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Benchmarks of Cuda-Based GMRES Solver for Toeplitz and Hankel Matrices and Applications to Topology Optimization of Photonic Components

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Generalized Minimal Residual Method (GMRES) was benchmarked on many types of GPUs for solving linear systems based on dense and sparse matrices. However, there are still no GMRES implementation benchmarks on Tesla V100 compared to GTX 1080 Ti ones or even for Toeplitz-like matrices. The introduced software consists of a Python module and a C++ library which enable to manage streams for concurrent computations of separated linear systems on a GPU (and GPUs). The GMRES solver is parallelized for running on a NVIDIA GPGPU accelerator. The parallelization efficiency is explored when GMRES is applied to solve (Helmholtz equation) linear systems based on the use of Green’s Function Integral Equation Method (GFIEM) for computing electric field distribution in the design domain. The proposed implementation shew the maximal speedup of 55 (\( \overline{t}=0.017\ \mathrm{s} \)) and of 125 (\( \overline{t}=0.77\ \mathrm{s} \)) for 1024 × 1024 (on GTX 1080 Ti) and 8192 × 8192 (on Tesla V100) dense Toeplitz matrices generated from GFIEM. 1024 × 1024 resolution provides accuracy 6.1% that can be acceptable according to testing and demonstrating on gradient computations and topology optimization. We open up possibilities for robust topology optimization of passive photonic integrated components. That has the advantage, e. g., of faster and more accurate designing photonic components on a PC without a supercomputer.

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Authors and Affiliations

  1. Skoltech Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology, Moscow, Russia

    Iu. B. Minin, S. A. Matveev, M. V. Fedorov, I. E. Zacharov & S. G. Rykovanov

  2. Fryazino Branch of Kotel’nikov Institute of Radio-Engineering and Electronics of Russian Academy of Sciences, Fryazino, Moscow, Russia

    Iu. B. Minin

  3. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

    S. A. Matveev

  4. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

    S. A. Matveev

  5. Sirius University of Science and Technology, Sochi, Krasnodar Krai, Russia

    M. V. Fedorov

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Minin, I.B., Matveev, S.A., Fedorov, M.V. et al. Benchmarks of Cuda-Based GMRES Solver for Toeplitz and Hankel Matrices and Applications to Topology Optimization of Photonic Components. Comput Math Model 32, 438–452 (2021). https://doi.org/10.1007/s10598-022-09545-2

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