Source-linked AI summary
Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet
Tobin Isaac, Noemi Petra, Georg Stadler, Omar Ghattas
TL;DR
Large-scale ice-sheet models require efficient methods to infer uncertain parameters from noisy observations, quantify posterior uncertainty, and propagate it to predictions. The paper develops a Gaussian, low-rank data-to-prediction framework for Antarctic ice flow and shows that its solve cost is independent of state, parameter, data, and processor-core dimensions. The approach is demonstrated by inferring basal sliding from InSAR velocities and predicting Antarctic ice mass flux with quantified uncertainty.
Problem
Large-scale nonlinear ice-sheet models make end-to-end inference, uncertainty quantification, and prediction computationally intractable for high-dimensional parameter fields.
Method
The paper linearizes parameter-to-observable and parameter-to-prediction maps around the MAP point, uses Gaussian approximations, and exploits low-rank randomized SVD structure.
Results
The data-to-prediction framework has a cost in forward and adjoint Stokes solves that is independent of parameter and data dimensions, and the complete workflow is demonstrated for Antarctic ice mass flux.
Takeaways & Limitations
Sparse information in observations can be exploited to make uncertainty-aware Antarctic ice-sheet prediction scalable across the full inference-to-prediction workflow.
Takeaways & Limitations
The uncertainty analysis relies on linearizations and Gaussian approximations, while scalable treatment of the resulting non-Gaussian distributions remains open.
Abstract
from arXiv · showhide
The majority of research on efficient and scalable algorithms in computational science and engineering has focused on the forward problem: given parameter inputs, solve the governing equations to determine output quantities of interest. In contrast, here we consider the broader question: given a (large-scale) model containing uncertain parameters, (possibly) noisy observational data, and a prediction quantity of interest, how do we construct efficient and scalable algorithms to (1) infer the model parameters from the data (the deterministic inverse problem), (2) quantify the uncertainty in the inferred parameters (the Bayesian inference problem), and (3) propagate the resulting uncertain parameters through the model to issue predictions with quantified uncertainties (the forward uncertainty propagation problem)? We present efficient and scalable algorithms for this end-to-end, data-to-prediction process under the Gaussian approximation and in the context of modeling the flow of the Antarctic ice sheet and its effect on sea level. The ice is modeled as a viscous, incompressible, creeping, shear-thinning fluid. The observational data come from InSAR satellite measurements of surface ice flow velocity, and the uncertain parameter field to be inferred is the basal sliding parameter. The prediction quantity of interest is the present-day ice mass flux from the Antarctic continent to the ocean. We show that the work required for executing this data-to-prediction process is independent of the state dimension, parameter dimension, data dimension, and number of processor cores. The key to achieving this dimension independence is to exploit the fact that the observational data typically provide only sparse information on model parameters. This property can be exploited to construct a low rank approximation of the linearized parameter-to-observable map.
1. Introduction
The paper addresses the computational difficulty of propagating uncertainty from observations through parameter inference to predictions in large-scale Antarctic ice-sheet models. It develops a scalable data-to-prediction framework that uses Gaussian approximations and low-rank structure to make this process dimension-independent.
- Motivation: High-dimensional Bayesian inference and uncertainty propagation are computationally intractable when complex nonlinear ice-sheet PDEs require repeated forward solves.Direct covariance calculations can require solves proportional to the parameter or data dimension.
- Motivation: Uncertainty in ice-sheet model predictions is essential for projections of future sea-level rise, but current models contain substantial parameter and structural uncertainties.Unknown fields such as basal sliding, topography, geothermal heat flux, and rheology must often be inferred from satellite observations through severely ill-posed inverse problems.
- Result: The number of forward and adjoint solves is independent of state, parameter, data, and processor-core dimensions when the underlying solver is scalable.The computational cost is dominated by forward and adjoint ice-sheet model solves.
- Approach: The framework linearizes parameter-to-observable and parameter-to-prediction maps around the maximum a posteriori point to obtain Gaussian approximations.These approximations support both posterior uncertainty quantification and prediction uncertainty propagation.
- Approach: A matrix-free randomized SVD constructs a low-rank approximation of the parameter-to-observable map because observations inform only a limited number of parameter-space directions.The resulting computational work depends on the information dimension rather than the full parameter or data dimension.
- Application: The framework applies the complete workflow to Antarctic ice flow, using satellite surface velocities to infer basal sliding and predict present-day ice mass flux with quantified uncertainty.The workflow includes inversion, Bayesian parameter uncertainty quantification, and forward propagation to the prediction quantity.
2. Forward problem: Modeling ice sheet flow
The forward problem models Antarctic ice as a nonlinear, viscous, incompressible, shear-thinning fluid and solves the resulting Stokes equations with scalable high-order discretization and iterative solvers. The solver remains effective across very large refined meshes and processor counts, although multigrid setup limits parallel scalability at larger core counts.
- Role in framework: The forward Stokes solver is the fundamental computational kernel repeatedly used by the inverse, uncertainty-quantification, and prediction stages.Its scalability is therefore required for the end-to-end data-to-prediction framework.
- Physical model: Ice flow is modeled as a viscous, shear-thinning, incompressible fluid governed by mass and momentum balance.The constitutive law uses strain-rate- and temperature-dependent effective viscosity.
- Boundary conditions: The basal sliding parameter enters a Robin boundary condition through exp(β), which represents resistance to basal sliding.This coefficient aggregates effects including friction, bedrock roughness, deforming till, and subglacial water conditions.
- Discretization: The discretization uses high-order, locally volume-conserving Qk × Qdisc k−2 finite elements on locally refined hexahedral meshes.The mesh strategy controls element aspect ratios and refines the Antarctic geometry with the mesh.
- Solver: An inexact Newton-Krylov method with matrix-free residual and operator applications solves the nonlinear systems, while block preconditioning supports iterative linear solves.The preconditioner combines algebraic multigrid for the viscous block with a viscosity-weighted lumped-mass Schur approximation.
3. Inverse problem: Inferring the basal sliding parameter from observed ice surface velocity
The paper formulates a regularized inverse problem to infer Antarctic basal sliding from satellite surface-velocity observations and solves it with adjoint-based, inexact Newton-CG methods. The inversion scales across increasing problem dimensions, reconstructs spatially varying sliding resistance, and fits velocity data well in sea-level-relevant regions, while the ill-posedness prevents exact recovery of the true solution.
- Problem formulation: The inverse problem infers the basal sliding field β from observed surface velocity using a nonlinear least-squares formulation with the forward Stokes model and a surface observation operator.The misfit is normalized because observed velocities span three or more orders of magnitude.
- Problem formulation: Regularization penalizes oscillatory basal sliding components, imposes smoothness, and addresses the ill-posedness caused by the smoothing map from β to surface velocity.The parameter γ controls the smoothness–misfit tradeoff, while δ makes the regularization operator invertible.
- Inverse problem solver: The inexact Newton-CG solver computes gradients through adjoint Stokes equations and Hessian actions matrix-free through second-order adjoints.This avoids forming the Hessian explicitly while retaining the operator action required by CG.
- Inversion results: Scaling from 10K to 1.5M inversion parameters and 96K to 23M states leaves Newton and CG iteration counts insensitive to parameter and data dimensions.Consequently, the number of Stokes solves does not increase with inverse-problem size.
- Inversion results: The inferred basal sliding coefficient varies over nine orders of magnitude, with weak resistance extending deep into the continental interior and correlating with faster flow.Low and high parameter values represent low and high resistance to basal sliding, respectively.
- Inversion results: The inversion fits observations particularly well in fast-flowing ice streams and outlet glaciers that deliver most mass flux to the ocean, but the ill-posed problem prevents exact recovery of the true solution.Bayesian inference is introduced to quantify uncertainty that the deterministic inversion cannot provide.
4. Bayesian quantification of parameter uncertainty: estimating the posterior pdf of the basal sliding parameter field
The paper formulates Bayesian inference for the basal sliding field and uses a low-rank Hessian approximation to make posterior uncertainty computationally tractable at massive parameter dimensions. The resulting posterior uncertainty is characterized through data-informed parameter directions and scalable covariance operations.
- Bayesian formulation: Bayesian inference combines a prior distribution over basal sliding parameters with a likelihood for the observed surface-velocity data to form a posterior distribution.The posterior need not be Gaussian because the parameter-to-observable map is nonlinear.
- Gaussian approximation: The Gaussian posterior approximation is accurate when the parameter-to-observable map is nearly linear over the posterior support, including weakly and strongly data-informed directions.The approximation is based on linearization around the MAP point.
- Low-rank covariance approximation: A dense posterior covariance is intractable because explicitly constructing the parameter-to-observable Jacobian would require n forward PDE solves, but Jacobian and adjoint actions require only linearized forward and adjoint solves.The data-misfit Hessian is typically compact, so its eigenvalues decay toward zero and only a limited number of parameter directions are strongly informed.
- Low-rank covariance approximation: The low-rank posterior update retains parameter modes that are both well informed by the data and assigned high probability by the prior.Posterior uncertainty is interpreted as prior uncertainty less the information gained from the data, filtered through the prior.
- Scalability: The randomized SVD requires matrix-vector products proportional to the effective rank, which can remain independent of parameter dimension when the prior-preconditioned Hessian retains spectral decay.After the eigenpairs are computed, posterior uncertainty queries require only low-rank inner products plus scalable prior-covariance solves.
- Computational results: 5,000 Hessian-matrix products produced the covariance approximation using 10,000 incremental forward/adjoint Stokes solves, about an order of magnitude less than finding the MAP point.Posterior samples show smaller basal-sliding variability in fast-flow regions and larger variability in central and West Antarctica.
5. Prediction with quantified uncertainty: forward propagation of basal sliding parameter uncertainty to mass flux prediction
The paper propagates uncertainty in the inferred basal sliding field to Antarctic ice-mass-flux predictions using a Gaussian approximation and linearization at the MAP point. Low-rank Hessian structure and adjoint solves make this prediction step tractable while distinguishing parameter sensitivity from uncertainty.
- Gaussian prediction approximation: The prediction pdf is approximated as Gaussian by linearizing the parameter-to-prediction map at the MAP estimate.The mean is obtained from the forward model at βMAP, while uncertainty uses the prediction Jacobian and inverse Hessian.
- Adjoint-based propagation: The prediction gradient is computed with a forward solve followed by an adjoint problem for the quantity of interest.The adjoint Stokes solver can be reused because the prediction uses the same ice-flow model as the inference problem.
- Low-rank computation: The inverse Hessian action is formed from its compact low-rank representation, avoiding incremental Stokes solves required by iterative linear-system methods.This reduces the cost of forming H^-1(βMAP)F*(βMAP) to linear algebra after the low-rank approximation is available.
- Prediction-influential directions: The influential prediction mode combines sensitivity of the mass-flux quantity with uncertainty in the basal sliding field.It is computed as W = Σ^-1/2H(βMAP)^-1F*(βMAP) and shown for multiple Antarctic outflow regions.
- Mass-flux predictions: The predicted mean ± standard deviation is 1170.83 ± 1080.09 Gt/a for all Antarctica, 359.60 ± 1.02 Gt/a for East Antarctica, and 71.24 ± 0.30 Gt/a for the Totten Glacier.Using the prior covariance instead gives standard deviations of 1327.22, 525.24, and 179.73 Gt/a, respectively.
- Mass-flux predictions: Sensitivity and uncertainty identify different influential regions: central and West Antarctica are more uncertain, while East Antarctic modes reflect mixed uncertainty and sensitivity.The top-row gradients show sensitivity alone; the bottom-row modes incorporate both sensitivity and posterior uncertainty.
6. Conclusions
The paper presents an end-to-end Antarctic ice-sheet data-to-prediction framework whose Stokes-solve cost is independent of parameter, data, and state dimensions, as well as processor count. This scalability relies on low-dimensional information content and low-rank Hessian structure, while the uncertainty analysis remains Gaussian and linearized.
- Conclusions: The framework infers parameters from observations, quantifies posterior uncertainty, and propagates that uncertainty to predictions with a constant number of forward or adjoint Stokes solves.With a scalable forward solver, the cost is also independent of state dimension and processor count.
- Conclusions: The algorithms exploit low-dimensional data information, enabling dimension-independent Newton-CG optimization and randomized-SVD Hessian approximation.Prediction uncertainty is then obtained from the inverse Hessian acting on an adjoint-derived prediction gradient.
- Conclusions: The uncertainty analysis uses linearized parameter-to-observable and parameter-to-prediction maps, producing Gaussian approximations.Extending the approach to scalable treatment of non-Gaussian distributions remains an open question.