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 2*S*.
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 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 's are computed, it may also be worth parallelizing the recurrence (8) [43].

Fri Dec 27 17:41:39 EST 1996

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