Basic grid refinement

[jbcaillau]

@PierreMartinon now that it is possible to pass a time_grid, we can rather easily combine solve with grid refinement and warm start to solve iteratively on a adaptive discretisation.

[ocots]

@PierreMartinon @jbcaillau @0Yassine0 In the Space shuttle problem from JuMP tutorial, the time steps are considered as variables to optimize. Do we agree that we cannot do that for the moment?

[PierreMartinon]

We can do

• a fixed uniform time grid (set grid_size=...) with fixed or free t0,tf.
• a fixed non-uniform time grid, also compatible with fixed or free t0,tf. More precisely the normalized grid in [0,1] is the one fixed, so if t0 and/or tf are free, all the time steps can change, but the 'repartition' of the steps is fixed.

We cannot do

• optimize the whole vector [t0, t1, ..., tf] with each t_i an independant variable

On the other hand, we could try to do that by using a fixed grid on [0,1] with a series of n positive optimization variables Dt_i, and manually rescale the dynamics. I expect the numerical cost to be horrendous though.

[jbcaillau]

@ocots @PierreMartinon @0Yassine0 yes, the usual manual trick works to glue things (e.g. multidynamics with unknown switching time). Actually, it is interesting and not completely standard to optimise the grid points themselves: interesting to compare with a uniform grid size (should be better, but price to pay = more vars).

@ocots I remember that you did sth similar, no?

@PierreMartinon @joseph-gergaud not very difficult to add this possibility in the transcription process (enforcing a lower bound on each $t_{i+1}-t_i$ to avoid stupid things), provided there is a demonstrated advantage.

[PierreMartinon]

I think I can do a simple example for this. @jbcaillau, you typically take positive Dt_i instead of the t_i themselves, easier constraint ;-)

[PierreMartinon]

@PierreMartinon now that it is possible to pass a time_grid, we can rather easily combine solve with grid refinement and warm start to solve iteratively on a adaptive discretisation.

A basic illustration here #158
(on uniform grids, but still passed as explicit grids...)

[PierreMartinon]

@ocots @PierreMartinon @0Yassine0 yes, the usual manual trick works to glue things (e.g. multidynamics with unknown switching time). Actually, it is interesting and not completely standard to optimise the grid points themselves: interesting to compare with a uniform grid size (should be better, but price to pay = more vars).

@ocots I remember that you did sth similar, no?

@PierreMartinon @joseph-gergaud not very difficult to add this possibility in the transcription process (enforcing a lower bound on each ti+1−ti to avoid stupid things), provided there is a demonstrated advantage.

Added an example in #158