Allow arbitrary initial psi estimates

[timothy-nunn]

Enables arbitrary initial psi estimates by providing a callable to (Equilibrium(..., psi=<some callable>)). The default behaviour remains unchanged where a Gaussian guess is used (now implemented in PsiGuessGaussian). Implemented a class to initialise psi from a parabolic Jtor assumption PsiGuessParabolicJtor (A.7 of Wai et al (2026))

Here is how the new initial psi guess compares to the default Gaussian:

The following slides show that, in this case, the initial guess does not have a large impact on the solvers ability to find a good solution. However, in some more complex equilibria this psi guess will be more favourable.

[kpentland]

This all looks great @timothy-nunn!

Can you remind me what the purpose was for having this alternate guess? Different options are always good I suppose, especially when you can input your approximate boundary and Ip estimate.

On a similar vein, could you show a plot(s) of the existing default guess (for one of the examples in FreeGSNKE) and then the same guess using your new Jtor function? If possible can you plot psi_plasma and jtor for each one? It's hard to gauge what the new function is doing without seeing it.

It would be great to get some examples of this in action in FreeGSNKE. Maybe we make another notebook showing these plots and how it works for both forward and inverse solves?

[kpentland]

All looking good @timothy-nunn, I'm happy to merge but perhaps we can make an example of how to use it in FreeGSNKE? Perhaps we could split the Example02 notebook into Example02a and 02b? We can show how to use this in 02b and any other more advanced features we come up with down the line? We can even check what effect this new guess has on the number of NK iterations taken until convergence...

Ready to go whenever you're happy @georgeholt1 !

[timothy-nunn]

All looking good @timothy-nunn, I'm happy to merge but perhaps we can make an example of how to use it in FreeGSNKE? Perhaps we could split the Example02 notebook into Example02a and 02b? We can show how to use this in 02b and any other more advanced features we come up with down the line? We can even check what effect this new guess has on the number of NK iterations taken until convergence...

Ready to go whenever you're happy @georgeholt1 !

Hi @kpentland will do. My tests don't show any significant difference in convergence for Example 1 in FreeGSNKE