oskooi
commented
1 year ago Here is an improved test which compares the adjoint gradients from the multi- vs. single-frequency simulations for five equally spaced frequencies spanning the full range of the source bandwidth. The gradient comparison uses the assertClose method which is based on the $L_\infty$ norm.
In order for this test to pass (using Meep compiled with double-precision floating point and a grid resolution of 50), the relative tolerance parameter of assertClose (tol_grad in the script) must be at least 1.1. A relative tolerance greater than one is an indication of unacceptably large errors in the broadband adjoint gradients.
def test_broadband_gradient(self):
print("*** TESTING BROADBAND GRADIENT ***")
nfrq = 5
multifreq = np.linspace(
self.fcen - 0.5*self.df,
self.fcen + 0.5*self.df,
nfrq,
)
multifreq_val, multifreq_grad = self.adjoint_solver_two_objfunc(
self.p,
multifreq,
)
tol_objfunc = 0.01 # relative tolerance for objective function comparison
tol_grad = 1.1 # relative tolerance for adjoint gradient comparison
for n in range(nfrq):
frq = multifreq[n]
singlefreq_val, singlefreq_grad = self.adjoint_solver_two_objfunc(
self.p,
[frq],
)
for m in [0, 1]:
s = n if m == 0 else n+nfrq
self.assertClose(singlefreq_val[m], multifreq_val[s], tol_objfunc)
self.assertClose(singlefreq_grad[:, m], multifreq_grad[:, s], tol_grad)
print(f"PASSED: frequency={frq}, m={m}.")Importantly, if the frequency range is reduced to 40% of the source bandwidth, the test passes with a smaller relative tolerance of tol_grad = 0.01. This is an indication that adjoint gradients computed at frequencies near the edge of the source bandwidth are the least accurate.
multifreq = np.linspace(
self.fcen - 0.2*self.df,
self.fcen + 0.2*self.df,
nfrq,
)To summarize, the main finding here is that in the current setup of the adjoint solver, accurate adjoint gradients over a broad bandwidth can only be obtained if the monitor bandwidth is significantly smaller than the source bandwidth.
stevengj
smartalecH
MarkMa1990
This is an attempt to investigate the errors in the adjoint gradient of a broadband simulation. The test involves comparing the adjoint gradients using two different configurations:
The adjoint gradient at wavelength $\lambda_m$ in (1) should match the adjoint gradient $\lambda_m$ in (2) for each $m=1,2,3$. However, we should expect some discrepancy in the results for $\lambda_1$ and $\lambda_3$ (the non-center wavelengths of the pulsed source) because the
EigenmodeSourceuses the mode profile for the center frequency only. While we have already investigated the errors introduced by the broadband mode source at off-center frequencies in Tutorial/Eigenmode Source, we do not seem to have investigated the effect of these errors on the adjoint gradient.The test involves two objective functions: reflectance and transmittance of a waveguide mode converter. A schematic of the simulation setup is shown in the first figure below. The design region is random. The results (second figure) demonstrate that the $L_2$ norm of the difference in the adjoint gradient for (1) and (2) is nearly one order of magnitude larger for the non-center wavelengths relative to the center wavelength. The relative error is larger for the reflectance objective function (blue data) than the transmittance objective function (red data). When the absolute magnitude of the gradients in (1) and (2) are compared side by side, they seem to be similar (third and fourth figures).
It would seem that a one order of magnitude reduction in the relative accuracy of the adjoint gradient is large enough that it could degrade the convergence of a topology optimization run.
1. Adjoint gradient of reflectance objective function at $\lambda$ = 1.55 μm (center wavelength)
2. Adjoint gradient of transmittance objective function at $\lambda$ = 1.55 μm (center wavelength)
The script used to generate these results will eventually be added as a separate unit test to
test_adjoint_solver.py.