Three-dimensional finite-difference frequency-domain analyses of partially periodic photonic structures result in large-scale ill-conditioned linear systems. Due to the lack of efficient preconditioner and reordering scheme, existed general-purpose iterative and direct solvers are inadequate to solve these linear systems in time or memory. We propose an efficient direct solver to tackle this problem. By exploring the physical properties, the coefficient matrix structure, and hardware computing efficiency, we extend the concepts of grid geometry manipulation and multi-level Schur method to propose the Compressed Hierarchical Schur algorithm (CHiS). The proposed CHiS algorithm can use less memory and remove redundant computational workloads due to the homogeneity and periodicity of photonic structures. Moreover, CHiS relies on dense BLAS3 operations of sub-matrices that can be computed efficiently with strong scalability by the latest multicore processors or accelerators. The implementation and benchmarks of CHiS demonstrate promising memory usage, timing, and scalability results. The feasibility of future hardware acceleration for CHiS is also addressed using computational data. This high-performance analysis tool can improve the design and modeling capability for various photonic structures.
"Compressed Hierarchical Schur Algorithm for Frequency-domain Analysis of Photonic Structures." Taiwanese J. Math. 23 (2) 473 - 501, April, 2019. https://doi.org/10.11650/tjm/181113