Hi Gunnar,
I would have thought that changing the value and gradient of the
target function had the potential to alter the minima?
Indeed, the target function is changed during the search, but
once a stable minimum is found, the DEN potential needs to
be zero by definition and the coordinates have to sit in a minimum
of the original target function.
Yes, I believe both statements are correct - I was referring to the nature of
the function during the procedure, and you refer to the nature after asymptotic
convergence. The nature of the target function and location of the minima are
changed during DEN refinement - at any given time step (before asymptotic
convergence) the minima of the target function may lie in locations different
to the original target function. However, upon convergence, dij ~= Dij
regardless of particular parameter values. With DEN, the target function
changes shape throughout the procedure, but results in the target function
being asymptotically equal to the original, thus refinement converges to a
minima of the original target function.
This behaviour is interesting and notable because it is different to other
terms used in refinement. Generally, prior information (e.g. geometry terms,
external structural information) in the form of restraints is determined
externally and remains static during refinement. These determine the nature of
the target function, but do not change it during refinement - this contrasts
with the DEN approach. Of course, the structure factors are also updated and
thus also alter the nature of the target function during refinement. Just some
interesting observations!
Here are my thoughts: since the DEN update formula is recursive, the
equilibrium distance can also be written in terms of the Dij alone (still
assuming gamma=1):
dij(t+1) = Dij(0)*(1-kappa)^(t+1) + kappa*sum_n=0^t{Dij(t+1-n)*(1-kappa)^n}
This means that the equilibrium distance is indeed dependent on the initial
distance Dij(0) for all times t. …
I hope I do not get you wrong, but with this argument
aren't you just saying that the path/trajectory (of both the atomic
coordinates
and the DEN potential) depends on the starting point?
No, this wasn't quite the point I was trying to make. For sure, we all
trivially know that any path during refinement depends on the starting point of
the parameter values.
However, it is interesting that the DEN restraint target (or DEN potential, or
equilibrium distance) depends on the starting point. Moreover, the DEN
restraint target can be expressed in a form that makes this dependency
explicit. This observation is not trivial, as it differs from other approaches.
As above, it is most interesting to acknowledge that this contrasts with other
terms used in refinement. For example, simple geometry/external restraints
representing prior information always remain static during refinement. At time
t, other restraints do not depend on their value at time t-1, and thus do not
depend on their value at time 0. Rather, they are pre-determined before
refinement begins.
The important point is that the decision on how to move the DEN
minimum from one iteration (at time t) to the next (at time t+1)
depends only on where the atoms are at t+1 and where the DEN minimum was
at time t.
Or equivalently, the decision on how to move the DEN minimum from one iteration
(at time t) to the next (at time t+1) depends on where the atoms are at t+1,
where they were at time t, where they were at time t-1, … , where they were at
time 0. Of course, the degree of dependency on distant history is controlled by
kappa. Very low values of kappa will result in DEN remembering more distant
historical values of the interatomic distance, and thus refinement will take
longer to converge. Very high values of kappa will result in DEN being
dependent only on the immediate history, and thus will have little effect on
refinement.
If we assume that there is a second starting point which results
in a minimization path that happens to cross exactly the path from the first
starting point (same atomic coordinates and same position of DEN minimum)
at some time t'. Then the new position of the DEN minimum at time t'+1 would
be
exactly at the same position that you get from the first path at time t+1.
Of course, if a second minimisation path happens to cross exactly the first
minimisation path, then they would both end up with the same final result.
There would be something wrong if they didn't! The property that two paths
within some neighbourhood of each other both converge to the same final
positions is a simple requirement for refinement robustness. Just to clarify, I
certainly did not make any incorrect/unsupported claims that DEN is not robust.
I was merely investigating the exact nature of the technique.
Interestingly, note that DEN requires both the atomic coordinates to be at the
same position AND the DEN
