Exact vs approximate second-order derivatives in vertically

Mathematics > Numerical Analysis
[Submitted on 20 Apr 2026]
Title:Exact vs approximate second-order derivatives in vertically-integrated ice sheet models
View PDF HTML (experimental)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.
Submission history
From: Trystan Surawy-Stepney [view email][v1] Mon, 20 Apr 2026 19:16:42 UTC (1,196 KB)
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Facts Only

Second order derivatives of model outputs with respect to input parameters are key to ice sheet modelling applications. Algorithmic differentiation (AD) allows for the computation of higher-order derivatives with relative ease in some models. The study presents an approximate second-order adjoint model for the shallow-stream approximation. This model is used to compute approximate second-order derivatives and is compared against full second-order derivatives found using AD. An analysis of the spectral decomposition of the Hessians calculated by both methods reveals that the subspaces spanned by their eigenvectors diverge after the leading four modes. The conclusion is that the utility of the approximate Hessian is case-dependent.

Executive Summary

Second-order derivatives of model outputs with respect to input parameters are crucial for applications in ice sheet modeling, such as computing Hessian-vector products and parametric uncertainty quantification. The study contrasts methods for computing these derivatives in vertically-integrated ice sheet models: full second-order derivatives found using algorithmic differentiation (AD) versus an approximate adjoint model approach. The research introduces an approximate second-order adjoint model for the shallow-stream approximation to compute derivatives, comparing its results with AD. While the resulting Hessian-vector products appear superficially similar, a deeper analysis of the spectral decomposition reveals that the subspaces spanned by the eigenvectors diverge after the leading four modes. The authors conclude that the utility of the approximate Hessian is case-dependent and that full Hessians, likely computed via AD, should be used where high fidelity is required beyond low rank.

Full Take

The research highlights a critical tension between computational efficiency and mathematical fidelity in large-scale physical modeling, specifically within ice sheet simulations. The finding that approximate methods yield results superficially similar to exact methods but diverge in their underlying spectral structure (eigenvector subspaces) points toward subtle, high-order nonlinear dependencies that are missed by the approximation. This suggests that while the approximate Hessian might suffice for certain low-rank applications, relying on it when high fidelity is demanded risks misrepresenting the true sensitivity landscape of the system. The divergence observed beyond the first four modes implies that structural information critical to the full dynamics is lost in the approximation, meaning the error in Hessian computation is not uniform across all modes but concentrated in the higher-order structure. This reinforces the necessity for a principled approach: computational methods must be carefully matched to the required physical fidelity, acknowledging that approximations may introduce systematic errors into parametric uncertainty quantification if used carelessly. The utility depends entirely on whether low-rank approximation is sufficient to capture physically relevant information or if the full Hessian is necessary to avoid large divergence in sensitivity analysis.

Sentinel — Human

Confidence

The text exhibits the characteristic density and nuanced argument structure of human-authored scientific research, focusing on the technical comparison of computational methods within a specialized field.

Signals Detected

Sentence structure is dense and follows academic conventions; complexity reflects domain expertise rather than mechanical rhythm.

The argument flows logically from problem definition (need for derivatives) to proposed solution comparison (approximate vs. full Hessian) to conclusion (case-dependent utility).

Use of specific technical terms (Hessian-vector products, algorithmic differentiation, spectral decomposition) is integrated smoothly without resorting to vague attribution.

The content structure matches a genuine scientific submission format; no immediate signs of LLM confabulation or manufactured citation patterns were detected in the provided metadata.

Human Indicators

Specific, high-level domain knowledge (Numerical Analysis, ice sheet modeling) is utilized effectively.

The conclusion relies on a nuanced analysis of mathematical properties (divergence of subspaces after 4 modes), which demonstrates specific research insight rather than generic synthesis.