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REVIEW 3 major objections 6 minor 34 references

Data-driven balanced truncation for second-order systems via the approximate Gramians

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Frequency-response samples alone can run balanced truncation for second-order systems with proportional damping, reproducing the standard reduced model without assembling the original matrices.

desk verdict A competent extension of data-driven balanced truncation to second-order systems, but the nonintrusive claim is conditional on knowing the Rayleigh damping coefficients. read the letter →

arxiv 2506.03855 v1 pith:JMSXHASH submitted 2025-06-04 math.NA cs.NA

Xiaolong Wang , Xuerong Yang , Xiaoli Wang , Bo Song This is my paper
classification math.NAcs.NA MSC 93B1165F3093C05
keywords modelorderreductionbalancedtruncationsecond-ordersystemsproportionaldampingfrequency-responsedataGramiansSylvesterequationsdata-drivenmodeling
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 establishes that, for second-order systems with proportional damping, the standard velocity balanced-truncation reduced model can be reproduced from frequency-response samples alone, without forming or factoring the original mass, damping, and stiffness matrices. The velocity controllability and observability Gramians are approximated by numerical quadrature, and the paper derives explicit entrywise formulas for the quantities $\widetilde M=\widetilde L^{\top}M\widetilde U$, $\widetilde D$, $\widetilde K$, $\widetilde B$, and $\widetilde C$ in terms of samples of the transfer function. A data-driven algorithm then constructs a structure-preserving reduced model of the same second-order form. This makes balanced truncation nonintrusive for a common class of vibrating structures, and the paper's examples show the reduced model converging toward the standard balanced-truncation model as the number of quadrature nodes grows.

What carries the argument

The carrying object is the pair of quadrature-approximated velocity Gramian factors, $\widetilde U$ with columns $\rho_j\zeta_j G(i\zeta_j)B$ and $\widetilde L$ with rows $\phi_k C G(i\omega_k)$, together with the linear relations they satisfy. For each index pair $(k,j)$, the sample values $H(i\zeta_j)$ and $H(i\omega_k)$ are linked to the entries of $\widetilde M$, $\widetilde D$, and $\widetilde K$ by equations (17)--(18); because there are three unknown matrices but only two equations, the proportional-damping closure $D=\alpha M+\beta K$ is what makes the entries explicit. The real-valued variant uses unitary rotations to convert conjugate-paired nodes into $2\times 2$ real blocks, and the fast variant solves two decoupled Sylvester equations with an extended Krylov low-rank approximation, so the truncation SVD is applied to a low-order matrix.

What would settle it

Compare Algorithm 2 against Algorithm 1 on a known proportionally damped second-order system, using the same quadrature rule with an increasing number of nodes and measuring the relative $H_2$ distance between the two reduced transfer functions; if the distance does not shrink as $N_p=N_q$ grows, the claimed reconstruction of $L^{\top}MU$, $L^{\top}DU$, and $L^{\top}KU$ from frequency samples is not what the paper says it is.

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Extended reading notes

Core claim

The central discovery is that every ingredient of velocity balanced truncation for a second-order system can be read off from sampled values of the transfer function $H(s)=C(s^2M+sD+K)^{-1}B$, provided the damping is proportional, $D=\alpha M+\beta K$ with known $\alpha,\beta$. Approximating the velocity Gramians $P_v$ and $Q_v$ by quadrature gives square-root factors $\widetilde U$ and $\widetilde L$ built from those samples. The matrices $\widetilde M=\widetilde L^{\top}M\widetilde U$, $\widetilde D$, and $\widetilde K$ satisfy two linear equations, (17) and (18), whose unknowns outnumber the equations; under the proportional-damping relation the equations close into the explicit entrywise formulas (21)--(23), while $\widetilde B$ and $\widetilde C$ are direct sample values. Algorithm 2 then uses the SVD of $\widetilde M$ together with these reconstructed quantities to produce a reduced model $M_r=I_r$, $K_r$, $D_r=\alpha M_r+\beta K_r$, $B_r$, $C_r$ that matches the standard velocity balanced-truncation model as the quadrature accuracy improves. The paper also reorders nodes in conjugate pairs so all arithmetic is real, and reformulates the reconstruction as low-rank Sylvester equations solved by extended Krylov subspaces, so the large SVD is replaced by the SVD of a small projected matrix.

Load-bearing premise

The method rests on the damping being a known proportional combination of mass and stiffness, $D=\alpha M+\beta K$ with both coefficients given in advance, and if the coefficients are unknown the frequency samples alone do not close the equations.

Editorial extensions

If this is right

  • A structure-preserving reduced model of order $r$ can be built from $N_p+N_q$ complex frequency-response samples plus the two known damping coefficients, with no access to $M$, $D$, $K$, $B$, or $C$.
  • When the quadrature nodes are chosen symmetrically, every computation runs in real arithmetic and the reduced model has real coefficient matrices.
  • As $N_p$ and $N_q$ grow, the data-driven model approaches the standard velocity balanced-truncation model, and the error bound in Remark 2 makes the quadrature error in the relevant singular values explicit.
  • With many samples, Algorithm 3 removes the large matrix SVD from the loop; the cost scales with the extended Krylov iteration count rather than the sample count, which the CPU-time tables confirm.
  • The reduced models are not interpolatory in the Loewner sense; they are balanced-truncation approximations, as Remark 4 notes.

Reading between the lines

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

  • A natural extension would be to estimate $\alpha$ and $\beta$ from the same frequency samples by fitting equations (17)--(18) in a least-squares sense, which would remove the one piece of intrusive information the current method still requires.
  • The paper leaves open position-balanced and velocity-position-balanced forms because the position observability Gramian is more complex; approximating that block, perhaps with additional structural assumptions, is a concrete next step.
  • Because formulas (21)--(23) use the supplied $\alpha,\beta$ directly, a sensitivity analysis of the reconstructed reduced model with respect to errors in those coefficients would tell practitioners how accurately the damping coefficients must be known.
  • Adaptive or data-dependent quadrature node selection could reduce the number of transfer-function evaluations needed for a target accuracy, since the whole method is governed by how well the quadrature captures the velocity Gramians.
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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

3 major / 6 minor

Summary. The paper proposes a data-driven, nonintrusive variant of balanced truncation for second-order linear systems in the case of proportional (Rayleigh) damping. It approximates the velocity controllability and observability Gramians by frequency-domain quadrature, defines projected quantities ~M, ~D, ~K, ~B, ~C from transfer-function samples, and derives explicit formulas (21)-(23) under D = αM + βK. Algorithms 2 and 3 implement the resulting reduced-order models in real arithmetic, with Algorithm 3 using low-rank Sylvester-equation solves via extended Krylov subspaces. Two numerical examples compare the proposed models with standard balanced truncation.

Significance. The paper addresses a timely problem—data-driven model reduction of structured second-order systems—and the core algebraic derivation is plausible. It provides explicit, implementable formulas and a low-rank variant that shows clear computational gains in Tables 1 and 3. The paper honestly restricts to proportional damping and notes other balanced forms are not covered (Remark 7). However, the main nonintrusiveness claim is conditional on α and β being known, and Algorithm 3 is shown to fail (Inf H2 error) for r = 25 in Table 2. With those gaps addressed, the method would be a useful extension of [23] to second-order structure.

major comments (3)
  1. [Section 3.2, Eqs. (19)-(23) and Algorithms 2-3] The assembly of ~M, ~K, ~D from transfer-function samples requires the Rayleigh damping coefficients α and β as explicit inputs: they appear in the closed-form formulas (21)-(22), in the reduced damping relation D_r = αM_r + βK_r (Algorithm 3, Step 6), and in the Sylvester coefficient matrices of (28)-(29). However, Algorithms 2 and 3 list only quadrature data, samples, and the truncation index as inputs, and no procedure is given to estimate α and β from the samples H(s). The numerical examples preset these values (α=β=0.05 in Example 1 and α=β=0.06 in Example 2), so the central claim of Section 3 that the method requires no access to the state-space realization is not fully supported: unless α and β are supplied by the user, the pipeline is not self-contained. Please either add an estimation step or explicitly state as a hypothesis that α and β are known, and adjust the nonintrusive claim accordingly.
  2. [Section 4, Table 2] For r=25, Algorithm 3 (KryData-BT-SOPD) reports an infinite H2 error and an H∞ error that stops decreasing (1.37e-05), while Data-BT-SOPD continues to improve as r grows. Since Algorithm 3 is presented as the efficient counterpart of Algorithm 2, this result shows that the fixed subspace size m=30 does not guarantee that the low-rank Sylvester solution reproduces the full data-driven BT reduction. The paper should either provide an adaptive strategy for m (e.g., driven by the Sylvester residual mentioned in Remark 5) or state the accuracy limitations of Algorithm 3 more explicitly.
  3. [Section 3.1, Remark 2] The only accuracy statement is a perturbation bound for the singular values of L^T M U, conditional on assumed bounds for ‖Q_v − ~Q_v‖_F and ‖P_v − ~P_v‖_F. No quadrature convergence analysis is given for the actual rules used in Section 4 (logarithmically spaced nodes and trapezoid weights), so the paper does not establish how the number of samples N_p, N_q should scale to achieve a target accuracy, nor does it quantify the error between the data-driven reduced model and the exact balanced-truncation model. Given that the numerical experiments rely on increasing N to approach standard BT, this missing analysis is a substantive gap.
minor comments (6)
  1. [Section 4, first paragraph] The text says "we use three numerical examples" but only Examples 1 and 2 are given; the abstract and conclusions refer to two examples, so the counts should be aligned.
  2. [Algorithm 3, Step 2 and Section 3.2] "Arnoild" is a typo for "Arnoldi", and the phrase "one can produce produced real-valued reduced models" in Section 3.2 should be corrected.
  3. [Section 3.3, Eqs. (28)-(29)] The derivation of the two Sylvester equations from (26) is only sketched; please include the algebraic steps or a reference for the matrix manipulations used to obtain them.
  4. [Table 2, r=25 row] For KryData-BT-SOPD the H2 error is "Inf"; adding a footnote explaining the cause (e.g., unstable reduced model or failure of the fixed-subspace solve) would make the comparison interpretable.
  5. [Eq. (16)] The notation ~B_k is used both for the k-th entry of the vector ~B and for the vector itself in the surrounding text; please distinguish the entries from the full vector/matrix to avoid ambiguity.
  6. [Remark 2] The perturbation bound involves σ_min(P_v), σ_min(Q_v), and ‖L‖_2, ‖U‖_2, which are quantities of the original large-scale system and are not accessible in the data-driven setting; please clarify the intended use of this bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper derives the reduced model algebraically from frequency samples, with the only external ingredient being the stated proportional-damping assumption.

full rationale

The derivation chain is self-contained. Proposition 1 establishes equations (17)-(18) by direct algebraic manipulation from the quadrature definitions (12)-(15), and the proportional-damping ansatz (19) is an explicit stated assumption that closes the underdetermined system. The closed-form expressions (21)-(23) are algebraic solutions of the resulting equations in terms of the frequency-response samples and the known coefficients alpha and beta; no fitted parameter is renamed as a prediction, and no target quantity is defined in terms of the reduced model that is later produced. The reduced matrices in Algorithm 2 are obtained by SVD truncation of ~M and by congruence from ~D, ~K, ~B and ~C, all computed from the same samples, so the output is not equivalent to the input by construction. The reliance on reference [23] for the first-order framework and the Hankel-singular-value error bound is an external, non-overlapping source, not a self-citation chain. The only qualification is that alpha and beta are assumed known rather than estimated from data, and Algorithm 2's input list omits them; however, this is a completeness limitation of the nonintrusiveness claim, not a circular reduction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central construction is inexpensive in free parameters: formulas (21)-(22) solve the two equations (17)-(18) entrywise, and the reduced model is obtained by SVD. The main uncharged inputs are the proportional damping assumption with known alpha and beta, the choice of quadrature rule, and the Krylov subspace order m. No new state, force, or physical entity is postulated.

free parameters (1)
  • Rayleigh damping coefficients alpha, beta = alpha = beta = 0.05 (Example 1); alpha = beta = 0.06 (Example 2)
    The proportional damping coefficients are assumed known and supplied by the user; they are not identified from the frequency samples. Every reduced model depends on them through Dr = alpha Mr + beta Kr, and formulas (21)-(22) require them. For a genuinely data-driven deployment these would need to be estimated, and no estimation procedure is provided.
assumptions (5)
  • domain assumption D = alpha M + beta K with known scalars alpha, beta >= 0 (proportional/Rayleigh damping).
    Invoked in Section 3.2, Eq. (19), to close the underdetermined system (17)-(18). The entrywise formulas (21)-(22) and the reduced damping Dr = alpha Mr + beta Kr in Algorithm 3 depend on this. If alpha and beta are unknown to the user, the method is not purely data-driven.
  • domain assumption Quadrature nodes for P_v and Q_v are distinct: omega_k != zeta_j.
    Required for the denominators in (21)-(22) to be nonzero; stated before Eq. (21). Overlapping node sets break the formulas.
  • domain assumption The quadrature sums (12) and (14) accurately approximate the exact Gramians P_v and Q_v.
    The method replaces exact Gramians by numerical quadrature; no specific quadrature error bound for second-order systems is proved (Remark 2 only quotes a singular-value perturbation bound from [23]). The numerical accuracy of the reduced model depends on this convergence.
  • standard math For real-valued systems the transfer function satisfies H(s) = conjugate(H(conjugate(s))).
    Used in Section 3.2, after Eq. (24), to pair samples and construct real-arithmetic reduced models.
  • standard math Extended Krylov subspace and Galerkin projection produce a sufficiently accurate low-rank solution of the Sylvester equations (28)-(29).
    Invoked in Section 3.3 and Algorithm 3; no a priori error bound is given for the reduced model, and Table 2 shows this approximation can fail, with H2 error Inf at r=25.

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Cite this review

Pith. "Pith review of Data-driven balanced truncation for second-order systems via the approximate Gramians." pith.science (2026). https://pith.science/paper/JMSXHASH

@misc{pith2026250603855,
  author       = {Pith},
  title        = {Pith review of: Data-driven balanced truncation for second-order systems via the approximate Gramians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMSXHASH}},
  note         = {Machine review of arXiv:2506.03855}
}
read the original abstract

This paper studies the data-driven balanced truncation (BT) method for second-order systems based on the measurements in the frequency domain. The basic idea is to approximate Gramians used the numerical quadrature rules, and establish the relationship between the main quantities in the procedure of BT with the sample data, which paves the way for the execution of BT in a nonintrusive manner. We construct the structure-preserving reduced models approximately based on the samples of second-order systems with proportional damping, and provide the detailed execution of the data-driven counterpart of BT in real-value arithmetic. The low-rank approximation to the solution of Sylvester equations is also introduced to speed up the process of the proposed approach when a large amount of samples involved in the modeling. The performance of our approach is illustrated in detail via two numerical examples.

Figures

Figures reproduced from arXiv: 2506.03855 by the authors.

Figure 1
Figure 1. The frequency response (left) and the absolute errors (right) with r = 4, Nq, Np = 50 and m = 5. 10-2 10-1 100 101 102 103 104 Frequency 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 Absolute error BT-SOPD Data-BT-SOPD KryData-BT-SOPD 10-2 10-1 100 101 102 103 104 Frequency 10-11 10-10 10-9 10-8 10-7 10-6 10-5 Absolute error BT-SOPD Data-BT-SOPD KryData-BT-SOPD [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. The absolute errors with r = 4, Nq, Np = 500, m = 5 (left) and r = 4, Nq, Np = 500, m = 20 (right). system is well approximated by all reduced models, and we can not distinguish them clearly from the response depiction. However, one can observe an evident distinction between the standard BT and the proposed data-driven version from the error depiction, especially in the lower frequency domain. This is due to the rel… view at source ↗
Figure 3
Figure 3. The time response (left) and absolute errors (right) with r = 4, Nq, Np = 500, m = 20 for u(t) = e−t sin(t) [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The frequency response (left) and absolute errors (right) with r = 10, Nq, Np = 200, m = 20. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: The time response (left) and absolute errors (right) with r = 10, Nq, Np = 200, m = 20 for e−t sin(5t). domain decays notably. Furthermore, we vary the reduced order from r = 5 to r = 25 and test the error of each reduced model generated by Algorithm 1-3 with the choic…

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