09/01/2015

# Scalable, Efficient Algorithms for Propagation of Uncertainty from Data Through Inference to Prediction for Large-Scale Problems

## Summary

Most of the research on efficient, 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 a recent study, researchers considered the broader question: given a model containing uncertain parameters, noisy observational data, and a prediction quantity of interest (QOI), how can efficient and scalable algorithms be constructed to (1) infer model parameters from the data (the deterministic inverse problem), (2) quantify uncertainty in the inferred parameters (the Bayesian inference problem), and (3) propagate the resulting uncertain parameters through the model for predictions with quantified uncertainties on the QOI (the forward uncertainty propagation problem)? The researchers developed efficient, scalable algorithms for this end-to-end, data-to prediction process in the context of modeling the flow of the Antarctic ice sheet and its effect on loss of grounded ice to the ocean. Ice is modeled as a viscous, incompressible, creeping, shear-thinning fluid, the observational data come from satellite measurements of surface ice flow velocity, and the uncertain parameter field inferred is a basal sliding parameter, represented by a heterogeneous coefficient in a Robin boundary condition at the ice sheet’s base. The QOI is the present-day ice mass flux from the Antarctic continent to the ocean. 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, despite their large size, observational data typically provide sparse information on model parameters. This property is exploited to construct a low rank approximation of the parameter-to-observable map via randomized singular value decomposition (SVD) methods and adjoint-based actions of Hessians of the data misfit functional.

## References

Isaac, T., N. Petra, G. Stadler, and O. Ghattas, 2015. “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,” *Journal of Computational Physics* **296**, 348–68. DOI: 10.1016/j.jcp.2015.04.047.