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Towards an Efficient Shifted Cholesky QR for Applications in Model Order Reduction using pyMOR

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that recomputing the shift from the current Gramian makes shifted Cholesky QR deliver machine-precision orthogonality even for ill-conditioned matrices with $\kappa_2(A)=10^{20}$, and that the resulting updating and…

desk verdict A genuinely useful, honestly benchmarked variant of shifted Cholesky QR, but the robustness claim rests on an unguarded eigenvalue estimate and should be tightened before it is advertised as generally robust. read the letter →

arxiv 2507.07788 v1 pith:MDGSVBJS submitted 2025-07-10 math.NA cs.NA

Maximilian Bindhak , Art J. R. Pelling , Jens Saak This is my paper
classification math.NAcs.NA MSC 65F2515A2315A1265F3568Q2565Y20
keywords shiftedCholeskyQRmodelorderreductionorthogonalizationupdatepanelschemeill-conditionedmatricespyMOR
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 claims that shifted Cholesky QR can be made robust enough for the extremely ill-conditioned matrices that arise in adaptive model order reduction, and that it can be updated and paneled efficiently under the vector-only interface of pyMOR. Its key change is to recompute the shift from the current Gramian whenever the Cholesky step breaks down, rather than reusing an initial estimate that can be too large by orders of magnitude and then be reapplied in every iteration. In benchmarks on synthetic matrices with condition numbers up to $\kappa_2(A) = 10^{20}$, far beyond the $1/u$ limit where earlier shifted variants fail, the new rsCholQR keeps the loss of orthogonality near machine precision, and the panel variant pnCholQR(r) keeps that robustness while reducing the flop count by up to half in the tall-skinny regime. The authors' practical point is that model order reduction codes do not have to sacrifice the communication-avoiding, BLAS-friendly structure of Cholesky QR to gain numerical reliability.

What carries the argument

The load-bearing object is the shifted Gramian identity $X + \sigma I = \tilde R^T \tilde R$, where $X = A^T A$; from the Cholesky factor $\tilde R$ the algorithm forms $Q \leftarrow Q\tilde R^{-1}$ and accumulates $R$. The robustness mechanism is the recomputed shift $\sigma = \max\{11(mn + n(n+1))u\|X\|_2, 2u\}$, evaluated from a Lanczos estimate of the largest eigenvalue of the current Gramian whenever the Cholesky step breaks down. For the update mode, Proposition 2.1 supplies the identity $R_2^T R_2 = A_2^T A_2 - A_2^T Q_1 Q_1^T A_2$, which converts the orthogonalization of new columns $A_2$ against an existing orthonormal basis $Q_1$ into a small $p \times p$ Cholesky problem. The panel scheme pnCholQR(r) applies that update block by block, keeping each Gramian small enough for cache and cutting the flop count by up to a factor of two when $n^3 \ll m$.

What would settle it

Construct a matrix with condition number $\kappa_2(A) = 10^{20}$ whose Gramian has a largest eigenvalue concentrated in a direction that the Lanczos process with $v_0 = \mathbf{1}$ underestimates, run rsCholQR with the Listing 1 eigsh call, and check the loss of orthogonality $\varepsilon_{\mathrm{LOO}}(Q)$; if the safeguard lets the Cholesky step break down or $\varepsilon_{\mathrm{LOO}}$ exceeds $10^{-13}$, the claimed robustness fails on that input.

Watch

Extended reading notes

Core claim

The central claim, stated in Section 4, is that rsCholQR and pnCholQR(r) are numerically robust even for matrix condition numbers as high as $\kappa_2(A) = 10^{20} \gg 1/u$, while the earlier sCholQR3 and iter-sCholQR variants work reliably only for $\kappa_2(A) < 1/u$. The mechanism is to recompute the shift $\sigma = \max\{11(mn + n(n+1))u\|X\|_2, 2u\}$ from the current Gramian iterate $X$ instead of reusing the initial estimate. With the shift matched to the current iterate, the repeated Cholesky correction drives $Q^T Q$ close to the identity in the reported experiments, with loss of orthogonality comparable to that of an LAPACK-based QR routine. The same mechanism is embedded in a column-update scheme and in a panel scheme that splits the input into $r$ blocks; the paper reports that both keep the robustness and, depending on hardware and matrix properties, can beat SciPy QR in runtime.

Load-bearing premise

Everything rests on the shift formula $\sigma = \max\{11(mn + n(n+1))u\|X\|_2, 2u\}$ staying in a narrow window, even though $\|X\|_2$ is only coarsely estimated by a Lanczos call with tolerance $10^{-2}$: too small an estimate can let the shifted Cholesky step break down, and a stale or oversized shift can quietly destroy orthogonality.

Editorial extensions

If this is right

  • If the claim holds, adaptive MOR loops can extend an orthonormal basis with appended vectors while keeping orthogonality near machine precision, using only vector-wise operations and no entrywise access to the underlying data.
  • The panel scheme gives a tunable cost-quality trade-off: as the number of panels grows, the flop count approaches half of rsCholQR's in the tall-skinny regime, so a MOR code can choose panel width to match cache size without losing robustness.
  • When the condition number of the input is unknown, the paper recommends rsCholQR and pnCholQR(r) as the safer defaults among Cholesky QR variants, while sCholQR3 and iter-sCholQR remain cheaper choices for $\kappa_2(A) < 1/u$.
  • On the notebook hardware the Cholesky variants can be faster than SciPy QR, and on the server the panel variants overtake SciPy QR for large column counts, so the communication-avoiding structure pays off in practice as well as in flop counts.

Reading between the lines

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

  • The authors leave implicit that the factor-10 escalation in Algorithm 2 is the weakest point of the update scheme; if that heuristic fails on some input, a provably safe shift bound would be needed, and the factor 11 in the shift formula is a plausible starting point for such a bound.
  • The authors leave implicit that their robustness evidence is entirely empirical: a testable follow-up is to certify the eigsh estimate with an upper bound, or to replace it with a deterministic power iteration, so that the safeguard $\max\{11(\cdots)u\|X\|_2, 2u\}$ is guaranteed rather than heuristic.
  • The authors leave implicit that the synthetic SVD-generated test matrices may not reproduce the nearly dependent snapshots of real PDE simulations; applying the panel scheme to snapshot vectors produced by different finite-element backends would test whether it can replace the rank-revealing behavior of the default modified Gram-Schmidt routine.
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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 three modifications to shifted Cholesky QR for MOR applications in pyMOR: rsCholQR recomputes the shift from the current Gramian instead of reusing an initial estimate; CholQRUpdate provides a QR-updating scheme; pnCholQR(r) splits the input into panels and applies the update iteratively. The authors benchmark these against SciPy QR, MGS2, sCholQR3, and iter-sCholQR on two hardware systems and several VectorArray backends, using random SVD-generated matrices with condition numbers up to 10^20. They claim that the new methods are numerically robust for these extreme condition numbers and can be competitive in runtime, while also giving a flop-count analysis of the proposed algorithms.

Significance. If the robustness claim holds, the paper offers a practically useful communication-avoiding orthogonalization method for MOR, especially in the adaptive/updating mode where only vector-wise access is available. The paper's strengths are its reproducible Zenodo artifacts, careful benchmarking over multiple backends and hardware, direct comparison against standard baselines, and a flop-count analysis. However, the robustness claim rests on an unanalyzed spectral-norm estimate and on a heuristic escalation loop, so the contribution is currently an experimental proposal rather than a fully established numerical method.

major comments (4)
  1. [§4 and Algorithm 1] The central claim that rsCholQR is numerically robust for κ2(A)=10^20 is load-bearing but depends on the unguarded norm estimate in Listing 1. If the starting vector v0=ones(n) has near-zero overlap with the dominant eigenspace of X=A^T A, the eigsh estimate can be orders of magnitude too small, the shift σ collapses to the 2u floor, and the call chol(X+σI) on line 7 of Algorithm 1 can break down with no retry or escalation in the outer loop. Since Section 4 recommends rsCholQR precisely when the condition number is unknown, the paper should either add an escalation/fallback analogous to Algorithm 2's inner while loop, use a norm estimate robust to starting vectors, or restrict the robustness claim to inputs for which the eigsh estimate is reliable.
  2. [§2.2, Algorithm 2 lines 8-10] The factor-10 escalation in the inner while loop is acknowledged to be a heuristic, and the manuscript offers no evidence or analysis that it terminates or that the resulting shift preserves accuracy. Because pnCholQR(r)'s robustness inherits from this escalation, the paper should at least report the distribution of inner-loop iterations and failure rates over the benchmark, or provide a heuristic justification with a bound on the number of escalations necessary before a Cholesky factor is obtained.
  3. [§3.3 and §4] The robustness claim is validated only on synthetic SVD-generated matrices with random orthogonal factors. For MOR data, structured linear dependence and a dominant eigenspace aligned away from the initial Lanczos vector are plausible, and those cases are exactly where the shift recomputation can fail. The paper should either include adversarial or application-derived test matrices, such as POD snapshots or IRKA bases, or add a formal convergence and backward-stability analysis for the recomputed shift; without one of these, the conclusion in Section 4 overstates the evidence.
  4. [§3.1 vs Algorithms 1-3] The pseudocodes for rsCholQR and pnCholQR(r) do not include the maximum of 10 outer iterations that the benchmark implementation enforces, nor do they state what is returned when the loop exits at the cap without satisfying the stopping criterion. This discrepancy matters because the robustness claim in Section 4 is effectively 'converges within 10 iterations on the tested random matrices'; the paper should specify the failure mode for non-convergence and align the pseudocode with the implementation.
minor comments (5)
  1. [Novelty statement] The novelty statement contains a typo: 'imrpove' should be 'improve'.
  2. [§2.1] The word 'Lanzcos' should be spelled 'Lanczos'.
  3. [Table 3 and Listing 1] Table 3 refers to 'scipy.linalg.eigsh', but Listing 1 uses 'scipy.sparse.linalg.eigsh'; the naming should be made consistent.
  4. [Figure 3b] The y-axis label 'Runtime [s] / m' is ambiguous; writing 'Runtime per vector length (s/m)' would be clearer.
  5. [Algorithm 1 line 5] Using an assignment inside the 'if' condition is unconventional; a try/catch or an explicit breakdown flag would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: robustness claims rest on external benchmarks and in-paper proofs.

full rationale

The central robustness claim for rsCholQR and pnCholQR(r) is an empirical finding supported by comparisons against external baselines (SciPy QR, MGS2, and sCholQR3 / iter-sCholQR from [21]); no fitted parameter is relabeled as a prediction. The shift formula in Algorithm 1 is a proposed safeguard motivated by experiments, and Algorithm 2's factor-10 escalation is explicitly acknowledged as an unresolved heuristic. The only self-citation is Algorithm 2's origin in [35] by the second author, but the correctness of the update is re-derived in this paper via Proposition 2.1, which contains a full proof, so [35] is not load-bearing. The coarse eigsh norm estimate in Listing 1 is a legitimate robustness risk, but it is a correctness and analysis gap, not a circular step, because success is not derived from that estimate by construction. Overall, the paper is self-contained against external benchmarks and exhibits no circular reduction of its claims to their own inputs.

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

The central robustness claim rests on hand-tuned shift and tolerance constants (factor 10 escalation, eigsh tolerance, stopping tolerance), on the transfer of the shift bound from [21], and on the representativeness of the synthetic test matrix family. No new entities are introduced, and no parameters are fit to data.

free parameters (4)
  • escalation_factor_10 = 10
    Algorithm 2 line 9: when the shifted Cholesky breaks down, the shift is multiplied by 10 until success. This hand-picked factor has no analysis; the authors state they tried other strategies without finding a cheaper robust one (Section 2.2).
  • eigsh_tolerance = 1e-2
    Listing 1: the largest eigenvalue of X is approximated with tol=1e-2 for speed. The robustness of the shift bound relies on this coarse estimate being sufficiently accurate.
  • stopping_tolerance = 1e-13
    Section 3.1: the stopping criterion is ||I - Q^T Q||_F <= 1e-13 rather than the u*sqrt(n) stated in Algorithm 1. This chosen tolerance defines what 'robust' means.
  • shift_floor = 2u
    Algorithm 1 line 6: the shift is floored at 2u to avoid a quasi-infinite loop if the recomputed shift becomes insignificant (Section 2.1).
assumptions (4)
  • domain assumption The shift bound σ = 11(mn+n(n+1))u||X||_2 from [21] guarantees that the shifted Cholesky factorization of X + σ I succeeds in floating-point arithmetic.
    Used in Algorithms 1 and 2 without proof in this paper; the paper transfers this bound from [21] and recomputes it on the fly.
  • domain assumption The test matrices A := U Σ V^T with log-equidistant singular values in [1, 10^20] are representative of matrices encountered in MOR applications.
    All benchmarks use this construction (Section 3.1); no real MOR test case (IRKA, POD, etc.) is run, so the MOR-relevance claim rests on this assumption.
  • standard math The IEEE double precision roundoff model with unit roundoff u ≈ 1.11e-16 applies, and the algorithms' operations follow standard BLAS/LAPACK rounding behavior.
    This is the backdrop for all floating-point error measures and the shift formulas.
  • domain assumption For MOR, linearly dependent vectors in the new block can be handled by shifting rather than dropped; the 'lucky breakdown' discussion in Section 2.2 argues rank-deficiency manifests as zero diagonal entries, and shifting is used as a workaround.
    The panel scheme does not implement rank-revealing pivoting, yet the MOR use case often requires detecting dependent vectors. The paper relies on shifting to avoid breakdown and defers pivoting to future work.

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Cite this review

Pith. "Pith review of Towards an Efficient Shifted Cholesky QR for Applications in Model Order Reduction using pyMOR." pith.science (2026). https://pith.science/paper/MDGSVBJS

@misc{pith2026250707788,
  author       = {Pith},
  title        = {Pith review of: Towards an Efficient Shifted Cholesky QR for Applications in Model Order Reduction using pyMOR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDGSVBJS}},
  note         = {Machine review of arXiv:2507.07788}
}
read the original abstract

Many model order reduction (MOR) methods rely on the computation of an orthonormal basis of a subspace onto which the large full order model is projected. Numerically, this entails the orthogonalization of a set of vectors. The nature of the MOR process imposes several requirements for the orthogonalization process. Firstly, MOR is oftentimes performed in an adaptive or iterative manner, where the quality of the reduced order model, i.e., the dimension of the reduced subspace, is decided on the fly. Therefore, it is important that the orthogonalization routine can be executed iteratively. Secondly, one possibly has to deal with high-dimensional arrays of abstract vectors that do not allow explicit access to entries, making it difficult to employ so-called `orthogonal triangularization algorithms' such as Householder QR. For these reasons, (modified) Gram-Schmidt-type algorithms are commonly used in MOR applications. These methods belong to the category of `triangular orthogonalization' algorithms that do not rely on elementwise access to the vectors and can be easily updated. Recently, algorithms like shifted Cholesky QR have gained attention. These also belong to the aforementioned category and have proven their aptitude for MOR algorithms in previous studies. A key benefit of these methods is that they are communication-avoiding, leading to vastly superior performance on memory-bandwidth-limited problems and parallel or distributed architectures. This work formulates an efficient updating scheme for Cholesky QR algorithms and proposes an improved shifting strategy for highly ill-conditioned matrices. The proposed algorithmic extensions are validated with numerical experiments on a laptop and computation server.

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    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der W alt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Lar- son, C. J. Carey, ˙I. Polat, Y. F...

    2020

Pith tools

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