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Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces.However, this nonlocality makes it challenging to design efficient solvers for such problems.In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains.We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs.In addition, we introduce both serial and parallel domain decomposition solvers.We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints.
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This content will become publicly available on June 9, 2026
Statistical Treatment of Variable MPI Latencies and MPI-Communication Hiding for Matrix-Free FEM
We consider large-scale implicit solvers for the numerical solution of partial differential equations (PDEs). The solvers require the high-bandwith networks of an HPC system for a fast time to solution. The increasing variability in performance of the HPC systems, most likely caused by variable communication latencies and network congestion, however, makes the execution time of solver algorithms unpredictable and hard to measure. In particular, the performance variability of the underlying system makes the reliable comparison of different algorithms and implementations difficult or impossible on HPC. We propose the use of statistical methods relying on hidden Markov models (HMM) to separate variable performance data into regimes corresponding to different levels of system latency. This allows us to, for ex- ample, identify and remove time periods when extremely high system latencies throttle application performance and distort performance measurements. We apply HMM to the careful analysis of implicit conjugate gradient solvers for finite-element discretized PDE, in particular comparing several new communication hiding methods for matrix-free operators of a PDE, which are critical for achieving peak performance in state-of-the-art PDE solvers. The HMM analysis allows us to overcome the strong performance variability in the HPC system. Our performance results for a model PDE problem discretized with 135 million degrees of freedom parallelized over 7168 cores of the Anvil supercomputer demonstrate that the communication hiding techniques can achieve up to a 10% speedup for the matrix-free matrix-vector product.
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- Award ID(s):
- 2343865
- PAR ID:
- 10620818
- Publisher / Repository:
- ACM
- Date Published:
- Format(s):
- Medium: X
- Location:
- ACM International Conference on Supercomputing 2025, Salt Lake City, UT, USA
- Sponsoring Org:
- National Science Foundation
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