Team Ecuador studies Algorithmic Differentiation (AD) of computer programs, blending :

We aim to produce AD code that can compete with hand-written sensitivity and adjoint programs used in the industry. We implement our algorithms into the tool Tapenade, one of the most popular AD tools at present.

Our research directions :

Algorithmic Differentiation (AD) differentiates
programs. The input of AD is
a source program

Any execution of

where each

which can be mechanically written as a sequence of instructions

The above computation of

Sensitivities, defined for a given direction

This expression is easily computed from right to left, interleaved with the original
program instructions. This is the tangent mode of AD.

Adjoints, defined after transposition (

This expression is most efficiently computed from right to left,
because matrixadjoint mode of AD, most effective for
optimization, data assimilation 35,
adjoint problems 28, or inverse problems.

Adjoint AD builds a very efficient program 30,
which computes the gradient in a time independent from the number of parameters tangent mode
would require running the tangent differentiated program

However, the inverse of their computation order. If the
original program overwrites a part of

Another research issue is to make the AD model cope with the constant evolution of modern language constructs. From the old days of Fortran77, novelties include pointers and dynamic allocation, modularity, structured data types, objects, vectorial notation and parallel programming. We keep developing our models and tools to handle these new constructs.

The most obvious example of a program transformation tool is certainly a compiler. Other examples are program translators, that go from one language or formalism to another, or optimizers, that transform a program to make it run better. AD is just one such transformation. These tools share the technological basis that lets them implement the sophisticated analyses 21 required. In particular there are common mathematical models to specify these analyses and analyze their properties.

An important principle is abstraction: the core of a compiler
should not bother about syntactic details of the compiled program.
The optimization and code generation phases must be independent
from the particular input programming language. This is generally achieved
using language-specific front-ends, language-independent middle-ends,
and target-specific back-ends.
In the middle-end, analysis can concentrate on the semantics
of a reduced set of constructs. This analysis operates
on an abstract representation of programs made of one
call graph, whose nodes are themselves flow graphs whose
nodes (basic blocks) contain abstract syntax trees for the individual
atomic instructions.
To each level are attached symbol tables, nested to capture scoping.

Static program analysis can be defined on this internal representation,
which is largely language independent. The simplest analyses on trees can be
specified with inference rules 24, 31, 22.
But many data-flow analyses are more complex, and better defined on graphs than on trees.
Since both call graphs and flow graphs may be cyclic, these global analyses will be solved iteratively.
Abstract Interpretation 25 is a theoretical framework to
study complexity and termination of these analyses.

Data flow analyses must be carefully designed to avoid or control
combinatorial explosion. At the call graph level, they can run bottom-up or top-down,
and they yield more accurate results when they take into account the different
call sites of each procedure, which is called context sensitivity.
At the flow graph level, they can run forwards or backwards, and
yield more accurate results when they take into account only the possible
execution flows resulting from possible control, which is called flow sensitivity.

Even then, data flow analyses are limited, because they are static and thus have very
little knowledge of actual run-time values. Far before reaching the very theoretical limit of
undecidability, one reaches practical limitations to how much information one can infer
from programs that use arrays 39, 26 or pointers.
Therefore, conservative over-approximations must be made, leading to
derivative code less efficient than ideal.

Scientific Computing provides reliable simulations
of complex systems. For example it is possible to simulate
the steady or unsteady 3D air flow around a plane that captures the physical phenomena
of shocks and turbulence. Next comes optimization,
one degree higher in complexity because it repeatedly simulates and
applies gradient-based optimization steps until an optimum is reached.
The next sophistication is robustness, that detects undesirable solutions which,
although maybe optimal, are very sensitive to uncertainty on design parameters or
on manufacturing tolerances. This makes second derivatives come into play.
Similarly Uncertainty Quantification can use second derivatives to evaluate how uncertainty on
the simulation inputs imply uncertainty on its outputs.

To obtain this gradient and possibly higher derivatives,
we advocate adjoint AD (cf3.1)
of the program that discretizes and solves the direct system.
This gives the exact gradient of the discrete function
computed by the program, which is quicker and more sound than differentiating
the original mathematical equations 28.
Theoretical results 27 guarantee convergence
of these derivatives when the direct program converges.
This approach is highly mechanizable. However, it requires
careful study and special developments of the AD model 32, 37
to master possibly heavy memory usage.
Among these additional developments, we promote in particular
specialized AD models for Fixed-Point iterations 29, 23,
efficient adjoints for linear algebra operators such as solvers, or exploitation
of parallel properties of the adjoint code.

Algorithmic Differentiation of programs gives sensitivities or gradients, useful for instance for :

A CFD program computes the flow around a shape, starting from a number of inputs that define the shape and other parameters. On this flow one can define optimization criteria e.g. the lift of an aircraft. To optimize a criterion by a gradient descent, one needs the gradient of the criterion with respect to all inputs, and possibly additional gradients when there are constraints. Adjoint AD is the most efficient way to compute these gradients.

Inverse problems aim at estimating the value of hidden parameters from other measurable values, that depend on the hidden parameters through a system of equations. For example, the hidden parameter might be the shape of the ocean floor, and the measurable values of the altitude and velocities of the surface. Figure 1 shows an example of an inverse problem using the glaciology code ALIF (a pure C version of ISSM 34) and its AD-adjoint produced by Tapenade.

One particular case of inverse problems is data assimilation 35
in weather forecasting or in oceanography.
The quality of the initial state of the simulation conditions the quality of the
prediction. But this initial state is not well known. Only some
measurements at arbitrary places and times are available.
A good initial state is found by solving a least squares problem
between the measurements and a guessed initial state which itself must verify the
equations of meteorology. This boils down to solving an adjoint problem,
which can be done though AD 38.
The special case of 4Dvar data assimilation is particularly challenging.
The 4th dimension in “4D” is time, as available measurements are distributed
over a given assimilation period. Therefore the least squares mechanism must be
applied to a simulation over time that follows the time evolution model.
This process gives a much better estimation of the initial state, because
both position and time of measurements are taken into account.
On the other hand, the adjoint problem involved is more complex,
because it must run (backwards) over many time steps.
This demanding application of AD justifies our efforts in
reducing the runtime and memory costs of AD adjoint codes.

Simulating a complex system often requires solving a system of Partial Differential Equations.
This can be too expensive, in particular for real-time simulations.
When one wants to simulate the reaction of this complex system to small perturbations around a fixed
set of parameters, there is an efficient approximation: just suppose that the system
is linear in a small neighborhood of the current set of parameters. The reaction of the system
is thus approximated by a simple product of the variation of the parameters with the
Jacobian matrix of the system. This Jacobian matrix can be obtained by AD.
This is especially cheap when the Jacobian matrix is sparse.
The simulation can be improved further by introducing higher-order derivatives, such as Taylor
expansions, which can also be computed through AD.
The result is often called a reduced model.

Some approximation errors can be expressed by an adjoint state. Mesh adaptation can benefit from this. The classical optimization step can give an optimization direction not only for the control parameters, but also for the approximation parameters, and in particular the mesh geometry. The ultimate goal is to obtain optimal control parameters up to a precision prescribed in advance.

Our research has an impact on environmental research: in Earth sciences, our gradients are used in inverse problems, to determine key properties in oceanography, glaciology, or climate models. For instance they determine basal friction coefficients of glaciers that are necessary to simulate their future evolution. Another example is to locate sources and sinks of CO2 by coupling atmospheric models and remote measurements.

For multithreaded parallel applications that use OpenMP, it is desirable to also compute the gradients in parallel. Last year we started extension of the AD model of Tapenade (source transformation, association by address, storage on tape of intermediate values) towards correct and efficient differentiation of OpenMP parallel worksharing loops. These loops are the most common idiom in OpenMP programs. This extension concerns the relatively easy tangent-linear mode of AD, but also the more sophisticated adjoint mode where the most interesting problems lie.

The major issue raised by the adjoint mode is the transformation of variable reads into adjoint variable overwrites, more accurately into increments. While there is no parallel conflict between two reads, two concurrent increments can cause a data race, unless they are both atomic. Classical automated detection of independence, for instance through the polyhedral methods used in parallelizing compilers, is as always limited. This year we have pushed forward the idea of extracting information about the memory access patterns of the original code (which we may assume correct and therefore free of data races) and of feeding this information to a theorem prover to which we then ask questions about independence of access patterns of the adjoint code. As a side benefit, we point out that the thorem prover can very well use other clues to answer the question, and can in particular implement its own polyhedral approach like in a parallellizing compiler. Therefore this approach nicely blends two sources of improvement.

Technically, all the information extraction phase is implemented inside Tapenade, as well as gathering the list of independence questions to be asked. The theorem proving part is delegated to a plugin of the Z3 tool, which is now systematically added into Tapenade. The issues of interfacing between different API's are thus reduced to a minimum. This new functionality is available in all distributions of Tapenade.

This extension was presented this year as a poster at the PPoPP'22 conference 20, and later as a full paper at the ICPP'22 conference 12.

We continued this year our collaboration with ONERA on differentiation of CUDA source. More generally we extended our AD model towards integration of AD as a fundamental component of SoNice. SoNice is the new CFD platform developed at ONERA by Bourasseau, Maugars, Content and colleagues, as a successor to Elsa. This work is supported by a contract between ONERA and INRIA. This contract officially completed in July 2022 but collaboration continues.

This year, we delivered to ONERA a version of Tapenade that can differentiate most of the test cases provided by ONERA, in tangent as well as adjoint mode.

On the issue of differentiation of the CUDA kernels of SoNice, we continued the work by investigating the interest of the formal methods studied in 7.1. This approach can bring the same benefits for CUDA code as for OpenMP, as we showed on example codes with regular array accesses. However, SoNice favors unsructured grids and therefore irregular array indices that do not lend themselves to this approach. The alternative approach is a mesh coloring technique. Unlike a classical mesh coloring that assigns colors to avoid read-write and write-write conflicts inside a given color, the adapted coloring also avoids read-read situations in the original source. This may slightly increase the number of colors, but in return guarantees the absence of conflicts in the adjoint code, and therefore avoids atomic pragmas in the adjoint CUDA code. The reason is that adjoint AD turns reads into increments, and thus read-read situations into write-write conflicts.

Still on the CUDA aspect, we improved the adjoint AD model to further reduce the need for storage of intermediate values in the adjoint CUDA kernels. Adjoint code reuses intermediate values from the primal computation in reverse order, and this is often arranged for through storage. However, CUDA storage capacity is limited and costly. One workaround is to promote recomputation instead of storage. We pushed the capacity of Tapenade to detect possible recomputation, to a level that is sufficient for SoNice. Still, finding an optimal mix of recomputation and storage is an open question.

Other developments required for SoNice have been:

A joint article has been presented at the ECCOMAS'22 conference in Oslo 13.

We support users with their first experiments of Algorithmic Differentiation of large codes. This concerned several Earth Science codes this year.

One set of target applications was addressed by Shreyas Gaikwad, University of Texas at Austin, PhD student supervised by Patrick Heimbach. His goal is to produce the adjoint of glaciology codes. Last year, Shreyas differentiated the SICOPOLIS code in adjoint mode with Tapenade. After some polishing this year, this work is now described in the main documentation of SICOPOLIS, hosted by readthedocs.io. An article was written and submitted to JOSS (Journal of Open Source Software).

This year, Shreyas successfuly differentiated the MIT GCM code, in tangent and adjoint mode, with Tapenade. All these codes are Fortran90 codes that have been previously differentiated with OpenAD, the former AD tool developed by Argonne National Lab. Krishna Narayanan provided crucial help and expertise, as he had been in charge of the previous differentiation with OpenAD. Indeed, differentiation with Tapenade did not encounter bugs in the AD tool itself. It also substantially simplified the tool chain that was necessary with OpenAD, a consequence of the different user interface choices of the two AD tools. The new tool chain was verified on most of the standard test cases of the MIT GCM.

A “pull request” of these new developments was submitted in December to the MIT GCM github repository, and is waiting for approval before final integration in the GCM. Documentation of this work was also prepared and submitted in the main documentation of the MIT GCM, hosted by readthedocs.io.

Shreyas Gaikwad presented this work at the Fall meeting of American Geophysical Union, 12-16 December 2022, in Chicago.

A sequel to this work is just starting, with Dan Goldberg, about differentiation of the STREAMICE package of the GCM, with Tapenade.

Independently from the above, we supported adjoint differentiation of the BEPS 33 code, a “biosphere” Earth Science simulation model taking vegetation into accout. This code is developed in part by the small company The Inversion Lab in Hamburg, Germany. We collaborate with Michael Vossbeck, engineer with The Inversion Lab, to provide the adjoint of BEPS by Tapenade. This year, we improved the algorithms and differentiation options of Tapenade to simplify the work chain that produces the adjoint of BEPS. This significantly simplifies future maintenance of the system. This work required extensions of Tapenade towards Fortran 2003 features that were not supported before. Differentiation with Tapenade is becoming a strategic part of the business of The Inversion Lab. Michael Vossbeck is progressively getting acquainted with the source of Tapenade, with the objective of contributing to Tapenade development, and of being able to maintain it.

The progress in highly accurate schemes for compressible flows on unstructured meshes (together with advances in massive parallelization of these schemes) allows to solve problems previously out of reach. The four-years programme Norma, associating:

is supported by the French ANR and by the Russian Science Foundation. Norma is a cooperation on the subject of extending Computational AeroAcoustics methods to simulate the noise emited by rotating machines (helicopters, aerial vehicles, unmanned aerial vehicles, wind turbines...). A detailed description with progress reports is available at this location.

Sections 7.5, 7.6, 7.7, 7.8 describe the 2022 contributions of Ecuador to the Norma programme.

Modelling turbulence is an essential aspect of CFD. The purpose of our work in hybrid RANS/LES (Reynolds Averaged Navier-Stokes / Large Eddy Simulation) is to develop new approaches for industrial applications of LES-based analyses. In the applications targeted (aeronautics, hydraulics), the Reynolds number can be as high as several tens of millions, far too high for pure LES models. However, certain regions in the flow can be predicted better with LES than with usual statistical RANS (Reynolds averaged Navier-Stokes) models. These are mainly vortical separated regions as assumed in one of the most popular hybrid models, the Detached Eddy Simulation (DES) model. Here, “hybrid” means that a blending is applied between LES and RANS. An important difference between a real life flow and a wind tunnel or basin is that the turbulence of the flow upstream of each body is not well known. The development of hybrid models, in particular DES in the litterature, has raised the question of the domain of validity of these models. According to theory, these models should not be applied to flows involving laminar boundary layers (BL). But industrial flows are complex flows and often combine in a single flow regions of laminar BL, regions of fully developed turbulent BL, and regions of non-equilibrium vortical BL. It is then mandatory for industrial use that the new hybrid models give a reasonable prediction for all these types of flow. We concentrated on evaluating the behavior of hybrid models for laminar BL and for vortical wakes. While less predictive than pure LES on laminar BL, some hybrid models still give reasonable predictions for rather low Reynolds numbers.

During 2022, several advances have been obtained. B. Sauvage developed a study of the NACA0021 test case at high angle of attack, an unsteady flow adressed with a DDES model and mesh adaptation. He also focused on the new test case chosen by the Norma consortium, the flow past a NACA0018 at various angles of attack, for which several measurements are available in the literature. A part of the work of B. Sauvage has been presented in CMMF'22 in Budapest and in DLES13 in Udine 15. B. Sauvage also made studies on approximation, with comparison of several Riemann solvers such as Roe flux difference splitting (FDS), HLLC flux vector splitting (FVS), and a new variant of the Toro-Vazquez FVS. For pure advection, the Toro-Vazquez FVS proved to be less diffusive than HLLC, of comparable diffusion to Roe, while being much more robust than Roe FDS. These properties are important when mesh adaptation algorithms are applied, since these algorithms are more challenging for the approximation.

The physical problem addressed by Norma involves a computational domain made of at least two components having different rotative motions. The numerical problem of their combination gave birth to many specialized schemes, such as the so-called sliding method, chimera method, immersed boundary method (IBM). The Ecuador team is studying a novel Chimera method, in cooperation with Lemma engineering (Sophia-Antipolis).

In 2022, B. Sauvage continued his study of mesh adaptive calculation of rotation flows. He developed a Multiple Reference Frame method 18 satisfying the Discrete Geometric Conservation Law. The new scheme together with the Chimera scheme are compared 19 with the calculation of the Caradonna rotor, both with mesh adaptation. The results have been presented in two communications, at ECCOMAS 14 in Oslo and CMFF22 16 in Budapest. B. Sauvage and Didier Chargy made progresses in the computation of the Caradonna-Tung rotor, with a new adapted calculation on 3 million vertices.

High order approximations for compressible flows on unstructured meshes are facing many constraints that increase their complexity i.e. their computational cost. This is clear for the largest class of approximation, the class of

-exact schemes, which rely on a local polynomial representation of degree

. We are investigating schemes which would solve as efficiently as possible the dilemma of choosing between an approximation with a representation inside macro-elements which finally constrains the mesh, and a representation around each individual cell, as in vertex formulations. For this purpose, we extend the Central Essentially Non Oscillating (CENO) family of schemes. We have developed a fourth-order accurate three-dimensional CENO.

In 2022, the consolidation and validation of the fourth-order accurate CENO in LEMMA's code NiceFlow has been advanced further. Alain Dervieux's article 11 on approximation on unstructured meshes has been finally published this year.

Reducing approximation errors as much as possible by modifying the mesh is a particular kind of optimal control problem. We formulate it exactly this way when we look for the optimal metric of the mesh, which minimizes a user-specified functional (goal-oriented mesh adaptation). In that case, the usual methods of optimal control apply, using adjoint states that can be produced by Algorithmic Differentiation.

The first volume of the book 17 on mesh adaptation by Alauzet, Loseille, Koobus and Dervieux has been published by Wiley this year.

B. Sauvage and A. Dervieux made advances in the theory of space-and-time mesh adaptation with the transient fixed point algorithm, in cooperation with F. Alauzet. Adapting the time step is an important issue for aerodynamic LES and hybrid RANS-LES calculations since implicit time advancing with large time steps is used for efficiency reasons while time truncation errors have to be mastered by identifying the best time step.

The team has a contract with ONERA to assist in the Algorithmic Differentiation of ONERA's new CFD platform “SoNICS”. The main objective is adjoint AD of the CUDA parts of the SoNICS source. The contract also includes support to the ONERA development team on advanced use of AD, e.g., fixed-point adjoints and binomial checkpointing. The contract ended formally in July this year.

Ecuador participates in the Joint Laboratory for Exascale Computing (JLESC) together with colleagues at Argonne National Laboratory. Laurent Hascoët gave a presentation at the 13th JLESC Workshop (virtual) Dec 14 to 16, 2021, about progress on AD of OpenMP.