Tuning Spectral Element Preconditioners for Parallel Scalability on GPUs

Phillips, Malachi; Kerkemeier, Stefan; Fischer, Paul

doi:10.1137/1.9781611977141.4

Cited by 7 publications

(8 citation statements)

References 23 publications

(74 reference statements)

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“…As a concluding remark, one can state that finding optimal sets of parameters for multigrid (smoothers) is a non-trivial task due to the large search space, the problem dependency as well as the hardware properties. In large production runs, an auto-tuning step—as proposed by Phillips et al (2022)—is unavoidable in order to reach the optimal usage of hardware and the minimum time to solution.…”

Section: Discussionmentioning

confidence: 99%

“…One issue with the definition (1) is that it relies on the assembly of the matrix A , which leads to prohibitive costs in the context of higher-order methods. As indicated above, low-rank approximations to trueA^i−1 are common, typically derived from an auxiliary operator defined on a nearby Cartesian mesh (Couzy, 1995; Phillips et al, 2022; Witte et al, 2021). This choice relies on the representation of the Laplacian Aicart on a Cartesian mesh as the tensor product of 1D stiffness and mass matrices.…”

Section: Preconditioners and Smoothersmentioning

confidence: 99%

“…Despite a wide range of previous contributions, the topics of data locality and possible performance optimizations of the different algorithms have not been studied before. On a related aspect, recent works in NekRS (Phillips et al, 2022; Phillips and Fischer, 2022) also compare these two smoothers but using a more basic implementation with several sweeps through data.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations

Munch,

Kronbichler

2023

The International Journal of High Performance Computing Applica

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This contribution presents data-locality optimizations of the additive Schwarz method (ASM) based on the fast-diagonalization method defined on overlapping cell-centric and vertex-star patches in the context of high-order matrix-free finite-element computations on modern CPU-based hardware. The developments are guided by detailed performance models of the ASM in the context of Chebyshev iterations when used as smoothers for p-multigrid. The proposed efficient implementation of ASM adopts concepts known from cell-loop infrastructures for efficient operator evaluation, in particular, the storage of information per geometric entity and the cache-friendly interleaving of cell loops and vector updates as a means to increase data locality. We use the latter concept for both applying the weights required by ASM and performing the vector updates required by the Chebyshev iteration, which are memory-bound operations with non-negligible costs in comparison to efficient operator evaluation. Furthermore, the solution of a scalar Poisson problem on a highly anisotropic and an unstructured mesh with p-multigrid using the developed smoothers indicates that efficient implementations of the additive Schwarz method can outperform optimized point-Jacobi preconditioners already for simple problems despite being more than twice as expensive per iteration. Even though ASM introduces additional communication steps per smoother application, the reduced number of iterations can lead to improved parallel scalability for intermediate problem sizes. At the scaling limit, the results are inconclusive due to these two opposing effects.

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Section: Discussionmentioning

confidence: 99%

Section: Preconditioners and Smoothersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations

Munch,

Kronbichler

2023

The International Journal of High Performance Computing Applica

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“…The pressure Poisson solve is treated with GMRES using p-multigrid as a preconditioner. Details of the formulation can be found in [28,27,29].…”

Section: Nek5000/rsmentioning

confidence: 99%

Simulating Atmospheric Boundary Layer Turbulence with Nek5000/RS

Min¹,

Tombouldies²

2022

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is a U.S. Department of Energy laboratory managed by UChicago Argonne, LLC under contract DE-AC02-06CH11357. The Laboratory's main facility is outside Chicago, at 9700 South Cass Avenue, Lemont, Illinois 60439. For information about Argonne and its pioneering science and technology programs, see www.anl.gov. DOCUMENT AVAILABILITYOnline Access: U.S. Department of Energy (DOE) reports produced after 1991 and a growing number of pre-1991 documents are available free at OSTI.GOV (http://www.osti.gov/), a service of the US Dept. of Energy's Office of Scientific and Technical Information.

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“…Combinations of these techniques with low-order-refined preconditioning are also possible (see Pazner, 2020). Detailed performance comparisons of matrix-free methods for high-order discretizations are available in Kronbichler and Wall (2018); Kronbichler and Ljungkvist (2019); Phillips et al (2022); and Kolev et al (2021a), among others.…”

Section: Introductionmentioning

confidence: 99%

End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations

Pazner

Kolev

Camier

2023

The International Journal of High Performance Computing Applica

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In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H(curl) and H(div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated.

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Tuning Spectral Element Preconditioners for Parallel Scalability on GPUs

Cited by 7 publications

References 23 publications

Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations

Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations

Simulating Atmospheric Boundary Layer Turbulence with Nek5000/RS

End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations

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