Issue 1015 algebraic solver

[valentinsulzer]

Description

I couldn't really get this working but thought I would merge what I have so far as it might be useful for another effort down the line (will require a better example)

• Added some new kinds of algebraic solvers (posing a minimization problem instead of root-finding), which can be constrained
• Added bounds attribute to the Variable object, to allow these constraints

The main problem is that the solvers tend to find local minima instead of roots.

Fixes #1015

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 #1059 into develop will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff            @@
##           develop    #1059   +/-   ##
========================================
  Coverage    97.73%   97.74%           
========================================
  Files          242      242           
  Lines        12476    12528   +52     
========================================
+ Hits         12193    12245   +52     
  Misses         283      283           

Impacted Files	Coverage Δ
pybamm/expression_tree/state_vector.py	100.00% <ø> (ø)
pybamm/discretisations/discretisation.py	99.77% <100.00%> (+<0.01%)	⬆️
pybamm/expression_tree/array.py	100.00% <100.00%> (ø)
pybamm/expression_tree/concatenations.py	97.45% <100.00%> (ø)
pybamm/expression_tree/matrix.py	100.00% <100.00%> (ø)
pybamm/expression_tree/variable.py	100.00% <100.00%> (ø)
pybamm/expression_tree/vector.py	100.00% <100.00%> (ø)
pybamm/models/standard_variables.py	100.00% <100.00%> (ø)
pybamm/solvers/algebraic_solver.py	100.00% <100.00%> (ø)
pybamm/solvers/base_solver.py	100.00% <100.00%> (ø)
... and 2 more

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[brosaplanella]

Looks great Tino, thanks! I assume that, apart from the solver getting stuck in a local minimum, the rest works fine, right?

[valentinsulzer]

It all works fine, yes, but I haven't yet found a case where rootfinding fails and minimization works instead. At least the bounds functionality is implemented, in case someone thinks of a hacky way to implement bounds for the rootfinder (e.g. artificially extending the function outside of the bounds)