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MITgcm's tangent linear and adjoint model
Automatic differentiation (AD), also referred to as algorithmic
(or, more loosely, computational) differentiation, is a technology for automatically augmenting computer programs,
including arbitrarily complex simulations, with statements for the
computation of derivatives, also known as sensitivities. AD tools
in our context provide source-to-source transformation, according
to a set of linguistic and mathematical rules, of a function (a
nonlinear prognostic model), given as computer code, to generate
efficient and accurate (truncation-free) readable code for
computing derivatives of the given function. Thus, 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. In the mid 1990's groups in the CMI
at MIT, SIO,
JPL and GFDL have begun to apply AD tools for generating tangent
linear and adjoint code for ocean circulation and climate studies.
The tools used comprised the Tangent linear and Adjoint Model
Compiler (TAMC) as well as its commercial successor,
Transformation of Algorithms in Fortran (TAF), developed by Ralf
Giering (\cite{gie-kam:98}, \cite{gie:99}, \cite{gie:00a}), for
which the MITgcm code has been adapted.
TAF 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.
This system has so far been used and is being applied in a
practical way to study five broad classes of problems:
Required AD tool features
The generation of
efficient tangent linear and adjoint code for the MITgcm has
pushed the limits of AD. It makes extensive use of advanced AD
tool features, and has driven, in the case of TAF, AD tool
development and improvement. Among the he required
features are:
 | handling of nonlinearities, switches, etc. |
| balancing of storing vs. re-computation via user
directives |
| n-level check-pointing (here, n=3) |
| flow directives for substituting/providing hand-written
derivative routines into the AD-generated derivative code, in
particular related to the WRAPPER |
| exploitation of self-adjointness |
| inclusion of the Message Passing Interface (MPI) library
for code execution in parallel environments |
| adjoint dump & restart, requiring the definition of the
full adjoint state |
ADM and TLM maintenance and improvement
Some key advantages of AD for derivative code generation are:
| up-to-date maintenance of derivative code along with the
ongoing forward code development as is the case for MITgcm;
at the time of writing, the available ADM and TLM are based
on code that lacks by only 3 weeks the latest available
MITgcm checkpoint. |
| incremental modifications of the forward code (rather
than extensive rewriting) when new features are added (new
'physics' packages, modified/improved numerics and dynamic,
etc.); as an example, the inclusion of the adjoint of the
Gent/McWilliams parameterization scheme (GM/Redi) requires
the adjoint state to propagate through vertical density
gradients, thus modifying the flow of adjoint sensitivities
compared to a configuration without GM/Redi. |
The benefits of such a strategy are illustrated in the
context of two directions of the ongoing model development:
Atmospheric code development
As described in
{mars-eta:03,adc-eta:03}, pressure vs. z-level
coordinate isomorphisms are being exploited to enable the
MITgcm to run as an atmospheric model. Relying on the
same hydrodynamical kernel for which an adjoint model is
available readily leads to an adjoint model for an
atmospheric configuration. Thus, an adjoint model for a
Held-Suarez type configuration representing a basis
atmospheric dynmics setup for the Atmospheric Model
Intercomparison Project (AMIP) is available and is currently
being used for sensitivity studies and entropy maximization
studies.
Coupled ocean - sea ice model
A new sea ice model has recently been implemented
{men-zha:03}, closing an important gap for global ocean modeling
and state estimation, enabling arctic modeling studies, and assimilation of large number hitherto
unexplored sea ice date to be assimilated into the MITgcm.
The development of the adjoint of the sea ice model has
greatly benefited from the availability of the adjoint of a
bulk formula package based on the \cite{lar-pon:82}
implementation to compute air-sea fluxes from atmospheric
fields. The existing bulk formula implementation
provided a natural link to the sea ice model and its adjoint. |
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