  We present a multiscale continuous Galerkin (MSCG) method for the fast and
accurate stochastic simulation and optimization of time-harmonic wave
propagation through photonic crystals. The MSCG method exploits repeated
patterns in the geometry to drastically decrease computational cost and
incorporates the following ingredients: (1) a reference domain formulation that
allows us to treat geometric variability resulting from manufacturing
uncertainties; (2) a reduced basis approximation to solve the parametrized
local subproblems; (3) a gradient computation of the objective function; and
(4) a model and variance reduction technique that enables the accelerated
computation of statistical outputs by exploiting the statistical correlation
between the MSCG solution and the reduced basis approximation. The proposed
method is thus well suited for both deterministic and stochastic simulations,
as well as robust design of photonic crystals. We provide convergence and cost
analysis of the MSCG method, as well as a simulation results for a waveguide
T-splitter and a Z-bend to illustrate its advantages for stochastic simulation
and robust design.
