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arxiv: 2602.16312 · v3 · pith:VENF7IKTnew · submitted 2026-02-18 · 🧮 math.NA · cs.NA

An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology

Marco Feder , Pasquale Claudio Africa This is my paper

Pith reviewed 2026-05-15 21:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discontinuous Galerkinmultigridagglomerationmonodomain modelcardiac electrophysiologypreconditionerpolytopic grids
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The pith

An agglomeration-based multilevel preconditioner accelerates iterative solvers for the discontinuous Galerkin discretization of cardiac electrophysiology models.

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

This paper develops an agglomeration-based multilevel preconditioner to speed up iterative solvers for the linear systems that come from using discontinuous Galerkin methods on the monodomain model of heart electrophysiology. The key is to create coarser grids by combining fine mesh elements into general polytopic shapes, forming a nested hierarchy without needing special coarse meshes. This matters because cardiac simulations on realistic 3D heart geometries involve large systems that are expensive to solve directly, and better preconditioners can make high-fidelity modeling feasible. Experiments on 2D and 3D domains with various ionic models demonstrate that the method scales well as the polynomial degree rises and as more multigrid levels are used.

Core claim

The central discovery is that agglomerating elements from a fine initial mesh produces a hierarchy of polytopic grids that supports an effective multilevel preconditioner for the DG-discretized monodomain equations. The preconditioner leverages these general coarse elements to achieve robust convergence of iterative solvers. Tests confirm strong performance and favorable scaling with respect to polynomial degree and number of levels on unstructured geometries and different ionic models.

What carries the argument

The agglomeration strategy for constructing nested hierarchies of polytopic grids from a fine mesh, which supports the multilevel preconditioner while preserving necessary stability properties.

If this is right

  • The solver remains effective across increasing polynomial degrees in the discontinuous Galerkin discretization.
  • Favorable scalability holds with respect to the number of levels in the multigrid preconditioner.
  • The approach applies successfully to realistic 3D unstructured geometries and multiple ionic models.
  • General polytopic elements at coarser levels maintain the approximation properties required for the preconditioner.

Where Pith is reading between the lines

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

  • Such preconditioners could support larger-scale simulations of cardiac wave propagation by reducing the computational cost of each solve.
  • The agglomeration technique might generalize to other partial differential equations discretized with discontinuous Galerkin methods on complex domains.
  • Integrating this with adaptive refinement could further improve efficiency in modeling heterogeneous heart tissue.

Load-bearing premise

The agglomeration procedure must produce polytopic elements regular enough to preserve the stability and approximation properties of the discontinuous Galerkin method across levels.

What would settle it

If numerical experiments on a realistic 3D heart geometry show iteration counts growing rapidly or divergence when using high polynomial degrees or many multigrid levels, that would indicate the claim does not hold.

Figures

Figures reproduced from arXiv: 2602.16312 by Marco Feder, Pasquale Claudio Africa.

Figure 1
Figure 1. Figure 1: (a) MPI partitioning of the realistic hexahedral ventricle mesh into 128 subdomains. Each color corresponds to a different MPI process. Bottom row: detailed views of an agglomerate on the boundary of the domain (b) and its 8 sub-agglomerates (c) and (d), each displayed in a different color. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) The computational mesh partitioned using parmetis among five MPI processes. To build the multigrid hierarchy, aggregates are generated using the locally owned cells within each MPI rank. Snapshots of the transmembrane potential U at (b) t = 0.04s and (c) t = 0.16s after the external application of the current Iapp(x, t) in Equation (14). as shown in Fig. 2a. As previously mentioned in Section 4.1, we g… view at source ↗
Figure 3
Figure 3. Figure 3: Number of PCG iterations per time step for Problem ( [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Idealized left ventricle test case: (a) hexahedral mesh of the ellipsoid and (b) time evolution of the transmembrane potential at a selected point. Idealized ventricle (5 processes) Level index l Card(Tl) l = 0 407,904 l = 1 50,990 l = 2 6,375 l = 3 800 l = 4 100 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fiber field computed using the Laplace-Dirichlet Rule-Based Methods [ [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Hexahedral mesh Th representing a realistic left ventricle. (b) Propagation of the trans￾membrane potential U. range from 8 to 13, far below the counts of AMG, which reach up to 21 iterations, and of Block-Jacobi, which reach up to 68 iterations. 6.4. Performance results on realistic three-dimensional test We now examine the performance of all the considered preconditioners for the realistic three-dime… view at source ↗
Figure 7
Figure 7. Figure 7: Snapshots of the transmembrane potential [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Number of PCG iterations per time step for Problem ( [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: PCG iteration time per time step for Problem ( [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of average wall-clock time (measured in seconds) to perform one time step for [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Breakdown of the setup phase cost for the agglomeration-based multigrid preconditioner on [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Scalability analysis with respect to the number of MPI processes for the three-dimensional [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Strong scaling results for the realistic ventricle test case with [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
read the original abstract

This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in cardiac electrophysiology. The proposed approach exploits general polytopic grids at coarser levels, obtained through the agglomeration of elements from an initial, potentially fine, mesh. By leveraging a robust and efficient agglomeration strategy, we construct a nested hierarchy of grids suitable for multilevel solver frameworks. The effectiveness and performance of the methodology are assessed through a series of numerical experiments on two- and three-dimensional domains, involving different ionic models and realistic unstructured geometries. The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner.

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.

Referee Report

1 major / 2 minor

Summary. The paper proposes an agglomeration-based multilevel preconditioner to accelerate iterative solvers for linear systems arising from discontinuous Galerkin discretizations of the monodomain model in cardiac electrophysiology. It constructs nested hierarchies of polytopic grids via element agglomeration from an initial mesh and evaluates performance through numerical experiments on 2D and 3D domains with different ionic models and realistic unstructured geometries, claiming strong solver effectiveness together with favorable scalability in both polynomial degree and number of multigrid levels.

Significance. If the agglomeration procedure reliably preserves the shape-regularity constants required for uniform DG stability and approximation properties, the method would supply a practical, scalable solver for high-order simulations of cardiac electrical propagation on complex geometries, addressing an important computational bottleneck in electrophysiology modeling.

major comments (1)
  1. [Abstract] Abstract: the claim of favorable scalability with respect to polynomial degree and number of levels rests on the unverified assertion that the agglomeration strategy produces polytopic coarse elements whose shape-regularity constants (minimum angles, diameter ratios) remain controlled. No a priori bound, quality metric, or analysis is supplied showing that the coercivity/continuity constants of the DG bilinear form for the monodomain operator stay uniform; numerical results alone therefore leave the central scalability claim dependent on an unproven assumption about the coarse-grid geometry.
minor comments (2)
  1. [Numerical experiments] Add a table or figure panel that explicitly reports iteration counts, convergence factors, and effective condition-number estimates versus polynomial degree and level count to make the scalability statements quantitatively transparent.
  2. Clarify the precise definition of the agglomeration criterion (e.g., any angle or aspect-ratio threshold) and state whether it is applied uniformly across all tested geometries and ionic models.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment below and have revised the manuscript to clarify the nature of our scalability claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of favorable scalability with respect to polynomial degree and number of levels rests on the unverified assertion that the agglomeration strategy produces polytopic coarse elements whose shape-regularity constants (minimum angles, diameter ratios) remain controlled. No a priori bound, quality metric, or analysis is supplied showing that the coercivity/continuity constants of the DG bilinear form for the monodomain operator stay uniform; numerical results alone therefore leave the central scalability claim dependent on an unproven assumption about the coarse-grid geometry.

    Authors: We agree with the referee that the manuscript provides no a priori analysis or bounds establishing that the agglomeration procedure preserves shape-regularity constants uniformly, and therefore does not prove that the DG coercivity and continuity constants remain independent of the number of levels or polynomial degree. The scalability statements in the original abstract were based solely on the numerical experiments reported in Sections 4 and 5. To correct this, we have revised the abstract to state that the observed scalability is empirical: “The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner, as observed across the numerical experiments.” We have also added a brief remark in the introduction and conclusion noting that a theoretical analysis of the agglomeration’s effect on mesh-regularity constants lies outside the scope of the present work. These changes remove the implication of a proven uniform bound while preserving the paper’s focus on practical performance. revision: yes

Circularity Check

0 steps flagged

No circularity: solver construction and numerical validation remain independent of fitted self-references

full rationale

The paper introduces an agglomeration-based multilevel preconditioner for DG discretizations of the monodomain equation. The core construction relies on standard agglomeration operators to generate coarse polytopic grids and on established multigrid theory for the hierarchy; neither the bilinear-form stability constants nor the reported p- and level-scalability are obtained by fitting parameters to the target performance metrics themselves. Numerical experiments on 2-D/3-D cardiac geometries and ionic models supply external evidence that the agglomeration preserves sufficient shape-regularity for the observed convergence rates. No self-definitional loop, fitted-input-renamed-as-prediction, or load-bearing self-citation chain appears in the derivation or the claimed results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from numerical PDE theory and cardiac modeling; no new free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption The monodomain model is a sufficient approximation for the electrical activity in cardiac tissue under the tested conditions.
    Invoked implicitly as the target PDE for discretization and solver testing.
  • domain assumption Agglomerated polytopic elements maintain the necessary approximation and stability properties for the DG method and multilevel preconditioner.
    Required for the hierarchy to function effectively but not proven in the abstract.

pith-pipeline@v0.9.0 · 5424 in / 1260 out tokens · 28215 ms · 2026-05-15T21:34:52.019404+00:00 · methodology

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