Automatic boxmap discretization

[cafaxo]

It would be really cool if the user could just do

g = boxmap(f)

Once the user applies g to a boxset, an automatic/adaptive discretization strategy is used. A naive approach would be to define an increasing sequence of point discretizations of the unit box and go along this sequence until the image stabilises. I imagine that this would work well for simple/easy problems. In this issue, we should decide:

• Do we want this?
• If yes, how rigoros should the discretization be?
• How exactly should it work?
• How would it be accessible? (If the automatic strategy is not very good in general, then boxmap might not be the most honest way)

This issue depends/extends on #30.

[gaioguy]

Yes! This is something we should definitely do. There are ideas on an adaptive version in Junge, O.: Rigorous discretization of subdivision techniques, Proceedings of EQUADIFF 99, World Scientific, 2000..

[cafaxo]

This method makes sense if f is locally Lipschitz. Do we want to require that for boxmap(f)? If it is locally Lipschitz, how would we determine L? Would the user have to supply that?

[gaioguy]

It seems like a reasonable general assumption to me. In practice, if one is not aiming for a rigorous computation of a box image (for which I would propose to use interval arithmetic anyway), one could simply evaluate the derivative of the map at the center of the box as an approximation to L - or rather do the SVD of the derivative and construct a properly aligned grid as proposed at the end of the above Ref.

[cafaxo]

We could have the case that the user maps a really coarse BoxSet (imagine a single big box) into a really fine partition. Only sampling the center for L could be inadequate in this case.

[gaioguy]

Right, it would be good to have some way to estimate how well the linearization approximates the map over the box. Maybe sampling at the vertices of the box?