Differentiating Fixed Point Iterations with ADOL-C: Gradient Calculation for Fluid Dynamics

Abstract

The reverse mode of automatic differentiation allows the computation of gradients at a temporal complexity that is only a small multiple of the temporal complexity to evaluate the function itself. However, the memory requirement of the reverse mode in its basic form is proportional to the operation count of the function to be differentiated. For iterative processes consisting of iterations with uniform complexity this means that the memory requirement of the reverse mode grows linearly with the number of iterations. For fixed point iterations this is not efficient, since any structure of the problem is neglected.

The method of reverse accumulation proposes for linear converging iterations an alternative, iterative computation of the gradient. The iteration of the gradient converges with the same rate as the fixed point iteration itself. The memory requirement for this method is independent of the number of iterations. Hence, it is also independent of the desired accuracy. We integrate the concept of reverse accumulation within the AD-tool ADOL-C to compute gradients of fixed point iterations.This approach decreases the memory requirement of the gradient calculation considerably yielding an increased range of applications. Runtime results based on the CFD code TAUij are presented.