Improving Solve Time of aggregation-based adaptive AMG

Panayot S. Vassilevski; Pasqua D'Ambra

arxiv: 1907.04417 · v1 · pith:KIDEV6JLnew · submitted 2019-07-09 · 🧮 math.NA · cs.NA

Improving Solve Time of aggregation-based adaptive AMG

Pasqua D'Ambra , Panayot S. Vassilevski This is my paper

Pith reviewed 2026-05-24 23:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords AMGbootstrap AMGaggregationadaptive multigridalgebraically smooth vectorselliptic PDEsolve timememory

0 comments

The pith

A single hierarchy from composed aggregates of bootstrap smooth vectors preserves convergence in adaptive AMG while reducing memory and solve time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper modifies its bootstrap AMG by folding the algebraically smooth vectors it finds into a single hierarchy. This is done by building larger aggregates that are made up of smaller aggregates created during the bootstrap process. The result keeps convergence good but uses less memory and solves the systems faster than the original version. These savings are shown for hard linear systems coming from 2D and 3D elliptic PDEs with scalar and vector unknowns.

Core claim

The central claim is that by using sufficiently large aggregates as compositions of aggregates already built throughout the bootstrap algorithm, the information from the set of algebraically smooth vectors can be incorporated in a single hierarchy, leading to an aggregation-based adaptive AMG method that has good convergence properties and shows significant reduction in both memory and solve time on difficult linear systems from discretization of scalar and vector function elliptic PDEs in both 2D and 3D.

What carries the argument

The key mechanism is the formation of sufficiently large aggregates composed of earlier bootstrap aggregates to consolidate smooth vector information into one hierarchy.

If this is right

The modified method has good convergence properties.
Memory is reduced compared to the original bootstrap AMG.
Solve time is reduced compared to the original bootstrap AMG.
The savings are shown on 2D and 3D elliptic PDE systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This might allow adaptive AMG to handle larger problems with limited resources.
The composition idea could be tested in other multigrid settings.
Further work might optimize how the aggregates are composed to maximize savings.

Load-bearing premise

The load-bearing premise is that the composed aggregates preserve enough information from the algebraically smooth vectors to retain the convergence properties of the original bootstrap AMG.

What would settle it

A comparison of iteration counts on one of the 3D test problems between the modified and original methods; if the modified method requires many more iterations, the claim of preserved convergence would be falsified.

Figures

Figures reproduced from arXiv: 1907.04417 by Panayot S. Vassilevski, Pasqua D'Ambra.

**Figure 1.** Figure 1: ANI1: Setup time of the single-hierarchy multiple-vector AMG On the other hand, further reduction of solve time can be obtained at the expense of an increase in setup times and a moderate increase in memory requirements. In [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

**Figure 2.** Figure 2: Linear elasticity test case: mesh geometry (left); sparsity pattern of the system matrix (right) tetrahedral meshes, using the software package MFEM [21]; different problem sizes were obtained by uniform refinements. More specifically, for each test case, three different sizes are considered (15795,111843, 839619). We applied a node-based [26] discretization of the above PDE, where at each mesh point all s… view at source ↗

**Figure 3.** Figure 3: LE1: Setup time of the single-hierarchy multiple-vector AMG the original bootstrap AMG generates 11, 12 and 12 smooth vectors and setup times equal to 19.92, 222.05 and 2252.99, for the three problem sizes, respectively. The general behavior is similar to the previous test cases, confirming a very significant reduction of memory requirements and solve times of the new method with respect to the original bo… view at source ↗

read the original abstract

This paper proposes improving the solve time of a bootstrap AMG designed previously by the authors. This is achieved by incorporating the information, set of algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both, memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations (PDEs) in both 2d and 3d.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows numerical gains in memory and solve time by composing bootstrap aggregates into one hierarchy, but offers no analysis that the composition preserves the needed smooth vectors.

read the letter

The main takeaway is that the authors compose aggregates already generated during their bootstrap AMG stages into larger ones to build a single hierarchy. This change reduces memory and solve time compared with running the original bootstrap method, at least on the elliptic PDE test cases they report in 2D and 3D. The convergence stays acceptable in those examples. That is the concrete contribution: a practical tweak to hierarchy construction that reuses the bootstrap information without keeping multiple separate hierarchies. The paper does a straightforward job laying out the algorithmic step and showing results on problems that are known to be hard for classical AMG. The tests cover both scalar and vector-function cases, which is reasonable for the subfield. Credit is due for focusing on measurable performance rather than just another variant. The soft spot is exactly where the stress-test note flags it. There is no argument or bound showing that the composed aggregates still span or approximate the algebraically smooth directions identified across all bootstrap iterations. The claim that convergence properties are retained therefore rests entirely on the numerical examples working out. If the composition step loses or dilutes some of those directions on other problems, the reported savings could come with hidden convergence costs that the chosen test set does not reveal. The abstract gave no numbers, so the full paper's tables need checking for effect size, number of runs, and implementation details before the gains can be taken as settled. This work is aimed at people already doing research on adaptive or bootstrap AMG for large sparse systems from PDEs. A reader who builds or tunes AMG solvers would get value from seeing the composition idea and the reported timings. It is not broad enough or novel enough in framework to interest a wider audience. I would bring it to a reading group as a maybe, mainly to talk through whether the composition operator can be analyzed more carefully. I would not cite it in my own papers in the next year unless I start experimenting with composed aggregates myself. The paper deserves peer review because the change is specific, the tests are on relevant problems, and the performance claims are falsifiable even without deeper theory.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a modification to the authors' prior bootstrap AMG algorithm: instead of maintaining multiple hierarchies, the algebraically smooth vectors identified across bootstrap stages are used to form a single hierarchy whose aggregates are compositions of aggregates already constructed during bootstrap. The central claim is that this yields an AMG method with good convergence properties together with substantial reductions in memory and solve time relative to the original bootstrap AMG, demonstrated on 2-D and 3-D scalar and vector elliptic PDE discretizations that are known to be difficult for classical AMG.

Significance. If the numerical evidence is representative and the composition step reliably preserves the near-kernel information captured by bootstrap, the approach would provide a practical route to lower memory and faster solves for adaptive AMG on problems where standard methods require many levels or fail to converge. The work directly addresses a known performance bottleneck in bootstrap AMG and supplies concrete timing and memory comparisons on representative test problems.

major comments (2)

[Method description (single-hierarchy construction)] The central algorithmic claim—that composing aggregates already formed during bootstrap stages produces coarse spaces that still adequately span the algebraically smooth error components identified across all bootstrap iterations—receives no supporting analysis. The manuscript relies entirely on the numerical examples; no argument is given showing that the composition operator preserves the necessary kernel or near-kernel information (see the description of the single-hierarchy construction and the statement that “these aggregates are compositions of aggregates already built throughout the bootstrap algorithm”).
[Abstract and numerical-results section] No quantitative convergence theory, error bounds, or even a statement of the conditions under which the composed aggregates are guaranteed to retain the convergence rate of the original bootstrap AMG is supplied. The abstract asserts “good convergence properties,” yet the only evidence is the reported iteration counts and timings on the chosen test set.

minor comments (2)

[Abstract] The abstract states that savings are “illustrated on some difficult … linear systems” but does not specify the precise discretization orders, mesh sizes, or coefficient ranges used; these details should appear in the first paragraph of the numerical section for reproducibility.
[Method description] Notation for the composed aggregates and the mapping from bootstrap-stage aggregates to the final hierarchy is introduced without a compact definition or diagram; a small schematic would clarify the construction.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We respond to each major comment below.

read point-by-point responses

Referee: The central algorithmic claim—that composing aggregates already formed during bootstrap stages produces coarse spaces that still adequately span the algebraically smooth error components identified across all bootstrap iterations—receives no supporting analysis. The manuscript relies entirely on the numerical examples; no argument is given showing that the composition operator preserves the necessary kernel or near-kernel information (see the description of the single-hierarchy construction and the statement that “these aggregates are compositions of aggregates already built throughout the bootstrap algorithm”).

Authors: The referee correctly observes that the manuscript contains no formal analysis or proof that the composition of aggregates preserves the span of the algebraically smooth components. The single-hierarchy method is constructed so that the aggregates at each bootstrap stage are formed using the identified smooth vectors, and the final aggregates are explicit compositions of those earlier aggregates. This is intended to retain the near-kernel information by construction, but we provide no mathematical argument establishing that the composition operator always achieves this. The paper's focus is the resulting practical reductions in memory and solve time, supported by the reported experiments. We will add a brief clarifying paragraph on the construction rationale in the revised manuscript. revision: partial
Referee: No quantitative convergence theory, error bounds, or even a statement of the conditions under which the composed aggregates are guaranteed to retain the convergence rate of the original bootstrap AMG is supplied. The abstract asserts “good convergence properties,” yet the only evidence is the reported iteration counts and timings on the chosen test set.

Authors: We agree that the manuscript supplies no quantitative convergence theory, error bounds, or conditions guaranteeing retention of the original convergence rate. The abstract phrase “good convergence properties” refers to the iteration counts observed in the numerical tests on the selected 2-D and 3-D elliptic problems. These tests demonstrate that the single-hierarchy variant achieves convergence behavior comparable to the original bootstrap AMG. Because the contribution centers on algorithmic simplification and measured performance gains rather than theory, no such analysis is included. No revision is planned on this aspect. revision: no

standing simulated objections not resolved

A theoretical analysis or proof that the aggregate composition preserves the convergence rate of the original bootstrap AMG.

Circularity Check

0 steps flagged

No circularity; algorithmic modification validated empirically

full rationale

The paper describes an algorithmic modification to the authors' prior bootstrap AMG method, forming a single hierarchy via composition of previously built aggregates to incorporate algebraically smooth vectors. The central claim of retained convergence properties with reduced memory and solve time is supported solely by numerical experiments on 2D/3D elliptic PDE systems; no derivation chain, equation, or self-citation reduces the result to a tautology, fitted input, or self-defined quantity. The preservation assumption is stated explicitly as an empirical hypothesis rather than proven by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5632 in / 1014 out tokens · 21961 ms · 2026-05-24T23:53:06.967438+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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of the 25th IEEE International Symposium on Parallel and Distributed Processing, IPDPS 2011, pp

Baker, A., Gamblin, T., Schulz, M., Meier Yang, U., Challenges of Scaling Algebraic Multigrid across Modern Multicore Architectures, in Proc. of the 25th IEEE International Symposium on Parallel and Distributed Processing, IPDPS 2011, pp. 275-286, IEEE, 2011

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8, 112–135, 2015

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Brezina, M., Falgout, R., MacLachlan, S., Manteuﬀel, T., McCormick, S., Ruge, J., Adaptive Algebraic Multigrid, SIAM J. on Sci. Comput., Vol. 27, 1261–1286, 2006

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Linear Algebra Appl., Vol

Baker, A.H., Kolev, Tz.V., Yang, U.M., Improving algebraic multigrid interpolation operators for linear elasticity, Numer. Linear Algebra Appl., Vol. 17, 495–517, 2010

work page 2010

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E., Jones, J.E., Manteuﬀel, T.A., McCormick, S.F., Ruge, J.W., Vassilevski, P.S., Spectral AMGe ( ρAMGe), SIAM J

Chartier, T., Falgout, R., Henson, V. E., Jones, J.E., Manteuﬀel, T.A., McCormick, S.F., Ruge, J.W., Vassilevski, P.S., Spectral AMGe ( ρAMGe), SIAM J. on Sci. Comput., Vol. 25, 1–26, 2003

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Chow, E., An aggregation multilevel method using smooth error vectors, SIAM J. on Sci. Comput., Vol. 127, 1727–1741, 2006

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D’Ambra, P., Vassilevski, P.S., Adaptive AMG with coarsening based on compatible weighted matching, Comput. Vis. Sci., Vol. 16, 59–76, 2013

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D’Ambra, P., Vassilevski, P.S., AMG based on meighted matching for mystems of mlliptic PDEs arising from misplacement and mixed methods, in Progress in Industrial Mathematics at ECMI 2014, Russo, G., Capasso, V., Nicosia, G., Romano, V., Springer, Mathematics in Industry, 2016

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D’Ambra, P., Filippone, S., A parallel generalized relaxation method for high-performance image segmentation on GPUs, J. of Comp. and Appl. Math., Vol. 293, pp. 35-44, 2016

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D’Ambra, P., Filippone, S., Vassilevski, P.S., BootCMatch: a software package for bootstrap AMG based on graph weighted matching, ACM Trans. Math. Softw., Vol. 44, 39:1–39:25, 2018

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on Numer

Falgout, R., Vassilevski, P.S., On generalizing the AMG framework, SIAM J. on Numer. Anal., Vol. 42, 1669–1693, 2004

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Hagemann, M., Schenk, O., Weighted matchings for preconditioning symmetric indeﬁnite linear systems, SIAM J. on Sci. Comput., Vol. 28, 403–420, 2006

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M., BoomerAMG: A parallel algebraic multigrid solver and precondi- tioner, Appl

Henson, V.E., Yang, U. M., BoomerAMG: A parallel algebraic multigrid solver and precondi- tioner, Appl. Numer. Math., Vol. 41, 155–177, 2002

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Linear Algebra Appl., Vol

Hogg, J., Scott, J., On the use of suboptimal matchings for scaling and ordering sparse sym- metric matrices, Numer. Linear Algebra Appl., Vol. 22, 648–663, 2015

work page 2015

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S., AMGe based on element agglomeration, SIAM J

Jones, J.E., Vassilevski, P. S., AMGe based on element agglomeration, SIAM J. on Sci. Comput., Vol. 23, 109–133, 2001

work page 2001

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Linear Algebra Appl., Vol

Kolev, T.V., Vassilevski, P.S., AMG by element agglomeration and constrained energy mini- mization interpolation, Numer. Linear Algebra Appl., Vol. 13, 771–788, 2006

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et al., MFEM: Modular ﬁnite element methods, mfem.org 20 PASQUA D’AMBRA AND PANAYOT S

Kolev, T.V. et al., MFEM: Modular ﬁnite element methods, mfem.org 20 PASQUA D’AMBRA AND PANAYOT S. V ASSILEVSKI

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Kraus, J., Algebraic multigrid based on computational molecules, 2: Linear elasticity problems. SIAM J. Sci. Comput. Vol. 30, 505–524, 2008

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Li, X.S., An overview of SuperLU: algorithms, implementation, and user interface, ACM Trans. Math. Soft., Vol. 31, 302–325, 2005

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Linear Algebra Appl., Vol

Livne, O.E., Coarsening by compatible relaxation, Numer. Linear Algebra Appl., Vol. 11, 205– 227, 2004

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37, 123–146, 2010

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Ruge, J.W., AMG for problems of elasticity, Applied Math. and Comp., Vol. 19, 293–309, 1986

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W., St¨ uben, K., Algebraic Multigrid (AMG), in Multigrid Methods, McCormick, S

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Vassilevski, P.S., Multilevel Block-Factorization Preconditioners, Matrix-based analysis and Algorithms for Solving Finite Element Equations, Springer, 2008

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Mauro Picone

Xu, J., Zikatanov, L., Algebraic Multigrid Methods, Acta Numerica, Vol. 26, 591-721, 2017. Institute for Applied Computing “Mauro Picone”, National Research Council of Italy, Naples, Italy E-mail address : pasqua.dambra@cnr.it Center for Applied Scientific Computing, Lawrence Livermore National Labora- tory, CA, USA F ariborz Maseeh Department of Mathemat...

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