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) Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.