High Performance Parallel Solvers for the time-harmonic Maxwell Equations

CNRS); Elise Fressart (CMAP); Lo\"ic Gouarin (X; Marc Massot (CMAP); Michel Nowak; Nicole Spillane (CMAP); S\'ebastien Dubois (CMAP)

REVIEW 3 major objections 5 minor 32 references

High Performance Parallel Solvers for the time-harmonic Maxwell Equations

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For large time-harmonic Maxwell systems, Hiptmair-Xu and Block Low-Rank preconditioning emerge as the two viable strategies among those tested.

desk verdict An honest, useful benchmark whose HX mesh-independence result is credible, but whose BLR conclusion rests on a single matrix and no code or data are shipped. read the letter →

arxiv 2507.13066 v1 pith:CMOMKY7V submitted 2025-07-17 math.NA cs.NA

Elise Fressart (CMAP) , Sébastien Dubois (CMAP) , Loïc Gouarin (X , CNRS) , Marc Massot (CMAP) , Michel Nowak , Nicole Spillane (CMAP) This is my paper · ORCID

classification math.NAcs.NA MSC 65F0865F1065N2278M10

keywords time-harmonicMaxwellequationspreconditionediterativesolversHiptmair-XupreconditionerBlockLow-RankfactorizationNédélecfiniteelementsFGMRESmesh-sizeindependencelarge-scaleelectromagneticscattering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to show which preconditioning strategies can solve the large, sparse, indefinite, non-Hermitian linear systems that arise when time-harmonic Maxwell equations are discretized with lowest-order Nédélec finite elements. It compares five approaches, including no preconditioner, sparse approximate inverse, restricted additive Schwarz, Hiptmair-Xu, and Block Low-Rank factorization, on problems with up to 38 million degrees of freedom. Its working conclusion is that Hiptmair-Xu preconditioning keeps outer iteration counts independent of mesh refinement for a fixed wavenumber, while Block Low-Rank preconditioning gives fast convergence with compressed memory requirements. This matters because direct LU factorization becomes too slow and memory-hungry for realistic large-scale electromagnetic scattering, and the underlying operator is neither Hermitian nor definite, so standard iterative methods struggle without preconditioning.

What carries the argument

The machinery has three parts. First, the complex Maxwell system is split into real and imaginary parts, turning the original equation into a $2 \times 2$ block system whose diagonal blocks are $C - M$ (a curl-curl term minus a mass term) with boundary coupling $B$. Second, the Hiptmair-Xu preconditioner for the positive Maxwell operator $C + M + B$ is used to approximately invert the diagonal blocks of a block-diagonal preconditioner $$P = \mathrm{diag}(C + M + B,\; C + M + B),$$ with FGMRES as the outer solver and either preconditioned conjugate gradient or Hiptmair-Xu itself as the inner solver. Third, the Block Low-Rank preconditioner is a compressed LU factorization that replaces eligible off-diagonal dense blocks by low-rank approximations, controlled by a threshold $\varepsilon$, and then applies the compressed factors inside FGMRES.

What would settle it

Run the same FGMRES plus Hiptmair-Xu pipeline on the L-shaped-scatterer geometry at a fixed wavenumber and fixed points per wavelength while refining the mesh, and record the outer iteration count; significant growth with refinement would contradict the claimed mesh independence. Independently, push the wavenumber above 15 on the enlarged domain, where the paper's own scaling already shows iteration growth at least like $k^{0.86}$; a falsifier is observing that growth become superlinear or that FGMRES stalls before reaching the $10^{-8}$ relative residual target.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a practical ranking of solver strategies for time-harmonic Maxwell scattering: among the tested methods, the Hiptmair-Xu preconditioner, adapted by splitting the field into real and imaginary parts and preconditioning each block by the positive Maxwell operator, and the Block Low-Rank compressed factorization, used as a preconditioner, are the two candidate strategies that deserve further development. Hiptmair-Xu keeps the outer FGMRES iteration count essentially constant as the mesh is refined for a fixed wavenumber, but the count grows with wavenumber and with the physical size of the domain. Block Low-Rank, applied inside FGMRES, converges in very few outer iterations at moderate compression thresholds and reduces both factorization time and memory relative to full-rank LU. The paper presents these as temporary conclusions from a work in progress, not as a proven theory.

Load-bearing premise

The load-bearing premise is empirical: the positive Maxwell operator $C + M + B$ used in the block-diagonal preconditioner is close enough to the indefinite split system that a few inner iterations still make the outer FGMRES converge. The paper does not prove this and explicitly notes that the $\sqrt{2}$ condition-number bound is not recovered, so higher wavenumbers, larger domains, or different scatterer geometries could break the mesh-independence and speed claims.

Editorial extensions

If this is right

If the observed mesh-size independence holds more broadly, users can refine the mesh to control discretization error without increasing the outer iteration count for a fixed wavenumber, making large simulations more predictable.
Block Low-Rank preconditioning lets practitioners keep some of the robustness of direct solvers while using less memory, and in the paper it succeeds on the 38-million-degree-of-freedom case where full-rank LU fails.
The parallel scaling results suggest that iterative approaches benefit more from additional CPU cores than the direct solver, so they are the more natural route to very large electromagnetic problems.
The growth of Hiptmair-Xu iterations with wavenumber and physical domain size signals that neither method will automatically handle high-frequency regimes, and the Block Low-Rank compression threshold sets a tunable trade-off among accuracy, speed, and memory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's real/imaginary split doubles the number of unknowns, so its time and memory comparisons would shift even further in favor of the iterative solvers if the formulation were taken into account in the accounting.
A natural untested combination is to use the Block Low-Rank factorization to invert the diagonal blocks of the positive Maxwell preconditioner inside the Hiptmair-Xu approach, potentially reducing both outer iterations and memory at once.
The strong deterioration on the 50-times-larger domain suggests that Hiptmair-Xu in this form may be limited to moderate electrical sizes, and that sweeping or absorption-based domain decomposition strategies, which the paper lists as untested, would be needed to extend it.
A concrete prediction from the paper's scaling is that for fixed points per wavelength the Hiptmair-Xu outer iteration count stays flat under mesh refinement but grows roughly like $k^{0.86}$ with wavenumber; testing other scatterer geometries would show whether that exponent is universal or geometry-dependent.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a benchmark paper, not a methods paper, and the most useful result is the HX one. The authors take the block-diagonal preconditioner from Grayver/Burg and Chen et al., apply it to a scattering problem with real permittivity and impedance boundary condition, and show empirically that FGMRES iteration counts are essentially flat as the mesh is refined for k=1 and k=10 on the unit cube. They are upfront that they do not recover the sqrt(2) condition-number bound. That honesty carries through the paper: the SAI and RAS results show them struggling to converge, and they say so plainly. The HX tables are the strongest part because they vary mesh size, wavenumber, domain size, and core count. The scaling results and the failure of the direct solver on the largest problems are also useful.

The soft spot is exactly where the stress-test note points. The BLR recommendation rests on Table 7, one matrix at k=1, one geometry, one core count. Table 8 compares BLR to HX only on that same case. The text says "with BLR, there is a compromise to be found" as if it were a general observation, but it is an observation about one matrix. At higher wavenumber the off-diagonal blocks are less likely to compress well, so the trade-off may not survive. The other issue is reproducibility: no code or data are included. For a benchmark whose whole value is the numbers, that is a real limitation. I don't think the missing sqrt(2) bound is itself a flaw—the paper never claims it—but it means the HX mesh-independence is empirical, and the range of tested wavenumbers is modest.

The citation pattern is fine, and the list of untested methods is a plus. No circularity: the parameters are set before the runs and the conclusions are observations.

Who is this for? Practitioners choosing a solver for time-harmonic Maxwell with Nedelec elements, and people working on HX-type preconditioners. It deserves a serious referee. I would send it to review, but I would ask for code/data and at least a second BLR test case—different wavenumber or geometry—before accepting. As is, it is a solid work-in-progress report, not yet a reference result.

Referee Report

3 major / 5 minor

Summary. The paper reports an empirical comparison of four preconditioning strategies—ParaSails sparse approximate inverse, one-level overlapping restricted additive Schwarz (RAS), a Hiptmair-Xu (HX) based block-diagonal preconditioner, and MUMPS Block Low-Rank (BLR) factorization—together with full-rank LU, for time-harmonic Maxwell systems discretized by lowest-order Nédélec elements. The real/imaginary split formulation (11) is used for the hypre-based solvers, and experiments range up to about 38 million degrees of freedom on up to 400 CPU cores. The principal claims are that HX yields mesh-independent FGMRES iteration counts for the unit-cube/L-shaped-scatterer test case, and that BLR provides an attractive compression/accuracy trade-off; the paper explicitly labels these as temporary conclusions.

Significance. If the reported results hold, this is a useful practitioner-oriented comparison because it exercises readily available library preconditioners at scale and honestly reports a negative theoretical result (the sqrt(2) condition-number bound from [26] is not recovered for the indefinite split system). No fitted constants enter the reported quantities, and the HX leg is supported by several tables with stable iteration counts across mesh sizes. The BLR leg, however, rests on a single large test case, and no code or data are provided for independent verification. The paper's empirical claims are therefore credible but not yet at the level of a general recommendation for the BLR strategy.

major comments (3)

[Section 3.4, Table 7] The BLR recommendation is supported by a single matrix (2x2,474,860 dofs, k=1, 10 ppw, 32 cores), and Table 8 only reuses that same case. The general statement in Section 4 that "with BLR, there is a compromise to be found" is stronger than this evidence: at higher wavenumbers, different mesh sizes, or different scatterer geometries, the off-diagonal blocks may become less low-rank and the compression/accuracy trade-off could shift unfavorably. I recommend either adding a parameter sweep in k, mesh size, and domain size for BLR, or explicitly limiting the BLR conclusion to the demonstrated regime.
[Section 3.4, Table 7] The reported "compression" is the ratio of factor sizes and the reported time is the total solve time, so the memory-savings and speed claims for BLR cannot be separated into factorization/setup cost versus FGMRES iteration cost. Please report factorization time, memory usage, and iteration time as separate quantities, or otherwise substantiate the memory-savings claim.
[Section 3.3, Eq. (15)] The mesh-independence conclusion for HX is empirical, and the only theoretical motivation cited for the block-diagonal preconditioner in (15) is the sqrt(2) condition-number bound from [26], which the authors state is not recovered. Because the preconditioner may be a poor approximation for other wavenumbers, domain sizes, or geometries, I ask for a quantitative diagnostic on at least one additional configuration (for example, estimated condition numbers of the preconditioned system or a second geometry) or a more explicit limitation statement in Section 4, so that the reader can assess how far the mesh-independence observation is expected to generalize.

minor comments (5)

[Section 3.3, Table 3] The table headers "HX" and "HX as solver" are not defined in the captions; the text explains them, but a brief caption note would help. Similarly, the column "#CG" should state whether it refers to inner CG iterations for both block solves or their total.
[Section 3.3] The statement that "the number of FGMRES iterations is at least proportional to k^0.86" appears to be based on only a few wavenumber values; please specify the fitting procedure or present the data points supporting this exponent.
[Section 3.1] The claim that "with more dofs (not shown here), the number of GMRES iterations reaches the upper limit of 1000" is not supported by any table; either include the data or rephrase this as a qualitative observation.
[Section 3.4] There is a typo in the text "preconditoner" which should be "preconditioner."
[General] No code or data availability statement is provided. Given that all central conclusions are empirical, a reproducibility statement or a link to the experimental data and scripts would substantially strengthen the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: this is an empirical benchmark whose conclusions are direct measurements, not derived predictions.

full rationale

This paper is a numerical benchmark, not a derivation. The central claims—that Hiptmair-Xu yields mesh-independent iteration counts and that Block Low-Rank offers a speed/memory compromise—are direct readings of Tables 3–6 and 7–8, with no parameter fitted to the reported outputs. Solver tolerances and thresholds (relative residual 10^-8, inner CG tolerance 10^-2, BLR epsilon values) are chosen before the runs and are not tuned to the conclusions. The adaptation of the positive Maxwell preconditioner P in (15) is explicitly flagged in Section 3.3 as not recovering the sqrt(2) condition-number bound, so the authors do not present a theoretical guarantee as their evidence; the mesh-independence statement is empirical. Citations to [7] and [24,25,26] supply the preconditioner construction, not the numerical conclusions, and none are self-citations of the present authors. The BLR conclusion rests on one test matrix (Table 7), which is an evidence-strength concern, not circularity: the compression ratio 2.05 and 10 FGMRES iterations are measured quantities, not constructed outputs. Section 4's 'work in progress' hedge further limits the scope of the claims. No load-bearing step reduces to its own input by definition, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on a few standard mathematical results (unique solvability, HX theory for positive Maxwell) and two domain assumptions specific to this paper: the effectiveness of the block-diagonal split preconditioner and the accuracy of BLR as a preconditioner. No new physical or mathematical entities are introduced.

free parameters (3)

BLR compression threshold epsilon = 5e-3, 1e-3, 1e-5, 1e-9 (tested values)
User-defined in MUMPS; the central claim that BLR is effective depends on choosing an appropriate epsilon, as shown in Table 7.
HX inner solver max iterations and tolerance = 20 iterations, 1e-2 relative residual
Used for the inner CG solves of the block-diagonal preconditioner (15); affects FGMRES iteration counts and timings in Tables 3-6.
ParaSails parameters m, thresh, filter = m=3; thresh and filter varied (e.g., 0.001, 0.01, 0.05)
Control the sparsity pattern of the sparse approximate inverse; performance of SAI in Table 1 depends on them, though this preconditioner is not central to the final recommendation.

assumptions (4)

domain assumption The weak formulation (2) of the time-harmonic Maxwell scattering problem has a unique solution.
The paper relies on [1, Theorem 4.17] for existence and uniqueness; no proof is given.
standard math The Hiptmair-Xu preconditioner (14) is an effective preconditioner for the positive Maxwell operator C+M+B.
This is the theoretical result from [7]; the paper uses it as a building block without reproving it.
domain assumption The block-diagonal matrix P in (15) is a suitable preconditioner for the split indefinite system (11).
Adapted from [24-26]; the paper explicitly notes the sqrt(2) condition-number bound is not recovered, so effectiveness is assumed and only empirically tested.
domain assumption The BLR factorization with threshold epsilon retains enough accuracy to serve as a preconditioner for FGMRES.
Standard behavior of BLR methods; the paper relies on this property for the numerical results in Table 7.

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Cite this review

Pith. "Pith review of High Performance Parallel Solvers for the time-harmonic Maxwell Equations." pith.science (2026). https://pith.science/paper/CMOMKY7V

@misc{pith2026250713066,
  author       = {Pith},
  title        = {Pith review of: High Performance Parallel Solvers for the time-harmonic Maxwell Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMOMKY7V}},
  note         = {Machine review of arXiv:2507.13066}
}

read the original abstract

We consider the numerical solution of large scale time-harmonic Maxwell equations. To this day, this problem remains difficult, in particular because the equations are neither Hermitian nor semi-definite. Our approach is to compare different strategies for solving this set of equations with preconditioners that are available either in PETSc, MUMPS, or in hypre. Four different preconditioners are considered. The first is the sparse approximate inverse, which is often applied to electromagnetic problems. The second is Restricted Additive Schwarz, a domain decomposition preconditioner. The third is the Hiptmair-Xu preconditioner which is tailored to the positive Maxwell equations, a nearby problem. The final preconditioner is MUMPS's Block Low-Rank method, a compressed block procedure. We also compare the performance of this method to the standard LU factorization technique, which is a direct solver. Performance with respect to the mesh size, the number of CPU cores, the wavelength and the physical size of the domain are considered. This work in progress yields temporary conclusions in favour of the Hiptmair-Xu and the Block Low-Rank preconditioners.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [26]

An adaptive finite element method for the eddy current model with circuit/field couplings

Junqing Chen, Zhiming Chen, Tao Cui, and Lin-Bo Zhang. An adaptive finite element method for the eddy current model with circuit/field couplings. SIAM Journal on Scientific Computing , 32(2):1020–1042, 2010

work page 2010

[2] [1]

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Peter Monk. Finite element methods for Maxwell’s equations . Numer. Math. Sci. Comput. Oxford: Oxford University Press, 2003

work page 2003

[3] [2]

Mixed finite elements in R3

Jean-Claude Nedelec. Mixed finite elements in R3. Numer. Math. , 35:315–341, 1980

work page 1980

[4] [3]

Iterative methods for sparse linear systems

Yousef Saad. Iterative methods for sparse linear systems. Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics, 2nd ed. edition, 2003

work page 2003

[5] [4]

O. G. Ernst and M. J. Gander. Why it is difficult to solve Helmholtz problems with classical iterative methods. In Numerical analysis of multiscale problems. Selected papers based on the presentations at the 91st London Mathematical Society symposium, Durham, UK, July 5–15, 2010., pages 325–363. Berlin: Springer, 2012

work page 2010

[6] [5]

A priori sparsity patterns for parallel sparse approximate inverse preconditioners

Edmond Chow. A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. , 21(5):1804–1822, 2000

work page 2000

[7] [6]

A restricted additive Schwarz preconditioner for general sparse linear systems

Xiao-Chuan Cai and Marcus Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. , 21(2):792–797, 1999

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Nodal auxiliary space preconditioning in H(curl) and H(div) spaces

Ralf Hiptmair and Jinchao Xu. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. , 45(6):2483–2509, 2007

work page 2007