GPU-Accelerated LOBPCG Method with Inexact Null-Space Filtering for Solving Generalized Eigenvalue Problems in Computational Electromagnetics Analysis with Higher-Order FEM

Dziekonski, Adam; Rewieński, Michał; Sypek, Piotr; Lamęcki, Adam; Mrozowski, Michał

doi:10.4208/cicp.oa-2016-0168

Cited by 14 publications

(5 citation statements)

References 21 publications

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“…Yet a few papers focused directly on the low-order FEM matvec. For example, (Dziekonski et al, 2017) used a conjugate gradient solver and optimized matvec product as its important part. (Dehnavi et al, 2010) and (Grigoraş et al, 2016) present similar findings and provided optimization strategies.…”

Section: Overview Of Published Literaturementioning

confidence: 99%

Acceleration of tensor-product operations for high-order finite element methods

Świrydowicz

Chalmers

Karakus

et al. 2019

The International Journal of High Performance Computing Applica

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This paper is devoted to GPU kernel optimization and performance analysis of three tensorproduct operators arising in finite element methods. We provide a mathematical background to these operations and implementation details. Achieving close-to-the-peak performance for these operators requires extensive optimization because of the operators' properties: low arithmetic intensity, tiered structure, and the need to store intermediate results inside the kernel. We give a guided overview of optimization strategies and we present a performance model that allows us to compare the efficacy of these optimizations against an empirically calibrated roofline.

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Section: Overview Of Published Literaturementioning

confidence: 99%

Acceleration of tensor-product operations for high-order finite element methods

Świrydowicz

Chalmers

Karakus

et al. 2019

The International Journal of High Performance Computing Applica

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“…This is in contrast to VFEM, where the normal discontinuity of edge elements leads to the formation of artificial charges at element interfaces, thus increasing the total divergence of the solutions; this problem worsens with mesh refinement. 50 Note that in VFEM, the zero-frequency spurious solutions are separated by filtering out the null-space of the curl operator from the spectrum using iterative techniques, 51,52 or by finding eigenvalues in the interior of the spectrum. This is necessary, since if the physical solution space is not normal to the null space, the physical solutions will be polluted by null vectors.…”

Section: B the Penalty Methods And The Zero-divergence Constraintmentioning

confidence: 99%

Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Pham,

Bharadwaj,

Ram-Mohan

2019

Preprint

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Present day electromagnetic field calculations have basic limitations since they employ techniques based on edge-based discretization methods. While these vector finite element methods (VFEM) solve the issues of tangential continuity of fields and the removal of spurious solutions, the resulting fields do not have a unique directionality at nodes in the discretization mesh. This review presents three calculations of electromagnetic fields: (i) waveguides, (ii) cavity fields, and (iii) photonic crystals. We have developed Hermite interpolation polynomials and node-based finite element methods in the framework of variational principles. We show that the Hermite-finite element method (HFEM) better accuracy with lower computational cost, and provides field directional continuity across the discretized space. It also permits multiscale calculations with mixed physics. For example, the nanoscale modeling of quantum well semiconductor laser structures has to be done together with cavity electrodynamics at the micron scale in vertical-cavity surface emitting lasers and other optoelectronic systems. These can be treated with high accuracy with the new HFEM, which is also applicable to quantum mechanical simulations in meso-and nano-scale systems. We use group representation theory to derive the HFEM polynomial basis set in two dimensions. In three dimensions we derive these polynomials using usual methods. We show that degeneracies in the frequency spectrum in a cubic cavity can be denumerably large even though the symmetry of the cube, O h , supports only singlets, doublets, or triplets. The additional operators available for the problem explains the origin of this "accidental degeneracy." We discuss this remarkable degeneracy and its reduction in detail. We consider photonic crystals corresponding to a 2D checkerboard superlattice structure, and the Escher drawing of "The Horsemen" which satisfies the nonsymmorphic group pg. We show that HFEM is able to deliver high accuracy in such spatially complex examples with far less computational effort than Fourier expansion methods. Finite element analysis employs geometric discretization and hence transcends geometrical limitations. Techniques explained here can be immediately extended to realistic and geometrically complex structures. The new algorithms developed here hold the promise of successful modeling of multi-physics systems. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.

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“…Dziekonski et. al [13] implement LOBPCG method with an inexact nullspace filtering approach to find eigenvalues in electromagnetics analysis.…”

Section: Background and Related Workmentioning

confidence: 99%

Evaluation of Directive-Based GPU Programming Models on a Block Eigensolver with Consideration of Large Sparse Matrices

Rabbi

Daley

Aktulga

et al. 2020

Accelerator Programming Using Directives

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Achieving high performance and performance portability for large-scale scientific applications is a major challenge on heterogeneous computing systems such as many-core CPUs and accelerators like GPUs. In this work, we implement a widely used block eigensolver, Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG), using two popular directive based programming models (OpenMP and OpenACC) for GPU-accelerated systems. Our work differs from existing work in that it adopts a holistic approach that optimizes the full solver performance rather than narrowing the problem into small kernels (e.g., SpMM, SpMV). Our LOPBCG GPU implementation achieves a 2.8x-4.3x speedup over an optimized CPU implementation when tested with four different input matrices. The evaluated configuration compared one Skylake CPU to one Skylake CPU and one NVIDIA V100 GPU. Our OpenMP and OpenACC LOBPCG GPU implementations gave nearly identical performance. We also consider how to create an efficient LOBPCG solver that can solve problems larger than GPU memory capacity. To this end, we create microbenchmarks representing the two dominant kernels (inner product and SpMM kernel) in LOBPCG and then evaluate performance when using two different programming approaches: tiling the kernels, and using Unified Memory with the original kernels. Our tiled SpMM implementation achieves a 2.9x and 48.2x speedup over the Unified Memory implementation on supercomputers with PCIe Gen3 and NVLink 2.0 CPU to GPU interconnects, respectively.

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GPU-Accelerated LOBPCG Method with Inexact Null-Space Filtering for Solving Generalized Eigenvalue Problems in Computational Electromagnetics Analysis with Higher-Order FEM

Cited by 14 publications

References 21 publications

Acceleration of tensor-product operations for high-order finite element methods

Acceleration of tensor-product operations for high-order finite element methods

Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Evaluation of Directive-Based GPU Programming Models on a Block Eigensolver with Consideration of Large Sparse Matrices

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