Issue 937 sensitivity

[valentinsulzer]

Description

Added sensitivity to CasadiAlgebraicSolver class. To do this, added a CasadiSolution and ProcessedCasadiVariable class, which can hold a symbolic solution (i.e. inputs are symbolic, not yet provided). The value and sensitivity of a ProcessedCasadiVariable can then be found for a specific input by calling .value(inputs) or .sensitivity(inputs).

I haven't added any examples or much extra docs for this, I think it can wait until we make this a more fleshed out feature. The next step would be to do the CasadiSolver.

Fixes #937

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.

• New feature (non-breaking change which adds functionality)
• Optimization (back-end change that speeds up the code)
• Bug fix (non-breaking change which fixes an issue)

Key checklist:

• No style issues: $ flake8
• All tests pass: $ python run-tests.py --unit
• The documentation builds: $ cd docs and then $ make clean; make html

You can run all three at once, using $ python run-tests.py --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

Merging #940 into develop will increase coverage by 0.05%.
The diff coverage is 100.00%.

@@             Coverage Diff             @@
##           develop     #940      +/-   ##
===========================================
+ Coverage    98.06%   98.11%   +0.05%     
===========================================
  Files          220      221       +1     
  Lines        11370    11518     +148     
===========================================
+ Hits         11150    11301     +151     
+ Misses         220      217       -3     

Impacted Files	Coverage Δ
...ybamm/expression_tree/operations/unpack_symbols.py	100.00% <ø> (ø)
pybamm/solvers/processed_variable.py	98.52% <ø> (ø)
pybamm/discretisations/discretisation.py	99.77% <100.00%> (+<0.01%)	⬆️
pybamm/expression_tree/input_parameter.py	100.00% <100.00%> (ø)
pybamm/expression_tree/operations/jacobian.py	100.00% <100.00%> (ø)
pybamm/expression_tree/symbol.py	97.44% <100.00%> (ø)
pybamm/models/base_model.py	98.67% <100.00%> (+0.08%)	⬆️
pybamm/parameters/parameter_values.py	99.08% <100.00%> (-0.15%)	⬇️
pybamm/parameters/print_parameters.py	97.14% <100.00%> (+0.17%)	⬆️
pybamm/simulation.py	98.41% <100.00%> (+1.36%)	⬆️
... and 5 more

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[martinjrobins]

looks great @tinosulzer, few comments on the api and testing below. very excited to try this out when you get it implemented using an ode solver :)

[TomTranter]

Hi Tino, I don't really get how to use the sensitivity functionality and how it's better than running solve with input parameters. It needs an example. I've read through the files changed and I see a few examples of how to use it on very basic examples using one liner models in the tests but I think it needs to be explained using the full battery models properly

[brosaplanella]

Hi Tino! Not my area of expertise (either sensitivity of CasADI) but looks really neat. If I understand correctly, CasADI allows to solve the model leaving one parameter unspecified so it can be passed later and the sensitivity wrt that parameter calculated.

However, I agree with @TomTranter that this feature really needs a tutorial notebook, especially with an application to a complex model such SPM(e) or DFN. Otherwise it is quite hard to understand what is going on.

[valentinsulzer]

To be honest, the lack of example/documentation was intentional as I want to keep this "semi-private" until we are settled on the API. Also, it currently only works with algebraic-only models, i.e. not any of the battery models

[brosaplanella]

Then it looks good to me :)

[TomTranter]

To be honest, the lack of example/documentation was intentional as I want to keep this "semi-private" until we are settled on the API. Also, it currently only works with algebraic-only models, i.e. not any of the battery models

But the issue is that I don't really know how to test it

[martinjrobins]

I'm happy with these changes, thanks @tinosulzer. There was some stuff in this PR about c-rate, should this be in here?

[martinjrobins]

are these changes to that parameters supposed to be in this PR?

[martinjrobins]

are these supposed to be in this PR, doesn't seem like these are anything to do with sensitivities?

[valentinsulzer]

I merged in another branch which was about to be merged to develop, thought those changes would disappear once that branch got merged