On 10/9/11 9:53 AM, Luc Maisonobe wrote: > 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.
Thanks, this is a great example. Why exactly is numeric differentiation not viable here? I assume it must have something to do with either the setup being almost as hard as just coding the analytic differentials or error. > > 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). Right. I considered exactly the same example and came to exactly the same conclusion :) > > 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. Here again, the key I guess is the reduced engineering cost over numerical differentiation or direct analytics. The simple example I was thinking about was computing the slope of an interpolating function, e.g. a cubic spline. Our own SplineInterpolator will provide the Polynomials, which give you the analytical derivatives; but to roll your own derivative for the whole function, you have to write code to locate the right knot points, pick out the interpolating polynomial, etc. Having [nabla] do what will amount to *the same thing* automatically saves engineering time. An interesting question for us in [math] is how we would feel about eventually using [nabla] internally for exactly this kind of thing. Providing a derivative for the function returned by the SplineInterpolator (or even more basically, replacing what we have now in Polynomial) is an example. Phil > > > Hope this helps > Luc > >> >> Thanks! >> >> Phil >> >> --------------------------------------------------------------------- >> >> To unsubscribe, e-mail: dev-unsubscr...@commons.apache.org >> For additional commands, e-mail: dev-h...@commons.apache.org >> > > > --------------------------------------------------------------------- > To unsubscribe, e-mail: dev-unsubscr...@commons.apache.org > For additional commands, e-mail: dev-h...@commons.apache.org > > --------------------------------------------------------------------- To unsubscribe, e-mail: dev-unsubscr...@commons.apache.org For additional commands, e-mail: dev-h...@commons.apache.org
