Here are some further ideas that I have not yet thought about in depth but should be considered. Some have already been mentioned briefly in previous chapters, but are collected here for convenience.

- An obvious item that has been overlooked is the course
grained parallelism inherent in the computation of the resolvents:
each resolvent is computed independently of all the others. Thus,
each could be computed on a separate processer, for example using the
PVM (
*P*arallel*V*irtual*M*achinne) software package [9]. - When using constant RUS, and iterations are failing, is there some way to decide if the problem is being caused by one (or a small number) of RUS 's? If so, then only these RUS 's need be recomputed, rather than the RUS for the entire trajectory. This is potentially a large savings. An obvious possibility is that the shadow step with the greatest 1-step error (or better, the one that has the least improvement) is one that needs its RUS recomputed. Or, perhaps a shadow step with a 1-step error that refuses to decrease is too long, making the resolvent invalid. Here, the shadow step could be split into smaller segments.
- There has been much work done in numerical analysis in the area of two-point boundary value problems. Shadowing trajectories have two ends, and each end has a boundary condition: at the last point, the growth of the expanding direction is pinched; the growth of the contracting direction (expanding in negative time) is pinched at the starting point. Thus there are two points, and boundary conditions at each end. The work on two-point boundary value problems may have bearing on shadowing.
- Thus far, perturbations are allowed only in the phase-space
co-ordinates of a trajectory. It seems reasonable to also allow
perturbations in
*time*. Would it help if perturbations in time were allowed as well? In other words, if the time at which the phase space point is measured is allowed to be perturbed by small amounts, will shadowing be any more successful? - It has been repeatedly stated that the GHYS refinement procedure
is similar to a Newton's method. It has many similarities: the
resolvent is a Jacobian; the method is iterative; when the Jacobian
is recomputed at each step, there is geometric converge; and finally,
it shows some of the same weaknesses as Newton's method. Is it possible
that the GHYS procedure
*is*a Newton's method in disguise, with boundary conditions? If it can be arranged to look exactly like a Newton's method, then the massive amount of literature and code devoted to Newton's method may be brought to bear on the problem.

The Newton's method item can be formalized a bit more. In general,
a one-dimensional Newton's method trying to find a zero of the
equation *y*=*f*(*x*), given an initial guess , is usually written
as . In the refinement problem, the
function being computed and for which we are trying to find a zero
is the function that computes the 1-step errors along the entire
trajectory. Let the entire trajectory be

Recall the equation for the 1-step errors is

where is the function that maps point at time to the true solution at time by integrating the solution of the ODE forwards in time. In the formulation I have in mind, we also need the backward errors as defined in the SLES refinement algorithm:

where integrates the ODE backwards in time. Then, I define a new
type of 1-step error, called the *total 1-step error* as the
sum of the forward and backwards 1-step errors:

with . Let the function that computes all the 1-step errors be

Let be the initial noisy orbit; will
represent the *k* iteration of the Newton's method. Then the Newton's
method for the GHYS refinement procedure may be written as

where is the Jacobian of the 1-step error function
, *i.e., * the derivative of the 1-step errors with respect to the
phase-space orbit. Writing out explicitly,

Taking partial derivatives, we see that

Finally, note that , and , the resolvents that we already compute.

Somewhere here is where the boundary conditions will need
to come into play: if the number of phase space dimensions is 2*D* (each
of the and vectors has 2*D* dimensions), then there are
*D* boundary conditions at each end of the trajectory, limiting the growth
of the stable and unstable components at their respective endpoints.
The corrections (computed from ) will probably also need to
be computed in a special order, as they are in the GHYS procedure.
I do not currently know how to include these boundary conditions in the
problem, but assuming they can be, the Newton iteration may look like
the scheme described above.

Recall that GHYS intended refinement simply as a method to
reduce noise in a trajectory, to give the more rigourous process
of *containment* a better chance to establish rigourous proof of
the existence of a true shadow. Clearly,
one avenue of research is to generalize containment to work on
arbitrary Hamiltonian systems. It may even prove to be about as
efficient, practically speaking, as refining to machine epsilon; and
most important, it is *rigourous.*

One interesting question, although it is not crucial to the proof of
existence of shadows, is the question of uniqueness of shadows. It is
already known that if one shadow exists, then infinitely many of
them exist, all packed into a small volume of phase space.
However, does choosing boundary conditions
for and give a unique true shadow from among
the infinite number of true shadows that exist, if one exists at all?
It seems to me that fixing the conditions probably produces a unique shadow,
because the number of boundary conditions
is exactly equal to the number of degrees of freedom. For example, fixing
the position and velocity at any given time produces a unique true solution,
so it seems reasonable to expect that fixing half the co-ordinates at
one time *a* and half at another time *b* would produce a unique solution
(if one
exists at all), as long as the image at time *b* of the
initial-condition-subspace at
time *a* is linearly independent from the final-condition-subspace at time *b*.
It could also be tested numerically by trying slightly different
first guesses for the shadow before refinement starts, or possibly
by using a different ``accurate'' integrator to find the shadow.
However, there may exist pathological boundary cases for which the
solution is not unique.

Sun Dec 29 23:43:59 EST 1996

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