[WIP] Trying to make MBAR work better adaptively when it fails.

[mrshirts]

• Changed the way defaults are set up. Now the default is BFGS + adaptive.
• Previously, it would run through solvers iteratively, assuming that multiple refinement is needed. However, if a solver is good, it should solve in one go. Therefore, we now loop through solvers until we find one that works.
• So now, automatically tries BFGS adaptive and if the first one chosen fails within tolerance.
• If multiple solvers are tried and fails, returns the best result.
• Tries to address Should have \sum_n W_nk = 1 error with pymbar3.0.5 while analysis is working with pymbar3.0.3 #419 and MBAR estimator automatically choosing the usable method #424
• Also makes the outputs and options of 'adaptive' more consistent with scipy methods to make the logic simpler.
• Provides more error information when sum of weights is not normalized.

TODO:

• scipy minimizer doesn't appear to be recognizing tolerance correctly - it will state success when the norm of the gradient is still relatively large (or at least larger than the tolerance). Different minimizers appear to use tolerance different ways? This needs to be analyzed.
• Decide the order of the methods in default - should BFGS be tried first?
• Maybe it's better to actually run multiple methods; start with something fast, then conclude with adaptive, which we know is more robust, once it's relatively close.
• Make the output good enough to use/understand for now (we will move to logging system in 4.0).
• See how different solvers behave and get tolerance to work correctly with them.

[orbeckst]

You're probably aware of @xiki-tempula 's implementation of "AutoMBAR" in alchemlyb. He did the testing and ultimately came up with the rationale for the order of methods that are tried.

[mrshirts]

Yes, I looked at that, but wasn't sure about logic. "hybr" is interesting since it is a root-finding method, which we haven't really looked at before.

Each method seems to handle tolerances a bit differently, and that's one thing that I've been trying to get to the bottom of. Calling adaptive at the end, which tends to continue a little further than other methods, seems to be a good way to makes sure that the results end up accurate, but I don't know if that's the "best" way to go.

[xiki-tempula]

@mrshirts I called hybr first as it is be super fast and can solve most cases when the simulation is sufficiently converged. So calling it first to save time on the easy cases.
If hybr failed, I will call adaptive which could solve 99% of the cases.
Lastly, I will call BFGS. It is not as robust as adaptive but could solve the cases that adaptive cannot.