Le 09/10/2011 16:51, Phil Steitz a écrit :
For the acon US talk I am giving, I would like to provide some
potential practical examples of how [nabla] might be used in
applications. So far, I have come up with some direct uses of
analytic derivatives in things like splines and projections. I
suspect there are quite a few numerical or optimization use cases
where having an exact derivative could improve speed or accuracy. I
would like to refer to some of those. Its OK if the needed
machinery does not exist yet in [nabla] - these are motivational
examples.
Here are the original example that lead me to start everything.
One of the classical process used in operational space flight dynamics
is orbit determination. Spacecraft orbits are not perfectly known due to
launcher errors, maneuvers errors and measurements errors. Error grows
from these initial errors due to the dynamics of the trajectory, so
after some time it becomes necessary to perform some measurements
(distances, angles, Doppler from some refence points like known ground
stations). Then we try to compute the orbit that fits the best these
measurements, taking into account many perturbing effects. This is
basically least squares or Kalman filtering, where we adjust some state
vector that evolves in time, using observation points and evolution
models. We need Jacobians matrices in this process, the Jacobians
associated with measurement at time ti with respect to initial orbit
state at time t0 or with respect to some model parameters like
atmospheric drag coefficients or thrusters efficiency. These models are
very complex. They are split in many classes, many methods, many
iterations so even if from a very high level we have for measurement m a
function like m(ti) = f(o(t0), t0, ti), the f function is in fact
several thousands lines of code from three different libraries.
Up to now, we had to compute the Jacobians by manual coding of the exact
formulas. This is a maintenance nigthmare. As we try to use more and
more complex force models for spacecraft dynamics, we have to compute
their differentials too. It would be nice if we could just focus on the
perturbation forces only and have some automatic engine that would
perform the heavy duty differentiation for us. There is really no added
value in the derivatives from a science or engineering point of view, so
it is a perfect target for automation, so we concentrate the human
efforts on something else: the perturbation forces by themselves.
This example (optimization and least squares) can also be reduced to one
dimension only, so derivatives would also be useful in root solvers (but
in fact, I know think solvers like the extended Nth order Brent I
implemented are in fact more efficient than Newton despite they don't
use derivatives).
Another example is sensitivity and error analysis. Having the
derivatives almost for free (in engineering costs, not in computation
costs) is a major advantage. Jacobians link an initial error to a final
error and hene can be used to convert covariance matrices. They can also
be used to set up convergence criterions in iterative processes or
margins for robustness analysis.
Hope this helps
Luc
Thanks!
Phil
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