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Reducing Operator Complexity of Galerkin Coarse-grid Operators with Machine Learning

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/23m1583533· OSTI ID:2504058
 [1];  [1];  [2];  [3];  [1]
  1. Emory Univ., Atlanta, GA (United States)
  2. Auburn Univ., AL (United States)
  3. Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
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Here, we propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in multigrid (MG) methods, addressing the well-known issue of increasing operator complexity. Guided by the MG theory on spectrally equivalent coarse-grid operators, we have developed novel machine learning algorithms that utilize neural networks combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall MG convergence for solving parametric partial differential equation problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and comparison with existing methods for computing non-Galerkin coarse-grid operators.
Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC52-07NA27344
OSTI ID:
2504058
Report Number(s):
LLNL--JRNL-851090; 1077417
Journal Information:
SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 5 Vol. 46; ISSN 1064-8275
Publisher:
Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
Country of Publication:
United States
Language:
English

References (13)

Parallel performance of algebraic multigrid domain decomposition journal October 2020
Black-box learning of multigrid parameters journal April 2020
Compact high-order stencils with optimal accuracy for numerical solutions of 2-D time-independent elasticity equations journal March 2020
Collocation Coarse Approximation in Multigrid journal January 2009
Bootstrap AMG journal January 2011
Non-Galerkin Coarse Grids for Algebraic Multigrid journal January 2014
Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation journal January 2015
Reducing Parallel Communication in Algebraic Multigrid through Sparsification journal January 2016
A New Class of AMG Interpolation Methods Based on Matrix-Matrix Multiplications journal July 2021
Learning Optimal Multigrid Smoothers via Neural Networks journal August 2022
Algebraic Multigrid Based on Element Interpolation (AMGe) journal January 2001
Reducing communication in algebraic multigrid with multi-step node aware communication journal June 2020
PyAMG: Algebraic Multigrid Solvers in Python journal April 2022

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Related Subjects

97 MATHEMATICS AND COMPUTING
machine learning
multigrid methods
neural networks
operator complexity