Choice of chi-squared functional for BMN Harmonic optimization

[lazersos]

It has been identified that in chisq_helicity.f90 the definition may need to be updated. The two key points are

• A undocumented radial weighting is applied to harmonics with respect to m (is not applied if sigma_helicity < 0)
• The harmonics are weighted by the energy in the desired spectrum as opposed to the B(n=0,m=0)

From an email by @pomphrey

It is possible to define a measure of quasi-axisymmetry many ways. For example, in their numerical studies Plunk and Helander (https://pure.mpg.de/rest/items/item_2574915_8/component/file_2598135/content) define the measure (see Eq 48) as the ratio of the non qas part to the full field whereas Landreman, Sengupta and Plunk (https://pure.mpg.de/rest/items/item_3020784/component/file_3039370/content) prefer the definition Aaron was referring to (Eq 5.2).
It seems from what you are saying, that ROSE adopts this second definition for the Chisq-Helicity. Neither one is "correct" ... they are just different choices. However, they will produce different cost function landscapes for the optimizer in question to explore, and therefore different results.

I do think that it would be a good idea to have the ability in STELLOPT to switch to the second definition, with B(0,0) on the denominator, and for some tests to be run seeing which is preferred (as Aaron suggests).

The radial weighting

            IF (sigma(ik) < 0.0) THEN
               rad_sigma = 1
            ELSE IF (m < 3) THEN
               rad_sigma = sj
            ELSE IF (m == 3) THEN
               rad_sigma = sj**1.5
            ELSE
               rad_sigma = sj*sj
            END IF

is rather curious and I wonder if Mike Z remembers why it is there? I suspect that this piece of coding is a residual of some monkeying around with the source code at some point long ago to cure a troublesome Bmn spectrum for a particular application. Perhaps Long-Poe Ku inserted this coding but we will never be able to confirm that. I don't think it should be there for a general application and I suspect it should be removed!