Fast Hermite interpolation and observables

[MarcBerliner]

Description

This PR adds Hermite interpolation to the IDAKLUSolver, which gives more accurate post-solve interpolation accuracy and improves performance by moving code from Python to C++. The high-frequency example below is 25x faster.

For simulations that save the adaptive time stepping states y, the new solver option hermite_interpolation uses the time derivatives yp to obtain a more accurate interpolator. Previously, our post-solve interpolation involved:

1. Computing a variable v for the given raw data sol.all_ts and sol.all_ys (like v_raw = sol[v].data)
2. Linear interpolating v_raw onto new time points t_new (like np.interp1d(t_new, sol.t, v_raw))

This was problematic because linear interpolation does not always provide an accurate solution (like in the example picture above). With the new Hermite interpolation, the two steps are combined into a single step that uses the state derivatives:

• Create a y_new matrix by interpolating sol.all_ys onto t_new accurately using sol.all_yps, then observe v at the t_new and y_new values. The Hermite interpolation API is just sol[v](t)
• We can still run sol[v].data to access the v_raw data points at the sol.t points

These changes are most visible with very large datasets. Here are the new timings to get an accurate version of the SPM with t_new = np.linspace(t0, tf, 1000000):

• Dense t_interp (old, Python): 10.6 s
• Hermite interp (new, C++): 0.421 s

giving a 25x speedup for this example.

Main changes in the code:

• Moves the solution indexing (including sol["Voltage [V]"].data or sol["Voltage [V]"](t)) from Python to C++
• Refactors the ProcessedVariable class and adds subclasses like ProcessedVariable0D, ProcessedVariable1D, etc.
• Adds an all_yps term to the Solution class
• Remove unused warn argument from the interpolator

Notes:

• Hermite interpolation can only work if we have all the adaptive time stepping y and yp values. This means simulations with output_variables or t_interp values cannot be supported.
• Sensitivity analyses are not currently using the C++ observable interface.
• There are a few places in ProcessedVariable that will be cleaned up once we move to a standalone IDAKLU package

Fixes #4419

Type of change

Please add a line in the relevant section of CHANGELOG.md to document the change (include PR #) - note reverse order of PR #s. If necessary, also add to the list of breaking changes.

• Optimization (back-end change that speeds up the code)

Key checklist:

• No style issues: $ pre-commit run (or $ nox -s pre-commit) (see CONTRIBUTING.md for how to set this up to run automatically when committing locally, in just two lines of code)
• All tests pass: $ python run-tests.py --all (or $ nox -s tests)
• The documentation builds: $ python run-tests.py --doctest (or $ nox -s doctests)

You can run integration tests, unit tests, and doctests together at once, using $ python run-tests.py --quick (or $ nox -s quick).

Further checks:

• Code is commented, particularly in hard-to-understand areas
• Tests added that prove fix is effective or that feature works

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.41%. Comparing base (08e5cf2) to head (4e708af).
Report is 1 commits behind head on develop.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop    #4464      +/-   ##
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- Coverage    99.46%   99.41%   -0.05%     
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  Files          293      293              
  Lines        22332    22554     +222     
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+ Hits         22212    22423     +211     
- Misses         120      131      +11     

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[kratman]

Pretty minor changes

[martinjrobins]

great stuff, thanks @MarcBerliner! I like the refactor of processed variable into subclasses. A few minor points below

[kratman]

From my side it looks good, but lets wait for @martinjrobins to approve of the changes

Changes were made

[martinjrobins]

happy with this, thanks @MarcBerliner