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Section: Research Program

Algorithmic Differentiation and Scientific Computing

Participants : Alain Dervieux, Laurent Hascoët, Bruno Koobus, Eléonore Gauci, Emmanuelle Itam, Olivier Allain, Stephen Wornom.


Glossary
linearization

In Scientific Computing, the mathematical model often consists of Partial Differential Equations, that are discretized and then solved by a computer program. Linearization of these equations, or alternatively linearization of the computer program, predict the behavior of the model when small perturbations are applied. This is useful when the perturbations are effectively small, as in acoustics, or when one wants the sensitivity of the system with respect to one parameter, as in optimization.

adjoint state

Consider a system of Partial Differential Equations that define some characteristics of a system with respect to some parameters. Consider one particular scalar characteristic. Its sensitivity (or gradient) with respect to the parameters can be defined by means of adjoint equations, deduced from the original equations through linearization and transposition. The solution of the adjoint equations is known as the adjoint state.


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 (cf 3.1) 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  [25]. Theoretical results  [24] 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  [29], [32] to master possibly heavy memory usage. Among these additional developments, we promote in particular specialized AD models for Fixed-Point iterations  [26], [17], efficient adjoints for linear algebra operators such as solvers, or exploitation of parallel properties of the adjoint code.