ADJOINTS OF FIXED-POINT ITERATIONS

Adjoint algorithms, and in particular those obtained through the adjoint mode of Automatic Differentiation (AD), are probably the most efficient way to obtain the gradient of a numerical simulation. This however needs to use the flow of data of the original simulation in reverse order, at a cost that increases with the length of the simulation. AD research looks for strategies to reduce this cost, taking advantage of the structure of the given program. One such frequent structure is fixed-point iterations, which occur e.g. in steady-state simulations, but not only. It is common wisdom that the first iterations of a fixed-point search operate on a meaningless state vector, and that reversing the corresponding data-flow may be suboptimal. An adapted adjoint strategy for this iterative process should consider only the last or the few last iterations. At least two authors, B. Christianson and A. Griewank, have studied mathematically fixed-point iterations with the goal of defining an efficient adjoint. In this paper, we describe and contrast these two strategies with the objective of implementing the best suited one into the AD tool that we are developing. We select a representative application to test the chosen strategy, to propose a set of user directives to trigger it, and to discuss the implementation implications in our tool.