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

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 s...

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Published in:	Computational mathematics and modeling Vol. 32; no. 4; pp. 438 - 452
Main Authors:	Minin, Iu. B., Matveev, S. A., Fedorov, M. V., Zacharov, I. E., Rykovanov, S. G.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.10.2021 Springer Nature B.V
Subjects:	Applications of Mathematics Benchmarks Computational Mathematics and Numerical Analysis Electric fields Green's functions Hankel matrices Helmholtz equations Integral equations Linear systems Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Optimization Photonics Solvers Sparse matrices Topology optimization Python module C++ performance GMRES Helmholtz equation Toeplitz-like matrix topology optimization parallelization GPU
ISSN:	1046-283X, 1573-837X
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Abstract	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 ( t ¯ = 0.017 s ) and of 125 ( t ¯ = 0.77 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.
AbstractList	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 ( t ¯ = 0.017 s ) and of 125 ( t ¯ = 0.77 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. 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 (t¯=0.017s) and of 125 (t¯=0.77s) 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.
Author	Minin, Iu. B. Zacharov, I. E. Matveev, S. A. Fedorov, M. V. Rykovanov, S. G.
Author_xml	– sequence: 1 givenname: Iu. B. surname: Minin fullname: Minin, Iu. B. organization: Skoltech Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology, Fryazino Branch of Kotel’nikov Institute of Radio-Engineering and Electronics of Russian Academy of Sciences – sequence: 2 givenname: S. A. surname: Matveev fullname: Matveev, S. A. organization: Skoltech Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences – sequence: 3 givenname: M. V. surname: Fedorov fullname: Fedorov, M. V. organization: Skoltech Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology, Sirius University of Science and Technology – sequence: 4 givenname: I. E. surname: Zacharov fullname: Zacharov, I. E. organization: Skoltech Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology – sequence: 5 givenname: S. G. surname: Rykovanov fullname: Rykovanov, S. G. organization: Skoltech Center for Computational and Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology
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Cites_doi	10.1109/EnT-MIPT.2018.00040 10.1137/S0895479803437803 10.1016/S0167-8191(97)00004-5 10.1137/0907058 10.1007/978-3-642-33078-0_31 10.1145/1089014.1089021 10.1016/S0024-3795(00)00064-1 10.3389/fbuil.2018.00069 10.1115/1.4005491 10.1002/(SICI)1097-0207(19980830)42:8<1441::AID-NME428>3.0.CO;2-C 10.1515/nanoph-2019-0308 10.1109/IPDPS.2014.48 10.1016/j.laa.2005.03.040 10.2172/1614847 10.1364/JOSAA.21.002223 10.1007/s11227-012-0825-3 10.1515/eng-2019-0059 10.1109/ISQED.2012.6187484 10.1137/10078356X 10.1201/9781420049961 10.1007/BF01732607 10.1137/12086563X 10.1080/01468039308204219 10.1002/num.20484 10.1016/S0167-8191(98)00084-2 10.1364/JOSAA.13.002441
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Snippet	Generalized Minimal Residual Method (GMRES) was benchmarked on many types of GPUs for solving linear systems based on dense and sparse matrices. However, there...
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SubjectTerms	Applications of Mathematics Benchmarks Computational Mathematics and Numerical Analysis Electric fields Green's functions Hankel matrices Helmholtz equations Integral equations Linear systems Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Optimization Photonics Solvers Sparse matrices Topology optimization
Title	Benchmarks of Cuda-Based GMRES Solver for Toeplitz and Hankel Matrices and Applications to Topology Optimization of Photonic Components
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