A fast direct solver for two dimensional quasi-periodic multilayered medium scattering problems
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This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. The fast solver is built from the linear system that results from the discretization of a boundary integral formulation that is robust at Wood's anomalies. When the interface geometries are complex, the linear system is too large to be handled via dense linear algebra. The key building block of the proposed solver is a fast direct direct solver for the large sparse block system that corresponds to the discretization of boundary integral equations. The solver makes use of hierarchical matrix inversion techniques, has a cost that scales linearly with respect to the number of unknowns on the interfaces and the precomputation can be used for all choices of boundary data. By partitioning the remainder of the precomputation into parts based on their dependence on incident angle, the proposed direct solver is efficient for problems involving many incident angles like those that arise in applications. For example for a problem on an eleven layer geometry where the solution is desired for 287 incident angles, the proposed solution technique is 87 times faster than building a new fast direct solver for each new incident angle. An additional feature of the proposed solution technique is that solving a problem where an interface or layer property is changed require an update in the precomputation that cost linearly with respect to the number points on the affected interfaces with a small constant prefactor. The efficiency for modified geometries and multiple solves make the solution technique well suited for optimal design and inverse scattering applications.
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