Enable nonzero k_point in adjoint solver

[mochen4]

Fixes #1667. The test_adjoint_solver.py is updated correspondingly to include when k_point is False, zero, and nonzero.

[mochen4]

The original setup of the EigenmodeCoefficient test in the test_adjoint_solver.py has the source and monitor coincide with the sides of the PMLs, and extends into the PMLs of the other direction. The adjoint gradients and directional derivates won't match if k_point is nonzero; if we use the original calculate_fd_gradient() in optimization_problem.py that calculates the fd_gradients in unit directions, the fields become nan. (The forward and adjoint run are both okay, but the fields are nan after single pixel is perturbed during calculate_fd_gradient().)

Using the setup of the tutorial example, the gradients agree with nonzero k_point.

If the source and monitor don't extend into the PMLs, then the adjoint gradients will match the directional derivates, but the fields are still nan when using calculate_fd_gradient(). If we then moved the source and monitor a little bit away from the PMLs, everything will work fine; so I further modified the test this way.

[codecov-commenter]

Codecov Report

Merging #1705 (fb0b3b2) into master (83a10c5) will increase coverage by 0.02%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master    #1705      +/-   ##
==========================================
+ Coverage   73.32%   73.35%   +0.02%     
==========================================
  Files          13       13              
  Lines        4525     4530       +5     
==========================================
+ Hits         3318     3323       +5     
  Misses       1207     1207              

Impacted Files	Coverage Δ
python/adjoint/objective.py	75.30% <100.00%> (+0.30%)	⬆️
python/adjoint/optimization_problem.py	69.58% <100.00%> (+0.23%)	⬆️
python/simulation.py	76.25% <100.00%> (+0.01%)	⬆️

[mochen4]

In the context of the PR #1704, the factor of 2 is needed whenever force_complex_fields=False. However, when k_point is nonzero, we may not need the factor even if we don't force complex fields.

I wonder if there are other cases where we don't need the factor even if force_complex_fields=False; and what will be the best way to check whether indeed only real fields are used.

[oskooi]

Looks like the failing test is test_adjoint_solver.py for the case in which Meep is compiled with single precision (#1698). Perhaps you need to just lower the tolerance some place when mp.is_single_precision() is true?

[mochen4]

Thanks Ardavan! I lowered the tolerance for single precision, and it works!