Fix adjoint gradient with conductivities

[mochen4]

Replaces #1823. Fixes #1811.

[stevengj]

Test failure?

[mochen4]

Test failure?

The double precision tests past but single precision one failed. I am still looking at why.

[stevengj]

Maybe try changing FD_DEFAULT in python/adjoint/utils.py to see if it is the fault of the finite-difference dε/du.

[stevengj]

In discussion with @mochen4, it seems like simply increasing the runtime fixes the problem — there seems to be a problem with the stop_when_dft_decayed convergence criterion in single precision.

[mawc2019]

When I compile the current master branch with --enable-single and Python 3.7.3, the check at test_damping still fails as described in #1821. The finite-difference gradients change a lot between double precision and single precision. I increased minimum_run_time to 2000 in forward_simulation_damping and adjoint_solver_damping, but the check still failed. A tweak following @smartalecH 's suggestion is as follows, with which the check passes locally. Maybe this issue is negligible because the online check does not fail.

    def test_damping(self):
        print("*** TESTING CONDUCTIVITIES ***")

        for frequencies in [[1/1.58, fcen, 1/1.53]]:
            ## filter the design field
            filter_radius = 0.12
            p_filtered = mpa.conic_filter(
                np.reshape(p,(Nx,Ny)),filter_radius,design_region_size.x,design_region_size.y,design_region_resolution).flatten()
            dp_filtered = mpa.conic_filter(
                np.reshape(dp,(Nx,Ny)),filter_radius,design_region_size.x,design_region_size.y,design_region_resolution).flatten()
            
            ## compute gradient using adjoint solver
            adjsol_obj, adjsol_grad = adjoint_solver_damping(p_filtered, frequencies)

            ## compute unperturbed S12
            S12_unperturbed = forward_simulation_damping(p_filtered, 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_damping(p_filtered+dp_filtered, frequencies)

            ## compare gradients
            if adjsol_grad.ndim < 2:
                adjsol_grad = np.expand_dims(adjsol_grad,axis=1)
            adj_scale = (dp_filtered[None,:]@adjsol_grad).flatten()
            fd_grad = S12_perturbed-S12_unperturbed
            print("Directional derivative -- adjoint solver: {}, FD: {}".format(adj_scale,fd_grad))
            tol = 0.06 if mp.is_single_precision() else 0.03
            self.assertClose(adj_scale,fd_grad,epsilon=tol)

[mochen4]

test_damping is only added after this PR, so certainly you wouldn't see the failure for commits before this one. But that test is only about the added features in this PR, so I don't think it is the reason why #1821 fails locally on your machine.

Maybe this issue is negligible because the online check does not fail.

The CI test runs the single-precision case with python3.9. I tested the master branch locally under python3.9 in single-precision, and tests passed. So I think the issue is either something with the python version or something in your local machine.

@mawc2019