
# include ``checkpoint_lev1.h''
...
end do
end do
end do
All files checkpoint_lev?.h
are contained in directory pkg/autodiff/.
7.2.3.4. Changing the default AD tool flags: ad_options files¶
7.2.3.5. Handwritten adjoint code¶
7.2.4. The cost function (dependent variable)¶
The cost function \({\cal J}\) is referred to as the 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 are referred to as the independent variables or control variables. All aspects relevant to the treatment of the cost function \({\cal J}\) (parameter setting, initialization, accumulation, final evaluation), are controlled by the package pkg/cost. The aspects relevant to the treatment of the independent variables are controlled by the package pkg/ctrl and will be treated in the next section.
the_model_main

 initialise_fixed
 
  packages_readparms
 
  cost_readparms
 o

 the_main_loop
... 
 initialise_varia
 
  packages_init_variables
 
  cost_init
 o

 do iloop = 1,nTimeSteps
  forward_step
  cost_tile
  
   cost_tracer
 end do

 cost_final
o
7.2.4.1. Enabling the package¶
pkg/cost is enabled by adding the line cost
to your file packages.conf
(see Section ???).
In general the following packages ought to be enabled simultaneously: pkg/autodiff, pkg/ctrl, and pkg/cost. The basic CPP option to enable the cost function is ALLOW_COST. Each specific cost function contribution has its own option. For the present example the option is ALLOW_COST_TRACER. All costspecific options are set in ECCO_CPPOPTIONS.h Since the cost function is usually used in conjunction with automatic differentiation, the CPP option ALLOW_ADJOINT_RUN (file CPP_OPTIONS.h) and ALLOW_AUTODIFF_TAMC (file ECCO_CPPOPTIONS.h) should be defined.
7.2.4.2. Initialization¶
The initialization of pkg/cost is readily enabled as soon as the CPP option ALLOW_COST is defined.
 The S/R cost_readparms.F
reads runtime flags and parameters from file
data.cost
. For the present example the only relevant parameter read is mult_tracer. This multiplier enables different cost function contributions to be switched on (= 1.
) or off (= 0.
) at runtime. For more complex cost functions which involve model vs. data misfits, the corresponding data filenames and data specifications (start date and time, period, …) are read in this S/R.  The S/R cost_init_varia.F initializes the different cost function contributions. The contribution for the present example is objf_tracer which is defined on each tile (bi,bj).
7.2.4.3. Accumulation¶
The ’driver’ routine cost_tile.F is called at the end of each time step. Within this ’driver’ routine, S/R are called for each of the chosen cost function contributions. In the present example (ALLOW_COST_TRACER), S/R cost_tracer.F is called. It accumulates objf_tracer according to eqn. (ref:asktheauthor).
7.2.4.4. Finalize all contributions¶
At the end of the forward integration S/R cost_final.F is called. It accumulates the total cost function fc from each contribution and sums over all tiles:
The total cost function fc will be the ’dependent’ variable in the argument list for TAF, i.e.,
taf output 'fc' ...
*************
the_main_loop
*************

 initialise_varia
 
 ...
  packages_init_varia
  
  ...
   #ifdef ALLOW_ADJOINT_RUN
   call ctrl_map_ini
   call cost_ini
   #endif
  ...
  o
 ...
 o
...
 #ifdef ALLOW_ADJOINT_RUN
 call ctrl_map_forcing
 #endif
...
 #ifdef ALLOW_TAMC_CHECKPOINTING
do ilev_3 = 1,nchklev_3
 do ilev_2 = 1,nchklev_2
 do ilev_1 = 1,nchklev_1
 iloop = (ilev_31)*nchklev_2*nchklev_1 +
 (ilev_21)*nchklev_1 + ilev_1
 #else
 do iloop = 1, nTimeSteps
 #endif
 
  call forward_step
 
  #ifdef ALLOW_COST
  call cost_tile
  #endif
 
  enddo
 o

 #ifdef ALLOW_COST
 call cost_final
 #endif
o
7.2.5. The control variables (independent variables)¶
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 with respect to control variables do not vanish are called active variables. All subroutines whose derivative with respect to the control variables don’t vanish are called active routines. Read and write operations from and to file can be viewed as variable assignments. Therefore, files to which active variables are written and from which active variables are read are called active files. All aspects relevant to the treatment of the control variables (parameter setting, initialization, perturbation) are controlled by the package pkg/ctrl.
the_model_main

 initialise_fixed
 
  packages_readparms
 
  cost_readparms
 o

 the_main_loop
... 
 initialise_varia
 
  packages_init_variables
 
  cost_init
 o

 do iloop = 1,nTimeSteps
  forward_step
  cost_tile
  
   cost_tracer
 end do

 cost_final
o
7.2.5.1. genmake2 and CPP options¶
Package pkg/ctrl is enabled by adding the line ctrl
to your file packages.conf
.
Each control variable is enabled via its own CPP option in
ECCO_CPPOPTIONS.h.
7.2.5.2. Initialization¶
 The S/R ctrl_readparms.F
reads runtime flags and parameters from file
data.ctrl
. For the present example the file contains the file names of each control variable that is used. In addition, the number of wet points for each control variable and the net dimension of the space of control variables (counting wet points only) nvarlength is determined. Masks for wet points for each tile (bi,bj) and vertical layer k are generated for the three relevant categories on the Cgrid: nWetCtile for tracer fields, nWetWtile for zonal velocity fields, nWetStile for meridional velocity fields.  Two important issues related to the handling of the control variables in 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 fields during the model initialization phase. Similarly, the corresponding adjoint fields which represent the gradient of the cost function with respect to the control variables are written to file at the end of the adjoint integration. Second, in addition to the files holding the 2D and 3D control variables and the corresponding cost gradients, a 1D control vector and 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, an option is available (ALLOW_NONDIMENSIONAL_CONTROL_IO) to nondimensionalize 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.
The files holding fields and vectors of the control variables and gradient are generated and initialized in S/R ctrl_unpack.F.
7.2.5.3. Perturbation of the independent variables¶
The dependency flow for differentiation with respect to the controls starts with adding a perturbation onto the input variable, thus defining the independent or control variables for TAF. Three types of controls may be considered:
Consider as an example the initial tracer distribution pTracer as control variable. After pTracer has been initialized in ptracers_init_varia.F (dynamical variables such as temperature and salinity are initialized in ini_fields.F), a perturbation anomaly is added to the field in S/R ctrl_map_ini.F:
(7.11)¶\[\begin{split}\begin{aligned} u & = \, u_{[0]} \, + \, \Delta u \\ {\bf tr1}(...) & = \, {\bf tr1_{ini}}(...) \, + \, {\bf xx\_tr1}(...) \end{aligned}\end{split}\]xx_tr1 is a 3D 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 in the context of automatic differentiation since TAF 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 when calling TAF:
taf input 'xx_tr1 ...' ...
Now, as mentioned above, MITgcm avoids maintaining an array for each control variable by reading the perturbation to a temporary array from file. To ensure the symbolic link to be recognized by TAF, a scalar dummy variable
xx_tr1_dummy
is introduced and an ’active read’ routine of the adjoint support package pkg/autodiff is invoked. The readprocedure is tagged with the variablexx_tr1_dummy
enabling TAF to recognize the initialization of the perturbation. The modified call of TAF thus readstaf input 'xx_tr1_dummy ...' ...
and the modified operation (to perturb) in the code takes on the form
call active_read_xyz( & ..., tmpfld3d, ..., xx_tr1_dummy, ... ) tr1(...) = tr1(...) + tmpfld3d(...)
Note that reading an active variable corresponds to a variable assignment. Its derivative corresponds 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 adjoint active write operations (and vice versa).
The handling of boundary values as control variables proceeds exactly analogous to the initial values with the symbolic perturbation taking place in S/R ctrl_map_forcing.F. Note however an important difference: Since the boundary values are time dependent with a new forcing field applied at each time step, the general problem may be thought of as a new control variable at each time step (or, if the perturbation is averaged over a certain period, at each \(N\) timesteps), i.e.,
\[u_{\rm forcing} \, = \, \{ \, u_{\rm forcing} ( t_n ) \, \}_{ n \, = \, 1, \ldots , {\rm nTimeSteps} }\]In the current example an equilibrium state is considered, and only an initial perturbation to surface forcing is applied with respect to the equilibrium state. A time dependent treatment of the surface forcing is implemented in the ECCO environment, involving the calendar (pkg/cal) and external forcing (pkg/exf) packages.
This routine is not yet implemented, but would proceed proceed along the same lines as the initial value sensitivity. The mixing parameters diffkr and kapgm are currently added as controls in ctrl_map_ini.F.
7.2.5.4. Output of adjoint variables and gradient¶
Several ways exist to generate output of adjoint fields.
In ctrl_map_ini.F, ctrl_map_forcing.F:
 The control variable fields
xx\_«...»
: before the forward integration, the control variables are read from file«xx\_ ...»
and added to the model field.  The adjoint variable fields
adxx\_«...»
, i.e., the gradient \(\nabla _{u}{\cal J}\) for each control variable: after the adjoint integration the corresponding adjoint variables are written toadxx\_«...»
.
 The control variable fields
In ctrl_unpack.F, ctrl_pack.F:
 The control vector
vector_ctrl
: at the very beginning of the model initialization, the updated compressed control vector is read (or initialized) and distributed to 2D and 3D control variable fields.  The gradient vector
vector_grad
: at the very end of the adjoint integration, the 2D and 3D adjoint variables are read, compressed to a single vector and written to file.
 The control vector
In addition to writing the gradient at the end of the forward/adjoint integration, many more adjoint variables of the model state at intermediate times can be written using S/R addummy_in_stepping.F. The procedure is enabled using via the CPPoption ALLOW_AUTODIFF_MONITOR (file ECCO_CPPOPTIONS.h). To be part of the adjoint code, the corresponding S/R dummy_in_stepping.F has to be called in the forward model (S/R the_main_loop.F) at the appropriate place. The adjoint common blocks are extracted from the adjoint code via the header file adcommon.h.
dummy_in_stepping.F is essentially empty, the corresponding adjoint routine is handwritten rather than generated automatically. Appropriate flow directives (dummy_in_stepping.flow) ensure that TAMC does not automatically generate addummy_in_stepping.F by trying to differentiate dummy_in_stepping.F, but instead refers to the handwritten routine.
dummy_in_stepping.F is called in the forward code at the beginning of each timestep, before the call to model/src/dynamics.F, thus ensuring that addummy_in_stepping.F is called at the end of each timestep in the adjoint calculation, after the call to addummy_in_dynamics.F.
addummy_in_stepping.F includes the header files adcommon.h. This header file is also handwritten. It contains the common blocks addynvars_r, addynvars_cd, addynvars_diffkr, addynvars_kapgm, adtr1_r, adffields, which have been extracted from the adjoint code to enable access to the adjoint variables.
WARNING: If the structure of the common blocks dynvars_r, dynvars_cd, etc., changes similar changes will occur in the adjoint common blocks. Therefore, consistency between the TAMCgenerated common blocks and those in adcommon.h have to be checked.
7.2.5.5. Control variable handling for optimization applications¶
In optimization mode the cost function \({\cal J}(u)\) is sought to be minimized with respect to a set of control variables \(\delta {\cal J} \, = \, 0\), in an iterative manner. The gradient \(\nabla _{u}{\cal J} _{u_{[k]}}\) together 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. quasiNewton method, conjugate gradient, … (Gilbert and Lemaréchal, 1989 [gillem:89]) to compute an update in the control variable for iteration step \(k+1\):
\(u_{[k+1]}\) then serves as input for a forward/adjoint run to determine \({\cal J}\) and \(\nabla _{u}{\cal J}\) at iteration step \(k+1\). Figure 7.2 sketches the flow between forward/adjoint model and the minimization routine.
The routines ctrl_unpack.F and ctrl_pack.F provide the link between the model and the minimization routine. As described in Section ref:asktheauthor the ctrl_unpack.F and ctrl_pack.F routines read and write control and gradient vectors which are compressed to contain only wet points, in addition to the full 2D and 3D fields. The corresponding I/O flow is shown in Figure 7.3:
ctrl_unpack.F reads the updated control vector
vector_ctrl_<k>
. It distributes the
different control variables to 2D and 3D files
xx_«...»<k>
. At the start of the forward
integration the control variables are read from
xx_«...»<k>
and added to the field.
Correspondingly, at the end of the adjoint integration the adjoint
fields are written to adxx_«...»<k>
, again via
the active file routines. Finally,
ctrl_pack.F collects all adjoint
files and writes them to the compressed vector file
vector_grad_<k>
.
NOTE: These options are not set in their packagespecific headers such as COST_OPTIONS.h, but are instead collected in the single header file ECCO_CPPOPTIONS.h. The packagespecific header files serve as simple placeholders at this point.
7.3. The gradient check package¶
An indispensable test to validate the gradient computed via the adjoint is a comparison against finite difference gradients. The gradient check package pkg/grdchk enables such tests in a straightforward and easy manner. The driver routine grdchk_main.F is called from the_model_main.F after the gradient has been computed via the adjoint model (cf. flow chart ???).
The gradient check proceeds as follows: The \(i\)th component of the gradient \((\nabla _{u}{\cal J}^T)_i\) is compared with the following finitedifference gradient:
A gradient check at point \(u_i\) may generally considered to be successful if the deviation of the ratio between the adjoint and the finite difference gradient from unity is less than 1 percent,
7.3.1. Code description¶
7.3.2. Code configuration¶
The relevant CPP precompile options are set in the following files:
 CPP_OPTIONS.h  Together with the flag ALLOW_ADJOINT_RUN, define the flag ALLOW_GRADIENT_CHECK.
The relevant runtime flags are set in the files:
data.pkg
 Set useGrdchk= .TRUE.
data.grdchk
the_model_main

 ctrl_unpack
 adthe_main_loop  unperturbed cost function and
 ctrl_pack adjoint gradient are computed here

 grdchk_main

 grdchk_init
 do icomp=...  loop over control vector elements

 grdchk_loc  determine location of icomp on grid

 grdchk_getxx  get control vector component from file
 perturb it and write back to file
 grdchk_getadxx  get gradient component calculated
 via adjoint
 the_main_loop  forward run and cost evaluation
 with perturbed control vector element
 calculate ratio of adj. vs. finite difference gradient

 grdchk_setxx  Reset control vector element

 grdchk_print  print results
7.4. Adjoint dump & restart – divided adjoint (DIVA)¶
Authors: Patrick Heimbach & Geoffrey Gebbie, 07Mar2003*
*NOTE:THIS SECTION IS SUBJECT TO CHANGE. IT REFERS TO TAF1.4.26.
Previous TAF versions are incomplete and have problems with both TAF
options pure
and mpi
.
The code which is tuned to the DIVA implementation of this TAF version
is checkpoint50
(MITgcm) and ecco_c50_e28
(ECCO).
7.4.1. Introduction¶
Most high performance computing (HPC) centers require the use of batch jobs for code execution. Limits in maximum available CPU time and memory may prevent the adjoint code execution from fitting into any of the available queues. This presents a serious limit for large scale / long time adjoint ocean and climate model integrations. The MITgcm itself enables the split of the total model integration into subintervals through standard dump/restart of/from the full model state. For a similar procedure to run in reverse mode, the adjoint model requires, in addition to the model state, the adjoint model state, i.e., all variables with derivative information which are needed in an adjoint restart. This adjoint dump & restart is also termed ’divided adjoint (DIVA)’.
For this to work in conjunction with automatic differentiation, an AD tool needs to perform the following tasks:
 identify an adjoint state, i.e., those sensitivities whose
accumulation is interrupted by a dump/restart and which influence the
outcome of the gradient. Ideally, this state consists of
 the adjoint of the model state,
 the adjoint of other intermediate results (such as control variables, cost function contributions, etc.)
 bookkeeping indices (such as loop indices, etc.)
 generate code for storing and reading adjoint state variables
 generate code for bookkeeping , i.e., maintaining a file with index information
 generate a suitable adjoint loop to propagate adjoint values for dump/restart with a minimum overhead of adjoint intermediate values.
TAF (but not TAMC!) generates adjoint code which performs the above specified tasks. It is closely tied to the adjoint multilevel checkpointing. The adjoint state is dumped (and restarted) at each step of the outermost checkpointing level and adjoint integration is performed over one outermost checkpointing interval. Prior to the adjoint computations, a full forward sweep is performed to generate the outermost (forward state) tapes and to calculate the cost function. In the current implementation, the forward sweep is immediately followed by the first adjoint leg. Thus, in theory, the following steps are performed (automatically)
 1st model call: This is the case if file
costfinal
does not exist. S/Rmdthe_main_loop.f
(generated by TAF) is called. calculate forward trajectory and dump model state after each
outermost checkpointing interval to files
tapelev3
 calculate cost function
fc
and write it to filecostfinal
 calculate forward trajectory and dump model state after each
outermost checkpointing interval to files
 2nd and all remaining model calls:
This is the case if file costfinal does exist. S/R
adthe_main_loop.f
(generated by TAF) is called. (forward run and cost function call is avoided since all values
are known)
 if 1st adjoint leg:
create index file
divided.ctrl
which contains info on current checkpointing index \(ilev3\)  if not \(i\)th adjoint leg: adjoint picks up at \(ilev3 = nlev3i+1\) and runs to \(nlev3  i\)
 if 1st adjoint leg:
create index file
 perform adjoint leg from \(nlev3i+1\) to \(nlev3  i\)
 dump adjoint state to file
snapshot
 dump index file
divided.ctrl
for next adjoint leg  in the last step the gradient is written.
 (forward run and cost function call is avoided since all values
are known)
A few modifications were performed in the forward code, obvious ones such as adding the corresponding TAFdirective at the appropriate place, and less obvious ones (avoid some reinitializations, when in an intermediate adjoint integration interval).
[For TAF1.4.20 a number of handmodifications were necessary to compensate for TAF bugs. Since we refer to TAF1.4.26 onwards, these modifications are not documented here].
7.4.2. Recipe 1: single processor¶
In ECCO_CPPOPTIONS.h set:
 #define ALLOW_DIVIDED_ADJOINT
 #undef ALLOW_DIVIDED_ADJOINT_MPI
Generate adjoint code. Using the TAF option
pure
, two codes are generated:mdthe_main_loop.f
: Is responsible for the forward trajectory, storing of outermost checkpoint levels to file, computation of cost function, and storing of cost function to file (1st step).adthe_main_loop.f
: Is responsible for computing one adjoint leg, dump adjoint state to file and write index info to file (2nd and consecutive steps).for adjoint code generation, e.g., add
pure
to TAF option listmake adtaf
One modification needs to be made to adjoint codes in S/R
adecco_the_main_loop.f
(generated by TAF):There’s a remaining issue with the
pure
option. Thecall ad...
betweencall ad...
and the read of thesnapshot
file should be called only in the first adjoint leg between \(nlev3\) and \(nlev31\). In the eccobranch, the following lines should be bracketed by anif (idivbeg .GE. nchklev_3) then
, thus:... xx_psbar_mean_dummy = onetape_xx_psbar_mean_dummy_3h(1) xx_tbar_mean_dummy = onetape_xx_tbar_mean_dummy_4h(1) xx_sbar_mean_dummy = onetape_xx_sbar_mean_dummy_5h(1) call barrier( mythid ) cAdd( if (idivbeg .GE. nchklev_3) then cAdd) call adcost_final( mythid ) call barrier( mythid ) call adcost_sst( mythid ) call adcost_ssh( mythid ) call adcost_hyd( mythid ) call adcost_averagesfields( mytime,myiter,mythid ) call barrier( mythid ) cAdd( endif cAdd) C C read snapshot C if (idivbeg .lt. nchklev_3) then open(unit=77,file='snapshot',status='old',form='unformatted', $iostat=iers) ...
For the main code, in all likelihood the block which needs to be bracketed consists of
adcost_final.f
(generated by TAF) only.Now the code can be copied as usual to
adjoint_model.F
and then be compiled:make adchange
then compile
7.4.3. Recipe 2: multi processor (MPI)¶
On the machine where you execute the code (most likely not the machine where you run TAF) find the includes directory for MPI containing
mpif.h
. Either copympif.h
to the machine where you generate the.f
files before TAFing, or add the path to the includes directory to your genmake2 platform setup, TAF needs some MPI parameter settings (essentiallympi_comm_world
andmpi_integer
) to incorporate those in the adjoint code.In ECCO_CPPOPTIONS.h set
 #define ALLOW_DIVIDED_ADJOINT
 #define ALLOW_DIVIDED_ADJOINT_MPI
This will include the header file
mpif.h
into the top level routine for TAF.Add the TAF option
mpi
to the TAF argument list in the makefile.Follow the same steps as in Recipe 1.
7.5. Adjoint code generation using OpenAD¶
Authors: Jean Utke, Patrick Heimbach and Chris Hill
7.5.1. Introduction¶
The development of OpenAD was initiated as part of the ACTS (Adjoint Compiler Technology & Standards) project funded by the NSF Information Technology Research (ITR) program. The main goals for OpenAD initially defined for the ACTS project are:
 develop a flexible, modular, open source tool that can generate adjoint codes of numerical simulation programs,
 establish a platform for easy implementation and testing of source transformation algorithms via a languageindependent abstract intermediate representation,
 support for source code written in C and Fortan, and
 generate efficient tangent linear and adjoint for the MIT general circulation model.
OpenAD’s homepage is at http://wwwunix.mcs.anl.gov/OpenAD. A development WIKI is at http://wiki.mcs.anl.gov/OpenAD/index.php/Main_Page. From the WIKI’s main page, click on Handling GCM for various aspects pertaining to differentiating the MITgcm with OpenAD.
7.5.2. Downloading and installing OpenAD¶
The OpenAD webpage has a detailed description on how to download and build OpenAD. From its homepage, please click on Binaries. You may either download prebuilt binaries for quick trial, or follow the detailed build process described at http://www.mcs.anl.gov/OpenAD/access.shtml.
7.5.3. Building MITgcm adjoint with OpenAD¶
17January2008
OpenAD was successfully built on head node of itrda.acesgrid.org
,
for following system:
> uname a
Linux itrda 2.6.22.242.fc6 #1 SMP Wed Aug 15 12:34:26 EDT 2007 i686 i686 i386 GNU/Linux
> cat /proc/version
Linux version 2.6.22.242.fc6 (brewbuilder@hs20bc24.build.redhat.com)
(gcc version 4.1.2 20070626 (Red Hat 4.1.213)) #1 SMP Wed Aug 15 12:34:26 EDT 2007
> module load ifc/9.1.036 icc/9.1.042
Head of MITgcm branch (checkpoint59m
with some modifications) was used for
building adjoint code. Following routing needed special care (revert
to revision 1.1): http://wwwcvs.mitgcm.org/viewvc/MITgcm/MITgcm_contrib/heimbach/OpenAD/OAD_support/active_module.f90?hideattic=0&view=markup.