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The majority of research on efficient and scalable algorithms incomputational science and engineering has focused on the forward problem: givenparameter inputs, solve the governing equations to determine output quantitiesof interest. In contrast, here we consider the broader question: given a(large-scale) model containing uncertain parameters, (possibly) noisyobservational data, and a prediction quantity of interest, how do we constructefficient and scalable algorithms to (1) infer the model parameters from thedata (the deterministic inverse problem), (2) quantify the uncertainty in theinferred parameters (the Bayesian inference problem), and (3) propagate theresulting uncertain parameters through the model to issue predictions withquantified uncertainties (the forward uncertainty propagation problem)? Wepresent efficient and scalable algorithms for this end-to-end,data-to-prediction process under the Gaussian approximation and in the contextof modeling the flow of the Antarctic ice sheet and its effect on sea level.The ice is modeled as a viscous, incompressible, creeping, shear-thinningfluid. The observational data come from InSAR satellite measurements of surfaceice flow velocity, and the uncertain parameter field to be inferred is thebasal sliding parameter. The prediction quantity of interest is the present-dayice mass flux from the Antarctic continent to the ocean. We show that the workrequired for executing this data-to-prediction process is independent of thestate dimension, parameter dimension, data dimension, and number of processorcores. The key to achieving this dimension independence is to exploit the factthat the observational data typically provide only sparse information on modelparameters. This property can be exploited to construct a low rankapproximation of the linearized parameter-to-observable map.

journal of computational physics

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