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 non-linear prognostic code (see Griewank and Walther, 2008 [GW08]). A software tool is used that parses and transforms source files according to a set of linguistic and mathematical rules. AD tools are like source-to-source translators in that they parse a program code as input and produce a new program code as output (we restrict our discussion to source-to-source tools, ignoring operator-overloading tools). However, unlike a pure source-to-source 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 [GK98], Giering, 2000 [Gie00]). The first application of the adjoint of MITgcm for sensitivity studies was published by Marotzke et al. (1999) [MGZ+99]. Stammer et al. (1997, 2002) [SWG+97] [SWG+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 mid-2007 we are also able to generate fairly efficient adjoint code of the MITgcm using a new, open-source AD tool, called OpenAD (see Naumann, 2006 [NUH+06] and Utke et al., 2008 [UNF+08]). This enables us for the first time to compare adjoint models generated from different AD tools, providing an additional accuracy check, complementary to finite-difference 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:

(7.1)\[\begin{split}\begin{aligned} {\cal M} \, : & \, U \,\, \longrightarrow \, V \\ ~ & \, \vec{u} \,\, \longmapsto \, \vec{v} \, = \, {\cal M}(\vec{u})\end{aligned}\end{split}\]

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

\[\vec{u} \, = \, \sum_{i=1}^{m} u_i \, {\vec{e}_i}, \qquad \vec{v} \, = \, \sum_{j=1}^{n} v_j \, {\vec{f}_j}\]

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

(7.2)\[\delta \vec{v} \, = \, M |_{\vec{u}^{(0)}} \, \delta \vec{u}\]

with resulting output perturbation \(\delta \vec{v}\). In components \(M_{j i} \, = \, \partial {\cal M}_{j} / \partial u_{i}\), it reads

(7.3)\[\delta v_{j} \, = \, \sum_{i} \left. \frac{\partial {\cal M}_{j}}{\partial u_{i}} \right|_{u^{(0)}} \, \delta u_{i}\]

(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 large-scale 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 CO2 in the Southern Ocean over a time interval, or a measure of some model-to-data misfit)

(7.4)\[\begin{split}\begin{aligned} \begin{array}{cccccc} {\cal J} \, : & U & \longrightarrow & V & \longrightarrow & \mathbb{R} \\ ~ & \vec{u} & \longmapsto & \vec{v}={\cal M}(\vec{u}) & \longmapsto & {\cal J}(\vec{u}) = {\cal J}({\cal M}(\vec{u})) \end{array}\end{aligned}\end{split}\]

The perturbation of \({\cal J}\) around a fixed point \({\cal J}_0\),

\[{\cal J} \, = \, {\cal J}_0 \, + \, \delta {\cal J}\]

can be expressed in both bases of \(\vec{u}\) and \(\vec{v}\) with respect to their corresponding inner product \(\left\langle \,\, , \,\, \right\rangle\)

(7.5)\[\begin{split}\begin{aligned} {\cal J} & = \, {\cal J} |_{\vec{u}^{(0)}} \, + \, \left\langle \, \nabla _{u}{\cal J}^T |_{\vec{u}^{(0)}} \, , \, \delta \vec{u} \, \right\rangle \, + \, O(\delta \vec{u}^2) \\ ~ & = \, {\cal J} |_{\vec{v}^{(0)}} \, + \, \left\langle \, \nabla _{v}{\cal J}^T |_{\vec{v}^{(0)}} \, , \, \delta \vec{v} \, \right\rangle \, + \, O(\delta \vec{v}^2) \end{aligned}\end{split}\]

(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 \(\delta {\cal J} = \left\langle \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{v} \, \right\rangle\), the definition of an adjoint operator \(A^{\ast}\) of a given operator \(A\),

\[\left\langle \, A^{\ast} \vec{x} \, , \, \vec{y} \, \right\rangle = \left\langle \, \vec{x} \, , \, A \vec{y} \, \right\rangle\]

which for finite-dimensional vector spaces is just the transpose of \(A\),

\[A^{\ast} \, = \, A^T\]

and from (7.2), (7.5), we note that (omitting \(|\)’s):

(7.6)\[\delta {\cal J} \, = \, \left\langle \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{v} \, \right\rangle \, = \, \left\langle \, \nabla _{v}{\cal J}^T \, , \, M \, \delta \vec{u} \, \right\rangle \, = \, \left\langle \, M^T \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{u} \, \right\rangle\]

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)\[\begin{split}\begin{aligned} \nabla _{u}{\cal J}^T |_{\vec{u}} & = \, M^T |_{\vec{u}} \cdot \nabla _{v}{\cal J}^T |_{\vec{v}} \\ ~ & = \, M^T |_{\vec{u}} \cdot \delta \vec{v}^{\ast} \\ ~ & = \, \delta \vec{u}^{\ast} \end{aligned}\end{split}\]

(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 \({\cal M}_{\Lambda} ( {\cal M}_{\Lambda-1} ( ...... ( {\cal M}_{\lambda} (...... ( {\cal M}_{1} ( {\cal M}_{0}(\vec{u}) ))))\), 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:

(7.8)\[{\cal J}({\cal M}(\vec{u})) \, = \, {\cal J} ( {\cal M}_{\Lambda} ( {\cal M}_{\Lambda-1} ( ...... ( {\cal M}_{\lambda} (...... ( {\cal M}_{1} ( {\cal M}_{0}(\vec{u}) )))))\]

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 \(\langle \, \nabla _{v}{\cal J}^T \, , \, \delta \vec{v} \, \rangle = \nabla_v {\cal J} \cdot \delta \vec{v}\) )

(7.9)\[\begin{split}\begin{aligned} \nabla_v {\cal J} (M(\delta \vec{u})) & = \, \nabla_v {\cal J} \cdot M_{\Lambda} \cdot ...... \cdot M_{\lambda} \cdot ...... \cdot M_{1} \cdot M_{0} \cdot \delta \vec{u} \\ ~ & = \, \nabla_v {\cal J} \cdot \delta \vec{v} \\ \end{aligned}\end{split}\]

whereas in reverse mode we have

(7.10)\[\begin{split}\boxed{ \begin{aligned} M^T ( \nabla_v {\cal J}^T) & = \, M_{0}^T \cdot M_{1}^T \cdot ...... \cdot M_{\lambda}^T \cdot ...... \cdot M_{\Lambda}^T \cdot \nabla_v {\cal J}^T \\ ~ & = \, M_{0}^T \cdot M_{1}^T \cdot ...... \cdot \nabla_{v^{(\lambda)}} {\cal J}^T \\ ~ & = \, \nabla_u {\cal J}^T \end{aligned}}\end{split}\]

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, \(M^T (\delta \vec{v}^{(\lambda) \, \ast}) = \delta \vec{v}^{(\lambda-1) \, \ast}\) 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

\[\begin{split}\boxed{ \begin{aligned} \nabla_{v^{(\lambda)}} {\cal J}^T |_{\vec{v}^{(\lambda)}} & = \, M_{\lambda}^T |_{\vec{v}^{(\lambda)}} \cdot ...... \cdot M_{\Lambda}^T |_{\vec{v}^{(\lambda)}} \cdot \delta \vec{v}^{\ast} \\ ~ & = \, \delta \vec{v}^{(\lambda) \, \ast} \end{aligned}}\end{split}\]

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

\[\begin{split}\begin{array}{rclcrcl} \delta \vec{u} & = & \left( \delta u_1,\ldots, \delta u_m \right)^T , & \qquad & \delta \vec{u}^{\ast} \,\, = \,\, \nabla_u {\cal J}^T & = & \left( \frac{\partial {\cal J}}{\partial u_1},\ldots, \frac{\partial {\cal J}}{\partial u_m} \right)^T \\ \delta \vec{v} & = & \left( \delta v_1,\ldots, \delta u_n \right)^T , & \qquad & \delta \vec{v}^{\ast} \,\, = \,\, \nabla_v {\cal J}^T & = & \left( \frac{\partial {\cal J}}{\partial v_1},\ldots, \frac{\partial {\cal J}}{\partial v_n} \right)^T \\ \end{array}\end{split}\]

denote the perturbations in \(\vec{u}\) and \(\vec{v}\), respectively, and their adjoint variables; further

\[\begin{split}M \, = \, \left( \begin{array}{ccc} \frac{\partial {\cal M}_1}{\partial u_1} & \ldots & \frac{\partial {\cal M}_1}{\partial u_m} \\ \vdots & ~ & \vdots \\ \frac{\partial {\cal M}_n}{\partial u_1} & \ldots & \frac{\partial {\cal M}_n}{\partial u_m} \\ \end{array} \right)\end{split}\]

is the Jacobi matrix of \({\cal M}\) (an \(n \times m\) matrix) such that \(\delta \vec{v} = M \cdot \delta \vec{u}\), or

\[\delta v_{j} \, = \, \sum_{i=1}^m M_{ji} \, \delta u_{i} \, = \, \sum_{i=1}^m \, \frac{\partial {\cal M}_{j}}{\partial u_{i}} \delta u_{i}\]

Then (7.7) takes the form

\[\delta u_{i}^{\ast} \, = \, \sum_{j=1}^n M_{ji} \, \delta v_{j}^{\ast} \, = \, \sum_{j=1}^n \, \frac{\partial {\cal M}_{j}}{\partial u_{i}} \delta v_{j}^{\ast}\]

or

\[\begin{split}\left( \begin{array}{c} \left. \frac{\partial}{\partial u_1} {\cal J} \right|_{\vec{u}^{(0)}} \\ \vdots \\ \left. \frac{\partial}{\partial u_m} {\cal J} \right|_{\vec{u}^{(0)}} \\ \end{array} \right) \, = \, \left( \begin{array}{ccc} \left. \frac{\partial {\cal M}_1}{\partial u_1} \right|_{\vec{u}^{(0)}} & \ldots & \left. \frac{\partial {\cal M}_n}{\partial u_1} \right|_{\vec{u}^{(0)}} \\ \vdots & ~ & \vdots \\ \left. \frac{\partial {\cal M}_1}{\partial u_m} \right|_{\vec{u}^{(0)}} & \ldots & \left. \frac{\partial {\cal M}_n}{\partial u_m} \right|_{\vec{u}^{(0)}} \\ \end{array} \right) \cdot \left( \begin{array}{c} \left. \frac{\partial}{\partial v_1} {\cal J} \right|_{\vec{v}} \\ \vdots \\ \left. \frac{\partial}{\partial v_n} {\cal J} \right|_{\vec{v}} \\ \end{array} \right)\end{split}\]

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)

\[\begin{split}M_{\lambda} \, = \, \left( \begin{array}{ccc} \frac{\partial ({\cal M}_{\lambda})_1}{\partial v^{(\lambda)}_1} & \ldots & \frac{\partial ({\cal M}_{\lambda})_1}{\partial v^{(\lambda)}_{n_{\lambda}}} \\ \vdots & ~ & \vdots \\ \frac{\partial ({\cal M}_{\lambda})_{n_{\lambda+1}}}{\partial v^{(\lambda)}_1} & \ldots & \frac{\partial ({\cal M}_{\lambda})_{n_{\lambda+1}}}{\partial v^{(\lambda)}_{n_{\lambda}}} \\ \end{array} \right)\end{split}\]

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

\[\begin{split}\begin{aligned} \left( \begin{array}{c} \delta v^{(\lambda) \, \ast}_1 \\ \vdots \\ \delta v^{(\lambda) \, \ast}_{n_{\lambda}} \\ \end{array} \right) \, = & \left( \begin{array}{ccc} \frac{\partial ({\cal M}_{\lambda})_1}{\partial v^{(\lambda)}_1} & \ldots \,\, \ldots & \frac{\partial ({\cal M}_{\lambda})_{n_{\lambda+1}}}{\partial v^{(\lambda)}_1} \\ \vdots & ~ & \vdots \\ \frac{\partial ({\cal M}_{\lambda})_1}{\partial v^{(\lambda)}_{n_{\lambda}}} & \ldots \,\, \ldots & \frac{\partial ({\cal M}_{\lambda})_{n_{\lambda+1}}}{\partial v^{(\lambda)}_{n_{\lambda}}} \\ \end{array} \right) \cdot % \\ ~ & ~ \\ ~ & % \left( \begin{array}{ccc} \frac{\partial ({\cal M}_{\lambda+1})_1}{\partial v^{(\lambda+1)}_1} & \ldots & \frac{\partial ({\cal M}_{\lambda+1})_{n_{\lambda+2}}}{\partial v^{(\lambda+1)}_1} \\ \vdots & ~ & \vdots \\ \vdots & ~ & \vdots \\ \frac{\partial ({\cal M}_{\lambda+1})_1}{\partial v^{(\lambda+1)}_{n_{\lambda+1}}} & \ldots & \frac{\partial ({\cal M}_{\lambda+1})_{n_{\lambda+2}}}{\partial v^{(\lambda+1)}_{n_{\lambda+1}}} \\ \end{array} \right) \cdot \, \ldots \, \cdot \left( \begin{array}{c} \delta v^{\ast}_1 \\ \vdots \\ \delta v^{\ast}_{n} \\ \end{array} \right) \end{aligned}\end{split}\]

(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 vector-valued function of dimension \(l > 1\), (7.10) has to be modified according to

\[M^T \left( \nabla_v {\cal J}^T \left(\delta \vec{J}\right) \right) \, = \, \nabla_u {\cal J}^T \cdot \delta \vec{J}\]

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 \({\bf f_{j}}\),

\[\nabla_u {\cal J}^T \, = \, M^T \cdot \nabla_v {\cal J}^T \, = \, \sum_{i} M^T_{ji} \, {\vec{e}_{i}}\]

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

\[\begin{split}\begin{aligned} \nabla_v {\cal J}^T & = \, 2 \, \, H \cdot \left( \, {\cal H}(\vec{v}) - \vec{d} \, \right) \\ ~ & = \, 2 \sum_{j} \left\{ \sum_k \frac{\partial {\cal H}_k}{\partial v_{j}} \left( {\cal H}_k (\vec{v}) - d_k \right) \right\} \, {\vec{f}_{j}} \\ \end{aligned}\end{split}\]

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:

\[\nabla_u {\cal J}^T \, = \, 2 \, M^T \cdot H \cdot \left( {\cal H}(\vec{v}) - \vec{d} \, \right)\]

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 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 \({\cal M}\), 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 [Gri92], Restrepo et al., 1998 [RLG98]). It is depicted in Figure 7.1 for a 3-level checkpointing (as an example, we give explicit numbers for a 3-day integration with a 1-hourly timestep in square brackets).

3-lvl checkpointing schematic figure

Figure 7.1 Schematic view of intermediate dump and restart for 3-level checkpointing.

  • 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 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 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 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 sub-subsection 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 \(k_{n-1}^{lev2}, \ldots, k_{1}^{lev2}\) carrying the adjoint computation to the initial time of the subsection \(k_{n}^{lev3}\). Then, the procedure is repeated for the previous subsection \(k_{n-1}^{lev3}\) 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 3-day 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 AD-related setups the following packages need to be enabled:

The packages are enabled by adding them to your experiment-specific configuration file packages.conf (see Section ???).

The following AD-specific 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 AD-unrelated options for the external forcing package pkg/exf. (NOTE: These options are not set in their package-specific headers such as COST_OPTIONS.h, but are instead collected in the single header file ECCO_CPPOPTIONS.h. The package-specific 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 3-level 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:

AD-target 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 re-generated
«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 re-generating 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:

  1. A header file AD_CONFIG.h is generated which contains a CPP option on which code ought to be generated. Depending on the make target, the contents is one of the following:
  2. A single file «MODE»_input_code.f is concatenated consisting of all .f files that are part of the list AD_FILES and all .flow files that are part of the list AD_FLOW_FILES.
  3. 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 platform-specific 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.
  4. 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.
  5. 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 hand-written AD code is available (see next section).

Each package directory contains its package-specific 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 package-specific 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 user-specified 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 hand-written adjoint code for the MITgcm WRAPPER (Section 6.2).

Flow directives for package-specific routines are contained in the corresponding package directories in the file «PKG»_ad.flow, e.g., ptracers-specific directives are in ptracers_ad.flow.

7.2.3.3. Store directives for 3-level checkpointing

The storing that is required at each period of the 3-level checkpointing is controlled by three top-level 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. Hand-written 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 cost-specific 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:ask-the-author).

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:

\[{\cal J} \, = \, {\rm fc} \, = \, {\rm mult\_tracer} \sum_{\text{global sum}} \sum_{bi,\,bj}^{nSx,\,nSy} {\rm objf\_tracer}(bi,bj) \, + \, ...\]

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_3-1)*nchklev_2*nchklev_1 +
 |                        (ilev_2-1)*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 C-grid: 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 2-D and 3-D control variables and the corresponding cost gradients, a 1-D 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 non-dimensionalize 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 3-D 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 read-procedure is tagged with the variable xx_tr1_dummy enabling TAF to recognize the initialization of the perturbation. The modified call of TAF thus reads

    taf -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 to adxx\_«...».
  • 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 2-D and 3-D control variable fields.
    • The gradient vector vector_grad: at the very end of the adjoint integration, the 2-D and 3-D adjoint variables are read, compressed to a single vector and written to file.
  • 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 CPP-option 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 hand-written 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 hand-written 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 hand-written. 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 TAMC-generated 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. quasi-Newton method, conjugate gradient, … (Gilbert and Lemaréchal, 1989 [GLemarechal89]) to compute an update in the control variable for iteration step \(k+1\):

\[u_{[k+1]} \, = \, u_{[0]} \, + \, \Delta u_{[k+1]} \quad \mbox{satisfying} \quad {\cal J} \left( u_{[k+1]} \right) \, < \, {\cal J} \left( u_{[k]} \right)\]

\(u_{[k+1]}\) 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.

flow between forward/adjoint model and the minimization

Figure 7.2 Flow between the 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:ask-the-author 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 2-D and 3-D fields. The corresponding I/O flow is shown in Figure 7.3:

forward/adjoint model I/O

Figure 7.3 Flow chart showing I/O in the forward/adjoint model.

ctrl_unpack.F reads the updated control vector vector_ctrl_<k>. It distributes the different control variables to 2-D and 3-D 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 package-specific headers such as COST_OPTIONS.h, but are instead collected in the single header file ECCO_CPPOPTIONS.h. The package-specific 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 finite-difference gradient:

\[\left(\nabla _{u}{\cal J}^T \right)_i \quad \text{ vs. } \quad \frac{\partial {\cal J}}{\partial u_i} \, = \, \frac{ {\cal J}(u_i + \epsilon) - {\cal J}(u_i)}{\epsilon}\]

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,

\[1 \, - \, \frac{({\rm grad}{\cal J})_i (\text{adjoint})} {({\rm grad}{\cal J})_i (\text{finite difference})} \, < 1 \%\]

7.3.1. Code description

7.3.2. Code configuration

The relevant CPP precompile options are set in the following files:

The relevant runtime flags are set in the files:

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, 07-Mar-2003*

*NOTE:THIS SECTION IS SUBJECT TO CHANGE. IT REFERS TO TAF-1.4.26.

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 sub-intervals 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:

  1. 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.)
  2. generate code for storing and reading adjoint state variables
  3. generate code for bookkeeping , i.e., maintaining a file with index information
  4. 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 multi-level 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/R mdthe_main_loop.f (generated by TAF) is called.
    1. calculate forward trajectory and dump model state after each outermost checkpointing interval to files tapelev3
    2. calculate cost function fc and write it to file costfinal
  • 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.
    1. (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 \(ilev3 = nlev3-i+1\) and runs to \(nlev3 - i\)
    2. perform adjoint leg from \(nlev3-i+1\) to \(nlev3 - i\)
    3. dump adjoint state to file snapshot
    4. dump index file divided.ctrl for next adjoint leg
    5. in the last step the gradient is written.

A few modifications were performed in the forward code, obvious ones such as adding the corresponding TAF-directive at the appropriate place, and less obvious ones (avoid some re-initializations, when in an intermediate adjoint integration interval).

[For TAF-1.4.20 a number of hand-modifications were necessary to compensate for TAF bugs. Since we refer to TAF-1.4.26 onwards, these modifications are not documented here].

7.4.2. Recipe 1: single processor

  1. In ECCO_CPPOPTIONS.h set:

  2. 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 list

      make 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. The call ad... between call ad... and the read of the snapshot file should be called only in the first adjoint leg between \(nlev3\) and \(nlev3-1\). In the ecco-branch, the following lines should be bracketed by an if (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)

  1. 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 copy mpif.h to the machine where you generate the .f files before TAF-ing, or add the path to the includes directory to your genmake2 platform setup, TAF needs some MPI parameter settings (essentially mpi_comm_world and mpi_integer) to incorporate those in the adjoint code.

  2. In ECCO_CPPOPTIONS.h set

    This will include the header file mpif.h into the top level routine for TAF.

  3. Add the TAF option -mpi to the TAF argument list in the makefile.

  4. 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:

  1. develop a flexible, modular, open source tool that can generate adjoint codes of numerical simulation programs,
  2. establish a platform for easy implementation and testing of source transformation algorithms via a language-independent abstract intermediate representation,
  3. support for source code written in C and Fortan, and
  4. generate efficient tangent linear and adjoint for the MIT general circulation model.

OpenAD’s homepage is at http://www-unix.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 pre-built 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

17-January-2008

OpenAD was successfully built on head node of itrda.acesgrid.org, for following system:

> uname -a
Linux itrda 2.6.22.2-42.fc6 #1 SMP Wed Aug 15 12:34:26 EDT 2007 i686 i686 i386 GNU/Linux

> cat /proc/version
Linux version 2.6.22.2-42.fc6 (brewbuilder@hs20-bc2-4.build.redhat.com)
(gcc version 4.1.2 20070626 (Red Hat 4.1.2-13)) #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.