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$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation

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Pith's one-line read An $\mathcal{H}_2$-optimal reduced linear quadratic-output model must tangentially interpolate the full system's linear and quadratic transfer functions at the mirror images of its own poles.

desk verdict Genuinely extends H2-optimal interpolation theory to LQO systems, but the main proof has a repairable yet central class-membership gap. read the letter →

arxiv 2505.03057 v2 pith:W3C4CKLD submitted 2025-05-05 math.NA cs.NAcs.SYeess.SYmath.DSmath.OC

Sean Reiter , Ion Victor Gosea , Igor Pontes Duff , Serkan Gugercin This is my paper · ORCID
classification math.NAcs.NAcs.SYeess.SYmath.DSmath.OC MSC 34C2041A0549K1565J0565F9993A1593C1093C80
keywords modelreductionH2-optimalitylinearquadratic-outputsystemstangentialinterpolationmultivariaterationaliterativeKrylovalgorithmPetrov-GalerkinprojectionVolterrakernels
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 derives first-order necessary conditions for a reduced model of a linear quadratic-output (LQO) system, meaning linear state dynamics with an output made of a linear term plus a quadratic term $M(x\otimes x)$, to minimize the $\mathcal{H}_2$ approximation error. The conditions say that at the mirror images of the reduced model's poles, the reduced linear and quadratic transfer functions must tangentially match the full-order ones, and certain weighted sums of the two must be matched in both value and derivative. These mixed-multipoint tangential interpolation conditions generalize the classical Meier-Luenberger conditions for linear systems. The paper also shows how to enforce all conditions by Petrov-Galerkin projection and gives an iterative rational Krylov algorithm, LQO-IRKA, that converges to models satisfying them. If correct, this turns $\mathcal{H}_2$-optimal quadratic-output model reduction into a multivariate rational interpolation problem that can be solved with shifted linear solves.

What carries the argument

The load-bearing object is the pole-residue expansion of the reduced transfer functions, $\tilde G_1(s)=\sum_j c_j b_j^T/(s-\lambda_j)$ and $\tilde G_2(s_1,s_2)=\sum_{j,k} m_{j,k}(b_j\otimes b_k)^T/((s_1-\lambda_j)(s_2-\lambda_k))$. Theorem 2.1 uses these expansions to express the $\mathcal{H}_2$ inner product and norm of an LQO system as finite evaluations of $G_1$ and $G_2$ at the mirrored poles $-\lambda_j$. Lemma 2.1's symmetry identities for $G_2$, inherited from the commutation matrix, let the authors group the left-tangential and derivative conditions into the compact form (29c)-(29d). Theorem 3.2 then constructs projection bases $V,W$ whose columns are the shifted solves of (31) and (32), so that Petrov-Galerkin projection enforces all $3r+r^2$ interpolation conditions simultaneously.

What would settle it

One could settle Theorem 3.1 by computing, for a small LQO example, a global $\mathcal{H}_2$-optimal order-$r$ reduced model with simple poles and evaluating whether (29a)-(29d) fail at its mirrored poles; any violation would refute the necessity claim. Alternatively, a stable full-order LQO system whose LQO-IRKA iterates converge to a model that still violates one of the four conditions would show the algorithm does not deliver the claimed optimality certificate.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.1: if an asymptotically stable order-$r$ LQO system $\tilde G$ with simple poles minimizes the squared $\mathcal{H}_2$ error against a full-order LQO system $G$, then the transfer functions $\tilde G_1$ and $\tilde G_2$ satisfy the four tangential interpolation conditions (29a)-(29d). In particular, $\tilde G_1$ and $\tilde G_2$ individually right-interpolate $G_1$ and $G_2$ at $-\lambda_k$ along residue directions, and linear combinations of the two transfer functions are interpolated in the left-tangential Lagrange and bi-tangential Hermite senses at all pairs of mirrored poles. This gives the LQO analogue of the Meier-Luenberger characterization and, because the proof only uses Hardy-space membership of the full-order functions, the paper notes the conditions extend beyond the specific LQO realization.

Load-bearing premise

The argument assumes the $\mathcal{H}_2$-optimal reduced model can be chosen with simple poles and that small pole/residue perturbations stay within the class of order-$r$ asymptotically stable LQO systems; the paper does not prove that a minimizer with simple poles always exists, and LQO-IRKA's convergence is reported empirically rather than proved.

Editorial extensions

If this is right

  • Every $\mathcal{H}_2$-optimal reduced LQO model with simple poles is a tangential interpolant, so the quest for optimal reduced models can be reposed as choosing poles and residue directions that satisfy (29a)-(29d).
  • Petrov-Galerkin projection with the bases (31) and (32) enforces all the optimality conditions at once, guaranteeing that any converged LQO-IRKA iterate meets the first-order necessary conditions.
  • LQO-IRKA requires only shifted linear system solves and matrix-vector products, making the framework applicable to large-scale systems where balancing-based LQO methods need costly Lyapunov solves.
  • When the quadratic output term is zero, conditions (29) reduce to the classical linear Meier-Luenberger interpolation conditions, so the paper's results contain the linear theory as a special case.
  • Because the $\mathcal{H}_2$ error bounds the time-domain $L^\infty$ output error via (20), satisfying these optimality conditions also controls worst-case output accuracy over time for finite-energy inputs.

Reading between the lines

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

  • The proof of Theorem 3.1 does not require the full-order model to have LQO structure, so the same interpolation conditions should characterize $\mathcal{H}_2$-optimal rational approximants for systems with second-order or delay dynamics; whether projection can construct them is left open by the paper.
  • No convergence proof for LQO-IRKA is given; a natural test is whether iterates always reach a stationary point from arbitrary initial data, and whether the simple-pole restriction ever excludes the true minimizer.
  • The mixed conditions couple every pair of poles through the $m_{k,\ell}$ residues, so a large-scale implementation may need to exploit sparsity or low-rank structure in the sums over $\ell$; the paper does not address this cost.
  • A data-driven analogue could identify the interpolation data directly from frequency samples of $G_1$ and $G_2$, bypassing the state-space realization entirely.
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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

1 major / 6 minor

Summary. This manuscript studies H2-optimal model reduction for linear systems with quadratic outputs (LQO systems). The main theoretical result, Theorem 3.1, states that any asymptotically stable order-r LQO reduced model with simple poles that minimizes the H2 error (22) must satisfy the four groups of tangential interpolation conditions (29a)-(29d), generalizing the Meier-Luenberger conditions from LTI systems. Theorem 3.2 gives explicit Petrov-Galerkin bases V and W that enforce these interpolation conditions at prescribed data, and Algorithm 4.1 (LQO-IRKA) iteratively updates the interpolation data using the current reduced model's poles and residue directions. The paper includes a detailed proof of the pole-residue H2 inner product formula (Theorem 2.1), a real-valued basis construction (Lemma 4.1), and numerical experiments on a 1D advection-diffusion problem with a quadratic cost, with code and data released on Zenodo.

Significance. The interpolatory optimality conditions are a natural and non-obvious generalization of the classical H2 interpolation theory, and the projection-based enforcement in Theorem 3.2 gives a constructive route to reduced models satisfying them. The numerical results suggest LQO-IRKA is competitive with balanced truncation for LQO systems and robust to initialization. Strengths include the detailed proofs of Theorem 2.1 and Theorem 3.2, the careful treatment of real-valued bases, the explicit complexity statement (shifted linear solves only), and the availability of reproducible code. The main caveat is the gap in the proof of Theorem 3.1 discussed below; if repaired, the paper would be a solid contribution to the model reduction literature.

major comments (1)
  1. [Appendix B / Theorem 3.1] The proof of Theorem 3.1 does not establish that the perturbed systems used to derive the necessary conditions lie in the admissible class of real order-r LQO systems. For (29b), equation (53) perturbs only the (j,k)-th quadratic residue direction m_{j,k}; when j≠k this violates the symmetry constraint m_{j,k}=m_{k,j} of equation (25), so ˇG is not of the form (2) and the sub-optimality inequality (30) cannot be invoked. For (29c), equation (54) perturbs only b_k, and for (29d), equation (58) shifts only λ_k; in both cases, for complex-conjugate pole pairs the necessary conjugate update of b_{k+1} or λ_{k+1} is omitted, so the perturbed system is complex-valued rather than a real admissible LQO system. These are not merely formalities: the contradiction argument proves stationarity over a larger unconstrained pole-residue class, which does not imply stationarity over the constrained LQO set. The theorem is likely repairable by using symmetric two-index perturbations for (29b) and real-structure-preserving perturbations for (29c)-(29d), but the proof of the central claim must be revised.
minor comments (6)
  1. [Proof of Theorem 3.2] In the sentence following equation (37), "to prove (33b)" should read "to prove (33c)", since the argument establishes the left-tangential Lagrange condition.
  2. [Lemma 4.1] The block matrix Q stated as (1/√2)[1 1; i −i] does not correctly relate W and W_p; the transformation should be (1/2)[1 −i; 1 i] up to column scaling, and "orthogonal" should be "unitary".
  3. [Section 5.2] The numerical computation of the relative H2 error in (47) is not described; the authors should state whether a full eigendecomposition of the order-3000 system or a Lyapunov solve was used.
  4. [Section 5.3] Only a single benchmark is tested, and no comparison is made with the Wilson/gramian-based H2-optimal method of [33] or the Riemannian BFGS method of [44]; such a comparison would strengthen the practical claims.
  5. [Section 2.3 / Eq. (17)] The notation \(\overline{G_1(s)}\) for conjugation of coefficients only is confusing; suggest defining it explicitly or using a different symbol.
  6. [Section 4.2.2] The absence of a convergence proof for LQO-IRKA is acknowledged, but the authors should state explicitly that convergence and attainment of (29) are empirical observations rather than guaranteed properties.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: Theorem 3.1's interpolatory optimality conditions are obtained by an independent variational perturbation argument; self-citations are contextual and non-load-bearing.

full rationale

The paper's central claim, Theorem 3.1, is not circular. The mixed-multipoint tangential interpolation conditions (29a)-(29d) are derived in Appendix B from the sub-optimality inequality (30) by explicit first-order perturbations of the reduced-model poles and residue directions, using the H2 inner-product and norm formulae (26)-(27) that are proved independently in Theorem 2.1 and Appendix A. The optimality conditions are not assumed as inputs; they are consequences of stationarity of the squared H2 error. The paper explicitly recognizes the apparent circularity of needing the optimal reduced model to set the interpolation data ('Of course, this assumes having access to the optimal reduced model. We resolve this circular causality issue in the section.') and resolves it in Section 4 with the LQO-IRKA fixed-point iteration, which updates interpolation data from the previous iterate. This is the standard IRKA self-consistency device, not a circular derivation: convergence to a fixed point satisfies independently derived necessary conditions. The projection construction in Theorem 3.2 is explicit and self-contained; the citation to the authors' prior work ([33, Theorem 3.2], 'any H2-optimal approximation ... is necessarily obtained via a Petrov-Galerkin projection') is contextual and not load-bearing, since Theorem 3.2 proves directly how to enforce the interpolation conditions by constructing V and W. Self-citations to [9,22,23,24] provide background and comparison algorithms, not the theorem's proof. Numerical benchmarking against the full-order time-domain output and against LQO-BT is external evidence, not a fit. The manuscript does flag genuine limitations: convergence of LQO-IRKA is reported empirically rather than proved (Section 4.2.2), and asymptotic stability is not guaranteed by the iteration. In addition, the Appendix B perturbation proof has a class-membership gap: perturbing a single residue m_{j,k} in (53), or applying a single complex pole shift in (58), may leave the class of real-valued LQO systems satisfying the symmetry constraint (25), so inequality (30) is invoked for systems not shown to be admissible. This is a correctness and rigor issue, not a circularity issue: the claimed conditions are not equivalent to the proof's inputs by construction. No step in the derivation reduces to a fitted parameter renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to force the choice.

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

No free parameters or invented entities appear in the derivation. The theory rests on standard complex analysis and on domain assumptions about stability, real-valued realizations, and simple poles of the reduced model. The algorithmic inputs (initial interpolation data, tolerance, order r) are user choices rather than fitted parameters.

assumptions (5)
  • domain assumption The full-order system (1) is asymptotically stable and E is nonsingular
    Assumed at the start of Section 1; required for the transfer functions G1 and G2 to lie in the Hardy spaces used to define the H2 norm.
  • domain assumption The reduced model has simple poles
    Assumed in Section 2.4 and Theorem 3.1; enables the pole-residue expansion (23) and the residue-based H2 inner product formula in Theorem 2.1. The paper notes repeated poles rarely appear in practice but does not prove the optimal minimizer has simple poles.
  • domain assumption Real-valued state-space realization, so G1(s)=G1(s) and G2(s1,s2)=G2(s1,s2)
    Used in the perturbation proofs in Appendix B to evaluate inner products via Theorem 2.1; this is standard for real dynamical systems.
  • standard math Residue theorem and contour integration results apply to the bivariate transfer functions
    The proof of Theorem 2.1 in Appendix A uses ML-estimates and the residue theorem to evaluate the double integrals; these are standard results in complex analysis.
  • domain assumption Perturbations of the reduced model parameters produce valid order-r LQO systems
    The proof of Theorem 3.1 perturbs poles and residue directions and implicitly assumes the resulting transfer functions correspond to some order-r LQO system; this holds because any function of the form (23) with the symmetry (25) is realizable, but the paper does not explicitly state this.

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Cite this review

Pith. "Pith review of $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation." pith.science (2026). https://pith.science/paper/W3C4CKLD

@misc{pith2026250503057,
  author       = {Pith},
  title        = {Pith review of: $\mathcalH_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W3C4CKLD}},
  note         = {Machine review of arXiv:2505.03057}
}
abstract

This paper addresses the $\mathcal{H}_2$-optimal approximation of linear dynamical systems with quadratic-output functions, also known as linear quadratic-output systems. Our major contributions are threefold. First, we derive interpolatory first-order optimality conditions for the linear quadratic-output $\mathcal{H}_2$ minimization problem. These conditions correspond to the mixed-multipoint tangential interpolation of the full-order linear- and quadratic-output transfer functions, and generalize the Meier-Luenberger optimality framework for the $\mathcal{H}_2$-optimal model reduction of linear time-invariant systems. Second, given the optimal interpolation data, we show how to enforce the interpolatory optimality conditions explicitly by Petrov-Galerkin projection of the full-order model. Third, to find the optimal interpolation data, we build on this projection framework and propose a generalization of the iterative rational Krylov algorithm for the $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems, called LQO-IRKA. Upon convergence, LQO-IRKA produces reduced linear quadratic-output systems that satisfy the interpolatory optimality conditions. The method only requires solving shifted linear systems and matrix-vector products, thus making it suitable for large-scale problems. Numerical examples are included to illustrate the effectiveness of the proposed method.

Figures

Figures reproduced from arXiv: 2505.03057 by the authors.

Figure 1
Figure 1. Output magnitudes and pointwise relative errors [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Relative H2 errors of the intermediate reduced models computed by LQO-IRKAeigs and LQO-IRKAimag for the first 50 iterations. LQO-IRKAeigs LQO-IRKAimag LQO-BT interponeStep,eigs interponeStep,imag Run time (s) 58.23 s 56.31 s 72.61 s 0.41 s 0.44 s Iteration count 124 110 N/A N/A N/A [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Relative H2 errors (47) due to the hierarchy of reduced models for orders r = 2, 4, . . . , 30. LQO-IRKAimag converge to the same local minimum for each order of reduction. 6. Conclusion We have presented a novel H2-optimality framework for the approximation of linear quadratic￾output systems (1) based on multivariate rational interpolation. In Theorem 3.1, we derive first￾order optimality conditions; these amount t… view at source ↗

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.