Fix adjoint gradient with conductivities

[mochen4]

Fixes #1811 after PR #1801

[smartalecH]

Would be good to add a simple test to test_adjoint_solver.py.

Maybe just borrow a simple test like this...

meep/python/tests/test_adjoint_solver.py

Lines 308 to 333 in d4b1e1a

	def test_adjoint_solver_eigenmode(self):
	print("*** TESTING EIGENMODE ADJOINT ***")
	## test the single frequency and multi frequency case
	for frequencies in [[fcen], [1/1.58, fcen, 1/1.53]]:
	## compute gradient using adjoint solver
	adjsol_obj, adjsol_grad = adjoint_solver(p, MonitorObject.EIGENMODE, frequencies)
	## compute unperturbed S12
	S12_unperturbed = forward_simulation(p, MonitorObject.EIGENMODE, frequencies)
	## compare objective results
	print("S12 -- adjoint solver: {}, traditional simulation: {}".format(adjsol_obj,S12_unperturbed))
	self.assertClose(adjsol_obj,S12_unperturbed,epsilon=1e-6)
	## compute perturbed S12
	S12_perturbed = forward_simulation(p+dp, MonitorObject.EIGENMODE, frequencies)
	## compare gradients
	if adjsol_grad.ndim < 2:
	adjsol_grad = np.expand_dims(adjsol_grad,axis=1)
	adj_scale = (dp[None,:]@adjsol_grad).flatten()
	fd_grad = S12_perturbed-S12_unperturbed
	print("Directional derivative -- adjoint solver: {}, FD: {}".format(adj_scale,fd_grad))
	tol = 0.04 if mp.is_single_precision() else 0.01
	self.assertClose(adj_scale,fd_grad,epsilon=tol)

...and retrofit it to include a damping term.

[smartalecH]

It would be nice if you could eliminate these formatting changes from the PR. It makes review much easier and is important when comparing diffs.

Maybe check some of the default settings for your code editor -- it looks like it may be removing trailing whitespace automatically.

[smartalecH]

It would probably be better to refactor and use sigma_row where possible.

meep/src/meepgeom.cpp

Line 1632 in d4b1e1a

void geom_epsilon::sigma_row(meep::component c, double sigrow[3], const meep::vec &r) {

[smartalecH]

This doesn't seem to be the right adjoint formulation.

For one, this would get called even during the off-diagonal calculations (where there are no sigma off-diagonal terms).

Also, shouldn't there be an additional term independent of the primary recombination step? Something like

Dᵀ_a ε dσ/du D_f + σDᵀ_a dε/du D_f

Obviously, there are more scaling terms, but the point is that there is an additional inner product because dA/du yields a sum of tensors (like in your notes). You can do all of these in a single step, but it probably shouldn't be done here.