  Adjoint variable method in combination with gradient descent optimization has
been widely used for the inverse design of nanophotonic devices. In many of
such optimizations, the design region is only a small fraction of the total
computational domain. Here we show that the adjoint variable method can be
combined with the Schur complement domain decomposition method. With this
combination, in each optimization step, the simulation only involves the
degrees of freedom that are inside the design region. Our approach should
significantly improve the computational efficiency of adjoint variable method
based optimization of photonic structures.
