  We present a fabrication uncertainty aware and robust design optimization
approach that can be used to obtain robust design estimates for nonlinear,
nonconvex, and expensive model functions. It is founded on Gaussian processes
and a Monte Carlo sampling procedure, and assumes knowledge about the
uncertainties associated with a manufacturing process. The approach itself is
iterative. First, a large parameter domain is sampled in a coarse fashion. This
coarse sampling is used primarily to determine smaller candidate regions to
investigate in a second, more refined sampling pass. This finer step is used to
obtain an estimate of the expected performance of the found design parameter
under the assumed manufacturing uncertainties. We apply the presented approach
to the robust optimization of the Purcell enhancement of a photonic crystal
nanobeam cavity. We obtain a predicted robust Purcell enhancement of
$\overline{F}_{\mathrm{P}} \approx 3.3$. For comparison we also perform an
optimization without robustness. We find that an unrobust optimum of
$F_{\mathrm{P}} \approx 256.5$ dwindles to only $\overline{F}_{\mathrm{P}}
\approx 0.7$ when fabrication uncertainties are taken into account. We thus
demonstrate that the presented approach is able to find designs of
significantly higher performance than those obtained with conventional
optimization.
