On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

Southworth, Ben S.; Manteuffel, Thomas A.

doi:10.1137/23m1586069

On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

Journal Article · Mon Jul 01 00:00:00 EDT 2024 · SIAM Journal on Matrix Analysis and Applications

DOI:https://doi.org/10.1137/23m1586069· OSTI ID:2433953

^[1]; Manteuffel, Thomas A. ^[2]

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Univ. of Colorado, Boulder, CO (United States)

The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction Π will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning ∥Π∥ > 1. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to nonorthogonality, Π may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the nonorthogonal correction, or the algorithm may diverge. Here this paper develops analytic formulae on how to construct “compatible” transfer operators in nonsymmetric AMG such that ∥Π∥ = 1 in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between “ideal” transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2433953

Report Number(s):: LA-UR--23-26552

Journal Information:: SIAM Journal on Matrix Analysis and Applications, Journal Name: SIAM Journal on Matrix Analysis and Applications Journal Issue: 3 Vol. 45; ISSN 0895-4798

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

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