  We present a sweeping preconditioner for quasi-optimal Domain Decomposition
Methods (DDM) applied to Helmholtz transmission problems in periodic layered
media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission
operators that are approximations of Dirichlet-to-Neumann (DtN) operators.
Employing shape perturbation series, we construct approximations of DtN
operators corresponding to periodic domains, which we then use as transmission
operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators
that are the building blocks of DDM are expressed via robust boundary integral
equation formulations. We use Nystr\"om discretizations of quasi-periodic
boundary integral operators to construct high-order approximations of RtR.
Based on the premise that the quasi-optimal transmission operators should act
like perfect transparent boundary conditions, we construct an approximate LU
factorization of the tridiagonal QO DD matrix associated with periodic layered
media, which is then used as a double sweep preconditioner. We present a
variety of numerical results that showcase the effectiveness of the sweeping
preconditioners applied to QO DD for the iterative solution of Helmholtz
transmission problems in periodic layered media.
