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REVIEW 4 major objections 7 minor 31 references

Tensor-structured PCG for finite difference solver of domain patterns in ferroelectric material

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A pseudoinverse preconditioner built from the Kronecker structure of the finite-difference Laplacian lets PCG solve cuboid-grid Poisson problems in about two or three iterations.

desk verdict A sound, honest case study whose central speed claim is plausible but not demonstrated by the reported experiments. read the letter →

arxiv 2501.03377 v1 pith:H6QVTGSR submitted 2025-01-06 math.NA cs.NA

Věnceslav Chumchal , Pavel Marton , Martin Plešinger , Martina Šimůnková This is my paper · ORCID
classification math.NAcs.NA MSC 15A0615A3015A6965F0865F1065N06
keywords conjugategradientspreconditionertensorKroneckerproductLaplaceoperatorPoissonequationferroelectricdomainstructureMoore-Penrosepseudoinverse
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

This paper seeks to establish that preconditioned conjugate gradients (PCG) can solve the finite-difference Poisson equation on a rectangle or cuboid in essentially two or three iterations, provided the preconditioner is the Moore-Penrose pseudoinverse of the discrete Laplacian. Because the Laplacian separates into a sum of Kronecker products of small one-dimensional difference matrices, its full eigendecomposition is available analytically, and the pseudoinverse acts entrywise in that eigenbasis. The paper argues that this makes the method effectively a direct solver whose residual is polished by the outer CG iteration, at a per-iteration cost comparable to one multiplication by the Laplacian. It supports the claim with 2D and 3D experiments up to roughly a million unknowns, including singular periodic and Neumann cases where the pseudoinverse with thresholding handles the null space.

What carries the argument

The load-bearing object is the family of 1D finite-difference Laplacians $L_n(\alpha,\beta,\gamma)$, whose four boundary-condition variants (periodic, Dirichlet, Neumann, mixed) have closed-form sine-and-cosine eigen-decompositions (Lemmas 3.1-3.4). The 2D/3D discrete Laplacian is the Kronecker sum of these 1D matrices, so its eigenvalues are sums of 1D eigenvalues and its eigenvectors are Kronecker products of 1D eigenvectors. The Moore-Penrose pseudoinverse preconditioner $M_{\mathrm{p-inv}}$ exploits this by applying $V_n^T R V_q$, multiplying entrywise by the Hadamard pseudoinverse of the matrix of eigenvalue sums $(\lambda_i+\lambda_j)^{\dagger}$, and transforming back with $V_n F V_q^T$ (Algorithms 4-7), which the paper identifies as a Fourier-solver step for the residual.

What would settle it

Take a dense, null-space-orthogonal right-hand side on a cuboid grid with periodic boundary conditions in all directions and run PCG with the pseudoinverse preconditioner; if the true residual has not dropped to near machine precision relative to the initial residual within three iterations (the paper's stated 'two or three are sufficient'), the central convergence claim is refuted. A concrete target: on the $512\times 256\times 8$ problem described in Section 5, count iterations to reach $\|H-\mathcal{L}U_s\|_F/\|H\|_F \le 10^{-10}$ and check whether the count is at most three.

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Extended reading notes

Core claim

The central discovery is that the Moore-Penrose pseudoinverse of the Kronecker-structured finite-difference Laplacian is an almost perfect preconditioner for PCG on this problem. Writing the 2D operator as $(I_q\otimes L_n)+(L_q\otimes I_n)$, the eigenvectors are Kronecker products of the 1D Laplacian eigenvectors and the eigenvalues are sums $\lambda_i(L_n)+\lambda_j(L_q)$; the pseudoinverse preconditioner transforms the residual into this eigenbasis, divides by the nonzero eigenvalue sums (treating sums below $10^{-13}$ as zero), and transforms back. In the authors' experiments the true residual is essentially at the desired level after the first iteration and fully converged after two or three, so the paper proposes viewing the scheme as a pseudoinverse-based direct solver with refinement by CG iteration. The same construction extends to 3D with a three-way tensor and triples of eigenvalue sums.

Load-bearing premise

The claim rests on being able to store and multiply dense $n\times q$ (or $n\times q\times t$) arrays, because the right-hand side and all intermediate objects in the preconditioner are dense; if only sparse matrix storage fits in memory, this preconditioner's advantage over sparse iterative solvers disappears.

Editorial extensions

If this is right

  • For the simple Laplace operator on a cuboid, PCG with the pseudoinverse preconditioner needs only two or three iterations to converge; the authors run ten to be safe.
  • The method behaves as a direct solver: the pseudoinverse application produces an almost-exact solution and the CG outer loop removes the residual left by the eigenvalue-sum threshold.
  • The per-iteration cost is $O(nq(n+q))$ in 2D and $O(nqt(n+q+t))$ in 3D, comparable to the unpreconditioned PCG iteration itself, with initialization cost dominated by the eigendecompositions of the 1D Laplacians.
  • For singular Laplacians (periodic or Neumann in all directions), centering the data and thresholding the pseudoinverse keeps the iteration in the null-space-orthogonal subspace, so the solution is recovered up to the expected additive constant.
  • The approach has been implemented in C/C++ and used to produce published results on ferroelectric domain walls cited in the paper.

Reading between the lines

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

  • Because the pseudoinverse preconditioner is a trigonometric (Fourier-type) solver, on strongly anisotropic grids where one dimension is much smaller than the others the per-iteration cost $O(nqt(n+q+t))$ can exceed that of an FFT-based Poisson solver with $O(N\log N)$ work; the paper's method is best suited to near-cubic grids where the dense-array storage is already affordable.
  • The same pseudoinverse-of-the-Kronecker-sum construction should transfer to other separable operators on cuboids, for example anisotropic diffusion with different coefficients per axis or Schrodinger operators with separable potentials, whenever the 1D factors have known eigendecompositions; the paper's experiments only cover the scalar homogeneous Laplacian.
  • The $10^{-13}$ threshold means the method is not a true direct solver for systems with very small nonzero eigenvalues; the number of CG iterations would grow if the right-hand side had significant energy in near-null directions, so one could probe robustness by constructing such right-hand sides explicitly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The paper studies the preconditioned conjugate gradient (PCG) solution of finite-difference discretizations of Poisson's equation on rectangular or cuboid domains, where the discrete Laplacian is a sum of Kronecker products. The authors derive three preconditioners adapted to this structure: a Jacobi-like fixed-point preconditioner, a low-Kronecker-rank approximate inverse, and a Moore-Penrose pseudoinverse of the discretized Laplacian. The pseudoinverse preconditioner is built from analytic spectral decompositions of the one-dimensional Laplacians and is applied through basis transformations and Hadamard multiplication. Numerical experiments on six 2D and three 3D right-hand sides show rapid residual decrease, and the paper concludes that two or three PCG iterations are sufficient for this problem.

Significance. The mathematical derivation of the pseudoinverse preconditioner from the eigendecompositions of the 1D Laplacians is correct and standard, and the paper honestly identifies the method as a pseudoinverse-based direct solver with CG refinement. If the performance claims are substantiated, the method is a practical and simple Fourier-like preconditioner for Kronecker-structured Poisson problems with dense right-hand sides, and the connection to the Ferrodo2 application in ferroelectric simulation is valuable. The paper also provides explicit algorithms, analytical eigenvector formulas, and cost tables, which are useful for reproduction. However, the central efficiency claim is currently supported mainly by wall-clock plots with post hoc iteration counts and no comparison against standard baselines, so the significance of the numerical evidence is not yet at the level of the paper's conclusions.

major comments (4)
  1. [§5.2, Remark 4.2, Fig. 9] The central claim in Section 6 that two or three PCG iterations are sufficient is not supported by the reported data. Remark 4.2 states that the iteration number is fixed experimentally so that the true residual decreases sufficiently, but no target residual or achieved residual is reported. Section 5.2 selects iteration counts post hoc, and Figure 9 shows only wall-clock time after initialization, with no iteration counts for any of the nine test cases. Since the pseudoinverse preconditioner is the exact inverse on the range of the operator, one iteration is expected in exact arithmetic; the practical question is how many iterations are needed in floating-point arithmetic at a specified relative tolerance. The paper does not answer this question, and the claim is therefore not reproducible from the presented data. Please provide a table of iteration counts and true residual norms relative to the normalized initial residual for each test problem.
  2. [§5.2, Figs. 4–9] The performance comparison lacks essential baselines. Figure 6 compares preconditioners only on a single 50×100 problem, and the larger 2D and 3D runs in Figure 9 report only pseudoinverse-preconditioned PCG times. No comparison is made with unpreconditioned sparse CG, a sparse direct solver, or an FFT-based direct solver, even though Remark 4.3 identifies the pseudoinverse preconditioner as essentially a Fourier solver. Without such baselines, the conclusion that the method is extremely efficient for practical problem sizes is not established. Please report timings, iteration counts, and achieved residuals against at least one standard sparse solver and one FFT/direct solver for the nine test problems.
  3. [§5.2, Eq. (5.1)] The residual metric used in the experiments is not adequate for the application claim. Equation (5.1) normalizes the initial residual norm to 1/(nq) or 1/(nqt), so plots of absolute residual norms are not comparable across grids, and no relative tolerance is specified. In addition, the physically relevant output for the ferroelectric application is the electric field, i.e., the gradient of the potential, but the paper reports only residual norms and does not report error in the gradient. Please report relative residual norms and, where possible, the error in the computed electric field relative to a reference solution.
  4. [§6] The applicability of the method depends critically on the ability to store dense right-hand sides and dense intermediate matrices of size O(nq) or O(nqt). Section 6 acknowledges this, but the limitation should be stated more prominently, and the numerical experiments should include a memory comparison with sparse iterative solvers for the reported grid sizes. In regimes where dense storage is infeasible, the method loses its claimed advantage; the paper should make the scope of the efficiency claim explicit and quantify at which grid sizes the dense approach becomes impractical.
minor comments (7)
  1. [Problem 5.2] The phrase 'Newton condition e_R = -1/2' should be 'Neumann condition'.
  2. [Lemma 3.4] The two formulas for the eigenvectors v_k are not explicitly assigned to L_{n,DN} and L_{n,ND}; please label them clearly.
  3. [§4.2] The indices ζ and ξ are used for entries of W without being defined; please replace them with i and j or define them in the notation section.
  4. [§4.1 and Algorithm 1] There are small language issues: 'spaned' should be 'spanned', and 'Until stopping criterion match' should be 'Until stopping criterion matches'.
  5. [§4.3.2] The paper states that no algorithmic description of the low-Kronecker-rank preconditioner is provided, yet this preconditioner is used in the experiments reported in Figures 6–8. Please either provide the algorithm or describe more precisely how the experiments were implemented, so that the comparison is reproducible.
  6. [Abstract and Section 1] Please proofread for language issues such as 'preconditioners that fits', 'homogenous', and 'isotropous'.
  7. [Fig. 9 caption] The caption claims the convergence curve is 'almost problem independent', but the plot uses wall-clock time after initialization on a log scale and shows no iteration counts; please rephrase or provide the actual iteration counts and residual values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pseudoinverse preconditioner is derived from the operator's own spectral decomposition, and the paper explicitly identifies the result as a direct solver with CG refinement.

full rationale

The derivation chain is self-contained. The pseudoinverse preconditioner is constructed from the analytical eigendecompositions of the one-dimensional finite-difference Laplacians (Lemmas 3.1-3.4), and equations (3.5), (4.5)-(4.7), and Algorithms 4-7 then build L-dagger from those spectra; no output quantity is fed back into the construction. The rapid convergence of PCG with M = L-dagger is a mathematical consequence, not an empirical prediction: on the range of L the preconditioned operator is the identity, so exact arithmetic terminates in essentially one step, and the paper acknowledges this by calling the method 'a pseudoinverse-based direct solver with iterative refinement by CG iteration' (Section 6) and noting it is 'essentially a Fourier solver' (Remark 4.3). The claimed two-to-three iterations are therefore explained by construction, not discovered from fitted data. Self-citations ([15], [16]) are used only as external application context and are not load-bearing for the numerical analysis. The experimental protocol fixes iteration counts to make residuals 'sufficiently' small (Remark 4.2, Section 5.2) and normalizes initial residuals to 1/N, which weakens reproducibility of the few-iteration claim, but this is a reporting limitation, not a circular reduction of the derivation to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard linear algebra (spectral decomposition of symmetric matrices, pseudoinverse) and on the domain assumption that the Poisson operator on a homogeneous isotropic cuboid discretizes to a sum of Kronecker products with known 1D spectra. No new physical entities are introduced. The main hand-chosen parameters are the eigenvalue cutoff and the experimental iteration counts.

free parameters (4)
  • Pseudoinverse eigenvalue threshold = 1e-13
    Eigenvalue sums with modulus below this threshold are treated as zero when forming the pseudoinverse (Algorithms 4 and 5). This hand-chosen cutoff affects the accuracy of the preconditioner.
  • Jacobi preconditioner parameters (p, omega) = p=3, omega=1.3
    Chosen for Experiment 5.5 without a systematic search; used only for the Jacobi-like preconditioner comparison, not for the main pseudoinverse claim.
  • Low-rank approximation rank rK = rK=3
    Chosen for Experiment 5.5; the paper notes higher ranks approach the pseudoinverse but with higher initialization cost.
  • Number of PCG iterations in final experiments = 10
    The paper performs ten iterations 'to be sure' even though convergence is observed within 2-3; the iteration count is not determined by a stopping criterion but chosen experimentally.
assumptions (5)
  • standard math Spectral properties of 1D finite-difference Laplacians Ln(alpha,beta,gamma) stated in Lemmas 3.1-3.4, including eigenvectors for periodic, Dirichlet, Neumann, and mixed BCs.
    These are cited to [10] and described as commonly known; the pseudoinverse preconditioner construction depends on them.
  • standard math Moore-Penrose pseudoinverse properties and the Eckart-Young-Mirsky theorem for low-rank approximation.
    Used in Section 4.3.2 to derive the low-rank preconditioner and in Section 4.3.3 for the pseudoinverse.
  • domain assumption The material is homogeneous and isotropic with scalar permittivity, and the domain is a rectangle or cuboid with a regular finite-difference grid.
    This makes the discretized Laplacian a sum of Kronecker products (Section 3.3, Eqs. 3.2-3.3).
  • domain assumption The right-hand side H is dense with general rank(s), and its size allows storing several dense matrices of O(nq) or O(nqt) entries.
    Stated in Section 6 as the condition under which the pseudoinverse preconditioner is applicable.
  • ad hoc to paper Eigenvalue sums with modulus below 1e-13 are treated as zero when forming the pseudoinverse.
    Introduced in Sections 4.3.2 and 4.3.3 to handle near-zero eigenvalues; affects the accuracy and iteration count.

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Pith. "Pith review of Tensor-structured PCG for finite difference solver of domain patterns in ferroelectric material." pith.science (2026). https://pith.science/paper/H6QVTGSR

@misc{pith2026250103377,
  author       = {Pith},
  title        = {Pith review of: Tensor-structured PCG for finite difference solver of domain patterns in ferroelectric material},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6QVTGSR}},
  note         = {Machine review of arXiv:2501.03377}
}
read the original abstract

This paper presents a case study of application of the preconditioned method of conjugate gradients (CG) on a problem with operator resembling the structure of sum of Kronecker products. In particular, we are solving the Poisson's equation on a sample of homogeneous isotropic ferroelectric material of cuboid shape, where the Laplacian is discretized by finite difference. We present several preconditioners that fits the Kronecker structure and thus can be efficiently implemented and applied. Preconditioner based on the Moore--Penrose pseudoinverse is extremely efficient for this particular problem, and also applicable (if we are able to store the dense right-hand side of our problem). We briefly analyze the computational cost of the method and individual preconditioners, and illustrate effectiveness of the chosen one by numerical experiments. Although we describe our method as preconditioned CG with pseudoinverse-based preconditioner, it can also be seen as pseudoinverse-based direct solver with iterative refinement by CG iteration. This work is motivated by real application, the method was already implemented in C/C++ code Ferrodo2 and first results were published in Physical Review B 107(9) (2023), paper id 094102.

Figures

Figures reproduced from arXiv: 2501.03377 by the authors.

Figure 1
Figure 1. Four right-hand sides of the regular two-periodic prob￾lem (see Problem 5.1) [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Right-hand side of the problem with mixed boundary conditions (see Problem 5.2). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Four right-hand sides of problem with randomly dis￾tributed charge (see Problem 5.3). Notice that strips are of different width — negative charge is more localized. The last three images show the first frontal slices (k = 0) of three-way tensors. 5.2. Numerical experiments. In all experiments we started the PCG with zero initial guess, i.e., U0 = 0n,q in 2D, or U0 = 0n,q,t in 3D cases. All the problems, in particula… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: compares convergence of the unpreconditioned CG and Jacobi-like method for three different values of ω. One can easily see the dominance of CG. Notice that the (horizontal) time axis is in logarithmic scale, due to disproportional run-times of both methods. Notice also…
Figure 5
Figure 5. Figure 5: Behavior of unpreconditioned CG observed by several quantities (see Experiment 5.4) [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: PCG with various preconditioners applied on Problem 5.1 of size 50 × 100 (see Experiment 5.5). p = 3, with ω = 1.3), Mlow-rank(3) (Kronecker rank rK = 3 approximate inverse), and Mp-inv (pseudoinverse preconditioner). We see that the unpreconditioned CG starts fastest …
Figure 7
Figure 7. Figure 7: Performance of Jacobi-like preconditioner is only weakly involved by the number of inner iterations p or the value of the splitting parameter ω (compare with [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Low-Kronecker-rank approximation of inverse works as preconditioner significantly better with higher ranks rK (if it works, see the last paragraph of Section 4.3.2; compare with [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]
Figure 9
Figure 9. Figure 9: The convergence curve of PCG with the pseudoinverse￾based preconditioner is almost problem independent. Time axis is in logarithmic scale (see Experiment 5.6). 5.3. Solutions. Just for completeness we present the approximate solutions of all three problems, computed by…
Figure 10
Figure 10. Figure 10: Approximate solutions of all three problems. Vertical axis measures the electric potential (in the bottom plot, there is the vertical scale related to the fifth layer from the top). 30 [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]

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