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REVIEW 4 major objections 5 minor 46 references

A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes a parameter-free, matrix-free biorthogonal subspace iteration that computes the smallest positive excitation eigenpairs of the linear response eigenvalue problem, including problems with zero eigenvalues.

desk verdict A practical, well-tested LREP solver with a genuinely useful deflation/batching design; the stability story is partly conjectural, but the paper deserves refereeing. read the letter →

arxiv 2506.08355 v1 pith:MCUTNKFE submitted 2025-06-10 math.NA cs.NA

Yu Li , Zijing Wang , Yong Zhang This is my paper
classification math.NAcs.NA MSC 65F1565F5065Y05
keywords linearresponseeigenvalueproblembiorthogonalinvariantsubspacemodifiedGram-Schmidtbiorthogonalizationgeneralizednullspacestructure-preservingeigensolverdeflationmatrix-freeparallelscalability
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 linear response eigenvalue problem $H\xi = \lambda\xi$, where $H$ has off-diagonal blocks $K$ and $M$ with $K$ symmetric positive semi-definite and $M$ symmetric positive definite, is the numerical core of random-phase-approximation, Bethe-Salpeter, and Bogoliubov-de Gennes calculations. The paper's claim is that a subspace iteration called BOSP can solve this problem for large sparse or dense systems by preserving the biorthogonality of approximate eigenvector pairs, using a modified Gram-Schmidt biorthogonalization. The biorthogonality gives a natural deflation: once eigenpairs converge, the search moves to their biorthogonal complement, so no user-chosen shift or penalty parameters appear. A moving batch mechanism keeps the small projected problem independent of the number of requested eigenpairs, and only matrix-vector products are needed. The reported evidence includes near machine-precision eigenvalues for an SPD test, seventeen iterations for the SiH4 molecular problem at tolerance $10^{-10}$, and a 17.44-second run on 576 processes for a 1.7-million-dimensional sparse problem.

What carries the argument

The load-bearing object is the direct sum decomposition $\mathbb{R}^{2n} = V_0 \oplus V_1 \oplus \cdots \oplus V_{n-r}$ of the space into biorthogonal invariant subspaces, where $V_0$ is the generalized nullspace spanned by vectors with $K x_i^0 = 0$ and $M y_i^0 = x_i^0$, together with the minimization principle that the $k$ smallest positive eigenvalues in the biorthogonal complement of $V_0 \oplus \cdots \oplus V_\ell$ equal the minimum of $\tfrac12 \operatorname{tr}(U^\top K U + V^\top M V)$ subject to $U^\top V = I_k$. The numerical engine that makes the decomposition usable is MGS-Biorth, a modified Gram-Schmidt biorthogonalization that maintains $X^\top Y = I_m$; the paper conjectures in (3.3) that its biorthogonality residual is $O(\epsilon)\sqrt{\kappa(X^\top Y)}$. The algorithm projects $H$ onto the search subspace with an oblique projector, producing a small $2d \times 2d$ problem whose diagonal blocks $U^\top K U$ and $V^\top M V$ are positive definite, so zero eigenvalues cannot enter the projected problem.

What would settle it

On a problem with a known generalized nullspace and a nearly linearly dependent search subspace, record $\|U^\top V - I_d\|_2$ while Algorithm 2 runs: if this biorthogonality residual grows well above roundoff while the discarding rule of Remark 3.1 is active, then the biorthogonal-complement deflation that stops the iteration is no longer guaranteed, and the claim that BOSP reliably delivers the smallest positive eigenpairs is not established.

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Extended reading notes

Core claim

Algorithm 2 (BOSP) is claimed to compute the first $n_e$ smallest positive eigenpairs of the linear response eigenvalue problem with SPSD $K$ and SPD $M$, excluding the zero-eigenvalue generalized nullspace $V_0$ and any converged eigenpairs without artificial parameters. The solver maintains numerically biorthogonal bases $X,Y$ with $X^\top Y = I$ via MGS-Biorth, forms a small structure-preserving projected problem, and solves it with an existing structure-preserving method; residuals are then used to enrich the search subspace with a Newton-type correction. The paper reports that this scheme converges superlinearly in practice, reaches near machine precision for a 1000-dimensional SPD example, beats the comparison eigensolver in iteration count, and shows near-linear parallel speedup on a 1.7-million-dimensional finite-element problem and a dense Bogoliubov-de Gennes problem of up to 16.7 million unknowns.

Load-bearing premise

Everything rests on the biorthogonalization step actually keeping the paired search vectors numerically biorthogonal as the iteration proceeds, a property the paper states as a conjecture rather than a theorem.

Editorial extensions

If this is right

  • Problems with singular $K$ are handled by computing the generalized nullspace $V_0$ once, so the iteration proceeds entirely in the biorthogonal complement and never contends with zero eigenvalues.
  • Converged eigenpairs drop out of the search automatically, so no user needs to supply a spectral shift, a damping parameter, or any a priori information about the spectrum.
  • The moving mechanism makes the cost of the small projected eigenvalue problem $O(n\,n_b^2 + n_b^3)$ instead of $O(n\,n_e^2 + n_e^3)$, so requesting thousands of eigenpairs does not inflate the projection size.
  • A matrix-free interface suffices, which is what allows dense problems such as the Bogoliubov-de Gennes system to be solved up to $n = 16{,}777{,}216$ with a roughly $O(n \log n)$ cost per iteration.
  • The reported residual behavior indicates superlinear convergence for the molecular test problems, so high-precision solutions can be obtained in a handful of iterations.

Reading between the lines

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

  • A direct extension of the paper's argument, not pursued numerically, would test whether the same biorthogonal-complement deflation remains parameter-free for the generalized LREP with a general SPD $B$; Remark 3.4 sketches the idea but offers no experiments.
  • Because only matrix-vector products are required and the projected problem size stays bounded by the batch size, the solver should apply to operator-defined response problems in which $K$ and $M$ are never assembled, a direction the dense Bogoliubov-de Gennes example already gestures toward.
  • The conjecture (3.3) on the biorthogonality residual could become a practical safeguard: monitor $\|U^\top V - I_d\|_2$ and reject the iteration when it exceeds a tolerance, rather than silently discarding columns as Remark 3.1 does.
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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 / 5 minor

Summary. The paper proposes BOSP, a subspace-iteration eigensolver for the linear response eigenvalue problem (1.2) with K symmetric positive semidefinite and M symmetric positive definite. It first computes a biorthogonal basis of the generalized nullspace V0, then iterates on a 2d-dimensional search space spanned by [V O; O U] with U^T V = I_d, projects H as in (3.5), solves the small LREP (3.6), enriches the search space with Newton/Gauss-Seidel directions from (3.11)-(3.12), and deflates converged pairs by working in their biorthogonal complement. A batching and moving mechanism in Section 3.3 keeps the projected problem size independent of the requested number of eigenpairs. Numerical experiments report near-machine-precision accuracy (Table 2), comparison with an Advanpix quadruple-precision benchmark (Table 3), iteration counts against LOBP4dCG (Table 4), performance for ne = 5000 (Example 4.5), and parallel scaling up to 576 processes (Table 7).

Significance. If the claims are fully established, the paper's main contributions are a parameter-free deflation mechanism based on biorthogonality and a moving-batch strategy that keeps the small projected problem independent of ne, together with a matrix-free interface. The numerical evidence is a strength: eigenvalues are checked against analytical formulas, quadruple-precision benchmarks, and an external solver, and no fitted constants enter the reported accuracies. The parallel timings in Table 7 show near-linear scaling, and Table 8 indicates mesh-independent convergence for a dense three-dimensional BdG problem. However, the central stability and convergence claims are not rigorously supported: the biorthogonality bound is conjectural, column discarding is unquantified, and no convergence theorem is supplied for the subspace iteration. The contribution is therefore promising but conditional.

major comments (4)
  1. [Section 3.1, Eq. (3.3)] The correctness of Algorithm 2 depends on U^T V = I_d being maintained after every call to MGS-Biorth. The only quantitative statement offered is Eq. (3.3), which the authors explicitly label a conjecture. The identities used later, such as positivity of U^T K U and V^T M V, biorthogonality of the reconstructed eigenvectors, and residual biorthogonality, all presuppose this equality. Please provide a theorem with explicit assumptions or add a numerical safeguard such as reorthogonalization with a monitored loss-of-biorthogonality test; without one, the central stability claim is not established.
  2. [Remark 3.1] The exception handling in Remark 3.1 silently discards columns whenever X^T Y is ill-conditioned. This means the working dimension can fall below d = 3ne, and in extreme cases below ne, so the projected problem (3.6) cannot represent all requested eigenpairs. The moving mechanism in Section 3.3 also assumes the subspace dimension is fixed, but discarding breaks that bookkeeping. The paper does not report how often columns are discarded, does not monitor the condition of X^T Y, and does not provide a restart mechanism. Please quantify this behavior in the numerical experiments or add a safeguard that detects and recovers from excessive column loss.
  3. [Section 2.2, Theorem 2.9] Theorem 2.9 is stated in full generality, but its proof explicitly treats only simple eigenvalues and says the multiple-eigenvalue case 'can be derived analogously'. Repeated or tightly clustered eigenvalues are central to the target applications, as Table 5 itself shows near-degenerate SiH4 eigenvalues. Since the minimization principle in the biorthogonal complement is the theoretical foundation for the deflation and for the algorithm's targeting of the smallest positive eigenpairs, a complete proof for repeated eigenvalues, or a clearly stated additional hypothesis, is needed.
  4. [Section 3.2 and Section 3.3] Algorithm 2 has no convergence theorem. Theorem 2.9 is a minimization principle over invariant subspaces, but it does not imply that the random initial subspace refined by the Gauss-Seidel and Newton updates in Steps 5-6 converges to the smallest positive eigenpairs, nor that the moving mechanism preserves convergence after converged pairs are removed from the search space. The residual plots in Figures 2 and 5 are empirical evidence but not a proof. If the authors intend the convergence behavior to be an empirical contribution, the text should state this limitation explicitly; otherwise a convergence analysis under stated assumptions is required.
minor comments (5)
  1. [Definitions 2.2 and 2.4] Definition 2.2 says 'Two vectors ... is said to be biorthogonal', which should be 'are said', and Definition 2.4 uses 'Matrice' instead of 'Matrices'.
  2. [Section 3.2, Eq. (3.9)] The displayed block matrix in the definitions of P and Q appears dimensionally inconsistent: X and Y are d x ne while the bracket is written as a 2 x 2 block matrix. The intended factor is likely a 2ne x ne block column such as [I_ne; O].
  3. [Section 3.2, after Eq. (3.16)] There is a typo, 'the numbe of ne', which should read 'the number of ne'.
  4. [Section 3.2, complexity paragraph] The phrase 'Roughly specking' should be 'Roughly speaking'.
  5. [Table 3] The column header 'our algorithm εk' is ambiguous; the table would be clearer if the two accuracy columns were labeled 'εk (Matlab eigs)' and 'εk (BOSP)'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LREP-specific theory is proved in the paper and numerical outputs are checked against external or analytical benchmarks.

full rationale

The central derivation is self-contained: Theorem 2.8 and Theorem 2.9 are proved in the paper from Lemmas 2.1, 2.6, and 2.7, rather than imported from the authors' earlier symmetric-eigenvalue papers. Refs. [18,19,46] are used only as analogies and for the moving mechanism, whose subspace-dimension reduction is explicit by construction in Section 3.3. Numerical eigenvalues are validated against independent targets: exact formulas in Example 4.1, Advanpix quadruple-precision eigs in Example 4.2, the external LOBP4dCG solver in Example 4.3, and analytical BdG eigenpairs in Example 4.7. No fitted constant is later reported as a prediction; the residual regression in Table 5 is descriptive, not load-bearing. The genuine weaknesses are correctness risks rather than circularity: Eq. (3.3), the only quantitative biorthogonality bound, is introduced with 'we conjecture the following estimation holds for MGS-Biorth', and Remark 3.1 discards ill-conditioned columns ('such vectors x_l and y_l are simply discarded and the iteration continues') without a detection or restart mechanism, while Algorithm 2 has no convergence theorem for the subspace iteration. These are unsupported-stability concerns, not reductions of the output to the input by construction.

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

The central derivation restates known minimization principles from Bai-Li and uses standard spectral theory. The genuinely new algorithmic ingredients are parameter-free, but they depend on conjectured biorthogonality stability and on several hand-chosen implementation parameters (nb, s, ngs, CG settings). No new physical or mathematical entities are introduced.

free parameters (4)
  • batch size nb = min{ne/5, 150} (default)
    Section 3.3 sets the batch size; it controls the projected matrix dimension and memory use. It is hand-chosen and Remark 3.5 acknowledges the trade-off between convergence and efficiency.
  • moving mechanism factor s = 3 (default)
    Section 3.3: number of batches stored in X and Y before merging, controlling the subspace dimension 2(s+2)nb. It is a hand-chosen implementation parameter.
  • Gauss-Seidel inner sweeps ngs = not specified in the paper
    The number of block Gauss-Seidel iterations in Eq. (3.12) is left unspecified; this changes the approximate Newton search directions and affects convergence speed.
  • inner CG tolerance and maximum iterations = 1e-2 and 20
    Given in Section 3.2 after Eq. (3.12); used to solve the linear systems approximately. These are heuristics that affect efficiency and search quality.
assumptions (5)
  • domain assumption K is symmetric positive semi-definite (SPSD) and M is symmetric positive definite (SPD)
    Stated after Eq. (1.2) as the core setting; used in Lemma 2.7, Theorem 2.8, and to prove the projected matrices U^T K U and V^T M V are SPD in Section 3.2. Excludes indefinite LREP cases.
  • domain assumption The generalized nullspace V0 can be computed accurately enough from Kx=0 and My=x using standard symmetric eigensolvers and CG
    Remark 2.11 says accuracy of V0 is critical; errors in V0 propagate into all biorthogonal-complement computations and can prevent convergence.
  • ad hoc to paper MGS-Biorth maintains biorthogonality with error O(epsilon) sqrt(kappa(X^T Y))
    Eq. (3.3) is labeled a conjecture; Algorithm 2 and the deflation mechanism depend on this numerical property.
  • domain assumption The subspace iteration starting from random U and V converges to the requested smallest positive eigenpairs
    No convergence theorem is provided; the paper relies on Theorem 2.9 and observed residuals in Figures 2 and 5.
  • standard math Standard spectral facts: eigenvalues of KM are nonnegative and the Jordan or direct-sum decomposition of H holds
    Used in Lemma 2.1 and Theorem 2.8; accepted background in matrix analysis.

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Pith. "Pith review of A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem." pith.science (2026). https://pith.science/paper/MCUTNKFE

@misc{pith2026250608355,
  author       = {Pith},
  title        = {Pith review of: A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCUTNKFE}},
  note         = {Machine review of arXiv:2506.08355}
}
read the original abstract

The linear response eigenvalue problem, which arises from many scientific and engineering fields, is quite challenging numerically for large-scale sparse/dense system, especially when it has zero eigenvalues. Based on a direct sum decomposition of biorthogonal invariant subspaces and the minimization principles in the biorthogonal complement, using the structure of generalized nullspace, we propose a Bi-Orthogonal Structure-Preserving subspace iterative solver, which is stable, efficient, and of excellent parallel scalability. The biorthogonality is of essential importance and created by a modified Gram-Schmidt biorthogonalization (MGS-Biorth) algorithm. We naturally deflate out converged eigenvectors by computing the rest eigenpairs in the biorthogonal complementary subspace without introducing any artificial parameters. When the number of requested eigenpairs is large, we propose a moving mechanism to compute them batch by batch such that the projection matrix size is small and independent of the requested eigenpair number. For large-scale problems, one only needs to provide the matrix-vector product, thus waiving explicit matrix storage. The numerical performance is further improved when the matrix-vector product is implemented using parallel computing. Ample numerical examples are provided to demonstrate the stability, efficiency, and parallel scalability.

Figures

Figures reproduced from arXiv: 2506.08355 by the authors.

Figure 1
Figure 1. The batching and moving mechanism. To improve efficiency, for a fixed number of degrees of freedom n, we choose to decrease the dimension of search subspace V, or equivalently the size of small-scale eigenvalue problem (3.6), by computing eigenpairs batch by batch. This batching strategy is analogous to that for the symmetric eigenvalue problem [12, 18]. To be precise, only the first nb unconverged eigenpairs are ut… view at source ↗
Figure 2
Figure 2. Descending trend of the normalized residuals with [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. The efficiency performance with/without the moving mechanism in [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The convergence performance when the first [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: The convergence performance in Example 4.7 for different mesh sizes. ing large-scale sparse and dense discrete eigenvalue system that may admit zero ei￾genvalues. Our solver is based on a direct sum decomposition of biorthogonal in￾variant subspaces where the generaliz…

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