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REVIEW 2 major objections 8 minor 41 references

Exact vs approximate second-order derivatives in vertically-integrated ice sheet models

T0 review · 2 major / 8 minor · reviewed 2026-07-05 · glm-5.2

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Pith's one-line read Approximate ice-sheet Hessian breaks after 4 modes

desk verdict Paper derives a second-order self-adjoint (SOSA) approximation for SSA ice sheet Hessians and benchmarks it spectrally against AD. The derivation is clean, the comparison is rigorous, and the conclusions are appropriately cautious. The main limitation is generalizability from two idealized domains. read the letter →

arxiv 2606.23691 v1 pith:HOEYZGNC submitted 2026-04-20 math.NA cs.NA

Trystan Surawy-Stepney , Stephen L. Cornford This is my paper · ORCID
classification math.NAcs.NA
keywords icesheetmodellingadjointmethodsalgorithmicdifferentiationHessianshallow-streamapproximationuncertaintyquantificationinverseproblemsself-adjoint
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 paper derives a PDE-level second-order adjoint method for the shallow-stream approximation (SSA) of ice sheet flow, called SOSA (second-order self-adjoint), which neglects the nonlinear dependence of ice viscosity on velocity. The motivation is practical: many ice sheet models lack algorithmic differentiation (AD) infrastructure, and SOSA can be implemented as a drop-in using existing forward-model stencils. The authors compare SOSA Hessian-vector products against exact AD-computed Hessians on two synthetic domains. The key finding is that while the two Hessians look superficially similar in direct comparisons, spectral analysis reveals their eigenvector subspaces align only for the first 4 modes (principal angles below π/8), then diverge and become persistently orthogonal after mode 33. Additionally, SOSA eigenvalues are systematically smaller than AD eigenvalues by a factor of approximately 3, attributable to the neglected viscosity nonlinearity (Glen's law exponent n=3). The authors conclude that SOSA is case-dependent in utility, adequate only at very low rank, and that a full Hessian computed via AD should be preferred when high fidelity is required.

What carries the argument

The central objects are the SOSA (second-order self-adjoint) Hessian and the AD (algorithmic differentiation) Hessian for the SSA momentum balance. The comparison tool is principal angle analysis between eigenvector subspaces: the cosine of each principal angle is obtained from the SVD of the matrix formed by the inner products of the two sets of eigenvectors. Small angles mean the subspaces are nearly parallel; π/2 means orthogonal. The eigenvalue ratio of ~3 is traced to the Glen's flow law exponent n=3 entering the viscosity derivative.

What would settle it

If, on a domain with different flow regime (e.g., strong shear or grounding-line dynamics), the eigenvector subspaces were to diverge immediately at mode 1 rather than mode 4, the 'first 4 modes are safe' conclusion would not generalize.

Watch

Extended reading notes

Core claim

The central discovery is a quantitative characterization of how the self-adjoint approximation degrades second-order derivative information. The divergence is not catastrophic—the eigenvector subspaces start close and the divergence does not accelerate after approximately 50 modes—but it is real and structurally significant. The factor-of-3 eigenvalue compression traces directly to the Glen's law exponent, providing a mechanistic explanation for the systematic underestimation of curvature. The subspace alignment pattern (good for 4 modes, orthogonal by mode 33, stable thereafter) also retrospectively explains why first-order self-adjoint gradient methods have worked well in practice: there's

Load-bearing premise

The self-adjoint approximation assumes that the tangent linear operator of the SSA residual is self-adjoint, which holds only when the vertically-averaged effective viscosity is treated as independent of ice velocity. This is false for Glen's flow law with n=3, where viscosity depends nonlinearly on the strain rate and hence on velocity. The entire SOSA derivation and the resulting spectral comparison depend on this linearization.

Editorial extensions

If this is right

  • Ice sheet models using SOSA for Newton-type optimisation should expect reliable curvature information only in the first 3-4 search directions, with progressively degraded directions beyond that.
  • For Bayesian uncertainty quantification requiring the inverse Hessian for posterior covariance, SOSA will systematically underestimate uncertainty in every direction by roughly a factor of 3, and the eigenvector misalignment beyond mode 4 makes faithful covariance reconstruction unlikely without thousands of modes.
  • The factor-of-n eigenvalue compression suggests a simple scaling correction could partially rescue SOSA for leading-mode applications, though the subspace divergence limits its effectiveness at higher rank.
  • The subspace alignment pattern provides a retrospective explanation for the empirical success of self-adjoint approximations in first-order ice sheet inverse problems: gradient methods explore curvature directions sequentially, and the first few are the ones that matter most.

Reading between the lines

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

  • The 'first 4 modes are safe' threshold is established on only two synthetic domains. Domains with stronger shear margins, grounding-line proximity, or different flow regimes could shift this threshold in either direction—particularly if nonlinear viscosity terms contribute more to the Hessian structure in those settings.
  • The factor-of-3 eigenvalue ratio is tied to n=3 (Glen's law). If a different rheological exponent were used, the ratio would scale accordingly, suggesting the eigenvalue compression is a predictable structural feature rather than a domain-specific artifact.
  • The persistent orthogonal subspace after mode 33 suggests there exists a class of curvature directions that are entirely invisible to the self-adjoint approximation—directions driven by velocity-dependent viscosity feedbacks. Identifying what physical structures these eigenvectors correspond to could guide when the approximation is safe versus dangerous.
  • The methodology (principal angle analysis between approximate and exact Hessians) is generalizable beyond ice sheets to any PDE-constrained optimisation where a self-adjoint approximation is used, and could serve as a diagnostic tool for assessing approximation fidelity in other geophysical inverse problems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. This manuscript derives a second-order self-adjoint (SOSA) formulation for the shallow-stream approximation (SSA) of ice sheet flow, in which the nonlinear dependence of viscosity on velocity is neglected (the standard linear-viscosity approximation). The resulting approximate Hessian-vector products are compared against exact Hessians computed via algorithmic differentiation (AD) in a JAX-based finite-volume code. The comparison uses two synthetic test domains (an ice shelf and a snaking ice stream) and employs spectral diagnostics—eigenvalue residuals, orthogonality residuals, and principal angles between invariant subspaces—to quantify the fidelity of the approximation. The main finding is that the SOSA and AD Hessians share similar leading eigenvector structure for approximately the first 4 modes, after which the subspaces diverge, with SOSA eigenvalues systematically smaller by a factor of roughly 3. The authors conclude that the SOSA Hessian is case-dependent in utility and recommend the full AD Hessian where high-fidelity second-order information is needed above very low rank.

Significance. The paper addresses a practically important question for the ice sheet modeling community: whether the widely used self-adjoint (linear-viscosity) approximation, well-established for first-order adjoints, can be extended to second-order derivatives with acceptable fidelity. The derivation (Appendices A–B) is self-contained and follows standard adjoint methodology cleanly. The spectral comparison is rigorous: eigenvalue residuals (Eq. 20) and orthogonality residuals (Eq. 21) confirm numerical quality of both Hessians, and principal angles between subspaces provide a principled similarity metric. The implementation in JAX enables a clean AD ground truth. The finding that subspace divergence begins after approximately 4 modes and the eigenvalue ratio of ~3 are concrete, falsifiable results that provide actionable guidance to practitioners. The honest assessment—that the approximation is case-dependent and inferior to the full Hessian above low rank—is appropriately cautious.

major comments (2)
  1. §5.2, paragraph on the factor-of-3 eigenvalue ratio: the heuristic explanation that '∂uG is roughly a factor of n=3 smaller when derivatives of the viscosity are included' is stated without derivation or reference. This is a load-bearing claim because it is the primary physical explanation for the systematic eigenvalue discrepancy that motivates the recommendation against using SOSA for uncertainty quantification (§6.2). A more precise justification—showing how the viscosity nonlinearity enters ∂uG and why it produces a factor of n in the eigenvalues—should be provided, or the claim should be softened to a conjecture.
  2. §5.2 and §7: the conclusion that subspaces are safe 'up to the first 4 modes' is established on exactly two idealized domains, both with uniform thickness and no grounding line. The abstract and conclusion state this threshold as a general finding. Given that real ice sheet inverse problems involve grounding zones, variable thickness, and thermomechanical coupling, the generalizability of the '4 modes' threshold is a correctness-risk concern. The authors should add a brief statement in the conclusion acknowledging that the threshold is domain-specific and may shift under different flow regimes, or alternatively test a third domain with non-uniform thickness to strengthen the claim.
minor comments (8)
  1. Abstract: 'second order derivatives' and 'second-order derivatives' are both used; please unify hyphenation throughout.
  2. §4.1, Ice Shelf description: 'righ-hand-side' should be 'right-hand-side' (missing 't').
  3. §4.1, Twisty Stream: the formula for C contains nested parentheses that are hard to parse; please verify the closing delimiters match.
  4. Figure 5 caption: panels (a) and (b) labels in the caption text appear swapped relative to the figure layout described; please check consistency.
  5. §5.2: the introduction of the functional J in Eq. (19) comes after results from a different functional (J = ∫√(u·u)) are shown in §5.1; a brief note that the functional changes between subsections would help the reader.
  6. Eq. (9): the definition φ(q) = φ₀e^q is used without specifying the meaning of φ₀ or the units/dimensions of q; a brief clarification would help.
  7. Appendix B.2: the list of 9 terms (Eqs. B.11–B.19) is introduced as '12 terms' earlier in the text; please reconcile the count.
  8. §6.1: 'indetically zero' should be 'identically zero' (also appears in Appendix B.1, Eq. B.6 description).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. Both major comments are well-taken and will be addressed in revision. On the factor-of-3 eigenvalue ratio, we agree the heuristic explanation is insufficiently justified and will either provide a more precise derivation or soften the claim to a conjecture. On the generalizability of the '4 modes' threshold, we agree it is domain-specific and will add an explicit caveat in the conclusion.

read point-by-point responses

  1. Referee: §5.2, paragraph on the factor-of-3 eigenvalue ratio: the heuristic explanation that '∂uG is roughly a factor of n=3 smaller when derivatives of the viscosity are included' is stated without derivation or reference. This is a load-bearing claim because it is the primary physical explanation for the systematic eigenvalue discrepancy that motivates the recommendation against using SOSA for uncertainty quantification (§6.2). A more precise justification—showing how the viscosity nonlinearity enters ∂uG and why it produces a factor of n in the eigenvalues—should be provided, or the claim should be softened to a conjecture.

    Authors: The referee is correct that the factor-of-3 explanation as currently stated is a heuristic assertion without sufficient justification. We will revise this in one of two ways. Our preferred approach is to provide a more precise derivation: the SSA effective viscosity scales as |u|^{(1-n)/n} under Glen's law, so the derivative of viscosity with respect to velocity introduces a factor of (1-n)/n, which for n=3 yields a factor of -2/3. The way this factor enters the operator ∂uG, combined with the structure of the Hessian-vector product (Eq. 17), can plausibly account for the observed ratio. However, we acknowledge that a fully rigorous derivation showing this factor propagates unchanged into the eigenvalues is not straightforward, as it depends on which terms dominate in each eigenmode. If we cannot complete this derivation to our satisfaction, we will instead soften the claim to a conjecture, explicitly stating that the factor-of-3 ratio is consistent with the viscosity nonlinearity scaling but that a rigorous derivation is not provided. Either way, the recommendation in §6.2 against using SOSA for uncertainty quantification does not depend solely on this explanation—it rests on the observed eigenvalue discrepancy itself, which is an empirical result from the spectral comparison. revision: partial

  2. Referee: §5.2 and §7: the conclusion that subspaces are safe 'up to the first 4 modes' is established on exactly two idealized domains, both with uniform thickness and no grounding line. The abstract and conclusion state this threshold as a general finding. Given that real ice sheet inverse problems involve grounding zones, variable thickness, and thermomechanical coupling, the generalizability of the '4 modes' threshold is a correctness-risk concern. The authors should add a brief statement in the conclusion acknowledging that the threshold is domain-specific and may shift under different flow regimes, or alternatively test a third domain with non-uniform thickness to strengthen the claim.

    Authors: We agree that the '4 modes' threshold is established on only two idealized domains and should not be stated as a general finding without qualification. We will add an explicit caveat in both the abstract and the conclusion acknowledging that this threshold is domain-specific and may shift under different flow regimes, particularly in the presence of grounding zones, variable thickness, or thermomechanical coupling. We considered adding a third test domain with non-uniform thickness, but we feel this is beyond the scope of a minor revision and would not by itself resolve the generalizability question—two or three idealized domains cannot establish a universal threshold. The honest and appropriate response is to qualify the claim rather than to overstate it. We will also adjust the language in §5.2 to make clear that the threshold of approximately 4 modes is specific to the Twisty Stream domain and that the broader, more robust finding is that subspace divergence begins at low rank and does not accelerate thereafter. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SOSA derivation is self-contained and validated against an independent AD-computed Hessian benchmark

full rationale

The paper derives a second-order self-adjoint (SOSA) approximation for the SSA ice sheet equations and compares its Hessian against an independently computed AD Hessian. The self-adjoint approximation (Eq. A.12) is explicitly stated as a modeling choice — freezing the viscosity μ̄ so that G is linear in u — and is the central object of study, not a concealed premise. No parameters are fitted to make the comparison work; the factor-of-3 eigenvalue ratio emerges from Glen's exponent n=3 in the physics, not from calibration. The spectral comparison (principal angles, eigenvalue residuals, orthonormality checks) uses standard linear algebra diagnostics applied to two independently constructed matrices. The derivation chain (Appendices A–B) proceeds from standard variational calculus: the Lagrangian (A.4) yields the FOA system (A.8), differentiation yields the SOA system (B.9–B.10), and the self-adjoint approximation simplifies terms but does not define the result in terms of itself. Citations are to external literature (MacAyeal, Cacuci, Petra et al., etc.) for standard adjoint methodology, not to a self-citation chain that would be load-bearing. The paper's conclusions (divergence after ~4 modes, eigenvalue ratio ~3, case-dependent utility) are empirical findings from the comparison, not definitions restated as predictions.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new entities are introduced. The SOSA method is a reformulation of existing adjoint methodology applied to existing equations (SSA).

free parameters (2)
  • Glen's flow law exponent n = 3
    Standard physical choice for ice rheology, not fitted to data in this paper. Used throughout the forward model and explains the factor-of-3 eigenvalue scaling.
  • Domain geometry and slipperiness fields = Ice Shelf: 500m uniform thickness, C=0 or 10^4; Twisty Stream: 1km thickness, C formula with epsilon=5e-3, R=180km, m=1/
    Synthetic test domains chosen by the authors. Not fitted to real data but designed to represent typical ice shelf and ice stream configurations.
assumptions (4)
  • domain assumption Self-adjoint approximation: (∂uG(u))*[λ] = G(λ), i.e., the tangent linear operator of the SSA residual is self-adjoint
    Invoked in Appendix A.1 (Eq. A.12) and carried through the entire SOSA derivation (Eqs. 15-17, B.29-B.37). This is the central approximation that makes the adjoint equations tractable but neglects nonlinear viscosity dependence on velocity.
  • domain assumption Functional J is separable in u and q, so ∂q∂uJ = 0
    Stated in Section 2.2.2 and Appendix B.2. Simplifies the SOA system by eliminating cross-derivative terms of the cost functional.
  • standard math Standard first- and second-order adjoint methodology (Lagrange multiplier approach for PDE-constrained optimization)
    The derivation framework in Appendices A-B follows standard optimal control theory (Lions 1971, Cacuci 2015/2016). No novel mathematical framework is introduced.
  • domain assumption Boundary conditions on adjoint variables match those on the forward variables, causing boundary terms from integration by parts to vanish
    Invoked in Section 2.2.1 and Appendix B.2. Required for the self-adjointness to hold and for the gradient/HVP expressions to be valid.

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Cite this review

Pith. "Pith review of Exact vs approximate second-order derivatives in vertically-integrated ice sheet models." pith.science (2026). https://pith.science/paper/HOEYZGNC

@misc{pith2026260623691,
  author       = {Pith},
  title        = {Pith review of: Exact vs approximate second-order derivatives in vertically-integrated ice sheet models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HOEYZGNC}},
  note         = {Machine review of arXiv:2606.23691}
}
read the original abstract

Second order derivatives of model outputs with respect to input parameters are key to several applications in ice sheet modelling. For example, the ability to compute Hessian-vector products broadens the list of available optimisation methods, and facilitates certain kinds of parametric uncertainty quantification. Some modern ice sheet models are built on frameworks supporting algorithmic differentiation (AD), allowing for the computation of higher order derivatives with relative ease. However, many of our most widely-used models are not. A natural alternative might be to follow common practise in first order gradient computation and construct an approximate second-order adjoint model at the PDE level, which neglects the nonlinear dependence of ice viscosity on velocity. Here, we present such a model for the shallow-stream approximation allowing one to compute approximate second-order derivatives, and compare with full second-order derivates found using AD. We find that this produces Hessian-vector products that are superficially similar to those computed via AD. However, an analysis of the spectral decomposition of the Hessians calculated in each way reveals that the subspaces spanned by their eigenvectors diverge after the leading 4 modes, though divergence does not accelerate after this. We conclude that the utility of the approximate Hessian is case-dependent, and a full Hessian, likely computed using AD, should be used where high fidelity is required above very low rank.

Figures

Figures reproduced from arXiv: 2606.23691 by the authors.

Figure 1
Figure 1. Test domains for our experiments. a) The Ice Shelf experiment. Ice is uniform [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Visual comparison of Hessian-vector products calculated using the AD and SOSA [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The first 10 eigenvectors of the Hessian computed via the AD (upper) and SOSA [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The first 500 eigenvalues of the AD- and SOSA-derived Hessians, for the functional [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: The similarity of subspaces spanned by the first [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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Pith tools

Reviewed July 5, 2026 · model on record in the stance chip above.