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A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems

This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. When the interface geometries are complex, the dominant term in the computational cost of creating the direct solver scales $O(NI)$ where $N$ is the number of discretization points on each interface and $I$ is the number of interfaces. The bulk of the precomputation can be re-used for any choice of incident wave. As a result, the direct solver can solve over 200 scattering problems involving an eleven layer geometry with complex interfaces 100 times faster than building a new fast direct solver from scratch for each new set of boundary data. An added benefit of the presented solver is that building an updated solver for a new geometry involving a replaced interface or a change in material property in one layer is inexpensive compared to building a new fast direct solver from scratch.