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Discussion and Results of the GHYS/QT algorithm

There is no guarantee that refinement converges towards a true orbit; if there was, then all noisy orbits would be shadowable. In fact, even if some refinements are successful, numerical refinement alone does not prove rigorously that a true shadow exists; it only proves the existence of a numerical shadow, i.e.,  a trajectory with less noise than the original.

QT [63] argue that if the refinement algorithm fails, there is good reason to believe that no shadow exists. They apply two arguments. First, from the more rigourous study of simpler systems, glitches are known to exist and are not just a failure of any particular refinement algorithm. Second, QT's results are consistent with a conjecture by GHYS on the frequency of glitches. However, Hayes [33] frequently saw cases in which the algorithm failed to find a shadow for noisy orbits of length S, but succeeded in finding a shadow for the superset of length 2S. Hence, the algorithm failed to find a shadow of length S, even though one exists.

Hayes [33] has preliminary results that seem to indicate that shadow lengths scale inversely with the number of dimensions of the system, at least for the gravitational N-body problem. This would imply that shadows get shorter as the number of degrees of freedom increases, which is a very pessimistic conclusion for simulations of high-dimensional systems.

We note that the GHYS/QT refinement algorithm is trivially parallelizable, since the computation of each tex2html_wrap_inline2614 is completely independent of all the others. For the same reason, it also has excellent locality of reference in a serial implementation, so virtual memory paging is minimized. Once the tex2html_wrap_inline2614 's are computed, it may also be worth parallelizing the recurrence (8) [43].

Wayne Hayes
Fri Dec 27 17:41:39 EST 1996

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