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Fast Chaotic Artificial Time Integration

  • Uri Ascher
  • Kees van den Doel

Abstract

Gradient descent methods for large positive definite linear and nonlinear algebraic systems arise when integrating a PDE to steady state and when regularizing inverse problems. However, these methods may converge very slowly when utilizing a constant step size, or when employing an exact line search at each step, with the iteration count growing proportionally to the condition number. Faster gradient descent methods must occasionally resort to significantly larger step sizes, which in turn yields a strongly nonmonotone decrease pattern in the residual vector norm.

In fact, such faster gradient descent methods generate chaotic dynamical systems for the normalized residual vectors. Very little is required to generate chaos here: simply damping steepest descent by a constant factor close to 1 will do. The fastest practical methods of this family in general appear to be the chaotic, two-step ones. Despite their erratic behavior, these methods may also be used as smoothers, or regularization operators. Our results also highlight the need for better theory for these methods.

Keywords

Gradient descent Artificial time integration Dynamical system Stability Chaos Regularization 

Notes

Acknowledgements

The first author thanks IMPA, Rio de Janeiro, for support and hospitality during several months in 2011 when this work was completed.

U. Ascher supported in part by NSERC Discovery Grant 84306.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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