Optimal polynomial smoothers for parallel AMG

In this paper, we explore polynomial accelerators that are well-suited for parallel computations, specifically as smoothers in Algebraic MultiGrid (AMG) preconditioners for symmetric positive definite matrices. These accelerators address a minimax problem, initially formulated in Lottes (Numer. Lin....

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Bibliographic Details
Published in:Numerical algorithms Vol. 100; no. 4; pp. 1783 - 1812
Main Authors: D’Ambra, Pasqua, Durastante, Fabio, Filippone, Salvatore, Massei, Stefano, Thomas, Stephen
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2025
Springer Nature B.V
Subjects:
Accelerators
Algebra
Algorithms
Approximation
Basic converters
Chebyshev approximation
Computer Science
Effectiveness
Graphics processing units
Iterative methods
Matrices (mathematics)
Minimax technique
Numeric Computing
Numerical Analysis
Optimization
Partial differential equations
Polynomials
Propagation
Theory of Computation
65F10
AMG
Parallel smoothers
GPU
65F08
65N55
65Y05
Chebyshev polynomials
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:In this paper, we explore polynomial accelerators that are well-suited for parallel computations, specifically as smoothers in Algebraic MultiGrid (AMG) preconditioners for symmetric positive definite matrices. These accelerators address a minimax problem, initially formulated in Lottes (Numer. Lin. Alg. with Appl. 30 (6), e2518 2023 ), aiming to achieve an optimal (or near-optimal) bound for a polynomial-dependent constant involved in the AMG V-cycle error bound, without requiring information about the matrices’ spectra. Lottes focuses on Chebyshev polynomials of the 4 th -kind and defines the relevant recurrence formulas applicable to a general convergent basic smoother. In this paper, we demonstrate the efficacy of these accelerations for large-scale applications on modern GPU-accelerated supercomputers. Furthermore, we formulate a variant of the aforementioned minimax problem, which naturally leads to solutions relying on Chebyshev polynomials of the 1 st -kind as accelerators for a basic smoother. For all the polynomial accelerations, we describe efficient GPU kernels for their application and demonstrate their comparable effectiveness on standard benchmarks at very large scales.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-025-02117-6