

Differentiating Fixed Point Iterations with ADOLC: Gradient Calculation for Fluid Dynamics
Part of a collection
  

Area Computational Fluid Dynamics 
Author(s)
S. Schlenkrich
, A. Walther
, N. R. Gauger
, R. Heinrich

Published in Modeling, Simulation and Optimization of Complex Processes  Proceedings of 3rd HPSC 2006

Editor(s) H. G. Bock, E. Kostina, H. X. Phu, R. Rannacher 
Year 2008 
Publisher Springer 
Abstract The reverse mode of Automatic Differentiation (ad) allows the computation of gradients at a temporal complexity that is only a small multiple of the function evlauation itself. However, the memory requirement of the reverse mode in its basic form is proportional to the size of the computational graph of the function to be differentiated. Hence, for iterative processes consisting of iterations with uniform complexity this means that the memory requirement caused by the reverse mode of ad is proportional to the number of iterations. For fixed point iterations this is not efficient, since it neglegts any structure of the problem. 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 ADOLC to compute gradients of fixed point iterations. This appraoch decreases the memory requirement of the gradient calculation considerably resulting in an increased range of applications. Results for a large scale application based on the CFD code TAUij are presented. 
AD Tools ADOLC 
AD Theory and Techniques Reverse Mode 
Related Applications
 Shape Optimization in Aerodynamics

BibTeX
@INPROCEEDINGS{
Schlenkrich2008DFP,
title = "Differentiating Fixed Point Iterations with {ADOLC}: Gradient Calculation for Fluid
Dynamics",
author = "S. Schlenkrich and A. Walther and N.R. Gauger and R. Heinrich",
editor = "H.G. Bock and E. Kostina and H.X. Phu and R. Rannacher",
year = "2008",
booktitle = "Modeling, Simulation and Optimization of Complex Processes  Proceedings of 3rd
HPSC 2006",
pages = "499508",
publisher = "Springer",
abstract = "The reverse mode of Automatic Differentiation (AD) allows the computation of
gradients at a temporal complexity that is only a small multiple of the function evlauation itself.
However, the memory requirement of the reverse mode in its basic form is proportional to the size of
the computational graph of the function to be differentiated. Hence, for iterative processes
consisting of iterations with uniform complexity this means that the memory requirement caused by
the reverse mode of AD is proportional to the number of iterations. For fixed point iterations this
is not efficient, since it neglegts any structure of the problem. 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 ADOLC to compute gradients of fixed point iterations. This appraoch decreases the memory
requirement of the gradient calculation considerably resulting in an increased range of
applications. Results for a large scale application based on the CFD code TAUij are presented.",
ad_area = "Computational Fluid Dynamics",
ad_tools = "ADOLC",
ad_theotech = "Reverse Mode"
}
 
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