Scaling function for regularization

[CheukHinHoJerry]

In the old ACE package, there is a scaling function used in ACESchrodinger:

which the current package don't have.

[cortner]

I will get to this eventually, but if you are in a rush, here is what this does, maybe this can help you implement it:

The scaling function specifies how to regularize each parameter. It's best to think of it as follows: the parameter for the envelope has not natural scaling and maybe shouldn't be regularized at all -> hence the 0.

But the parameters associated with the linear ACE are associated to basis functions and those basis functions have a natural regularity attached to them. In the context of the 1d models each basis function Bi can be thought of as a symmetrized tensor product P_k1 ... P_kN where the kt are the degrees. The scaling attached to this basis function is morally speaking something like the H^p norm. For simplicity we just use

   k1^p + ... + kN^p

The p here is the 2 in scaling(ace, 2)

So to implement this one would need to recover the specification of each basis function and one can then reconstruct this scaling very easily.

[CheukHinHoJerry]

This should be done. Shall we close this issue?

CC @cortner

[cortner]

you know best, please close it if you think this has been taken care of.