### Abstract

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 loss of grounded ice to the ocean. The 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 to be inferred is the basal sliding parameter, represented by a heterogeneous coefficient in a Robin boundary condition at the base of the ice sheet. 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-measured in number of forward (and adjoint) ice sheet model solves-is independent of the state dimension, parameter dimension, data dimension, and the number of processor cores. The key to achieving this dimension independence is to exploit the fact that, despite their large size, 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 via randomized SVD methods and adjoint-based actions of Hessians of the data misfit functional.

Original language | English (US) |
---|---|

Pages (from-to) | 348-368 |

Number of pages | 21 |

Journal | Journal of Computational Physics |

Volume | 296 |

DOIs | |

State | Published - Sep 1 2015 |

### Fingerprint

### Keywords

- Adjoint-based Hessian
- Antarctic ice sheet
- Bayesian inference
- Data-to-prediction
- Ice sheet flow modeling
- Inexact Newton-Krylov method
- Inverse problems
- Low-rank approximation
- Nonlinear Stokes equations
- Preconditioning
- Uncertainty quantification

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Cite this

*Journal of Computational Physics*,

*296*, 348-368. https://doi.org/10.1016/j.jcp.2015.04.047

**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.** / Isaac, Tobin; Petra, Noemi; Stadler, Georg; Ghattas, Omar.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 296, pp. 348-368. https://doi.org/10.1016/j.jcp.2015.04.047

}

TY - JOUR

T1 - 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

AU - Isaac, Tobin

AU - Petra, Noemi

AU - Stadler, Georg

AU - Ghattas, Omar

PY - 2015/9/1

Y1 - 2015/9/1

N2 - 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 loss of grounded ice to the ocean. The 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 to be inferred is the basal sliding parameter, represented by a heterogeneous coefficient in a Robin boundary condition at the base of the ice sheet. 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-measured in number of forward (and adjoint) ice sheet model solves-is independent of the state dimension, parameter dimension, data dimension, and the number of processor cores. The key to achieving this dimension independence is to exploit the fact that, despite their large size, 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 via randomized SVD methods and adjoint-based actions of Hessians of the data misfit functional.

AB - 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 loss of grounded ice to the ocean. The 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 to be inferred is the basal sliding parameter, represented by a heterogeneous coefficient in a Robin boundary condition at the base of the ice sheet. 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-measured in number of forward (and adjoint) ice sheet model solves-is independent of the state dimension, parameter dimension, data dimension, and the number of processor cores. The key to achieving this dimension independence is to exploit the fact that, despite their large size, 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 via randomized SVD methods and adjoint-based actions of Hessians of the data misfit functional.

KW - Adjoint-based Hessian

KW - Antarctic ice sheet

KW - Bayesian inference

KW - Data-to-prediction

KW - Ice sheet flow modeling

KW - Inexact Newton-Krylov method

KW - Inverse problems

KW - Low-rank approximation

KW - Nonlinear Stokes equations

KW - Preconditioning

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=84929377879&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929377879&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2015.04.047

DO - 10.1016/j.jcp.2015.04.047

M3 - Article

VL - 296

SP - 348

EP - 368

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -