Noemi Petra

Ruanui Nicholson, Noemi Petra, Jari Kaipio

We consider the reconstruction of a heterogeneous coefficient field in a Robin boundary condition on an inaccessible part of the boundary in a Poisson problem with an uncertain (or unknown) inhomogeneous conductivity field in the interior of the domain. To account for model errors that stem from the uncertainty in the conductivity coefficient, we treat the unknown conductivity as a nuisance parameter and carry out approximative premarginalization over it, and invert for the Robin coefficient field only. We approximate the related modelling errors via the Bayesian approximation error (BAE) approach. The uncertainty analysis presented here relies on a local linearization of the parameter-to-observable map at the maximum a posteriori (MAP) estimates, which leads to a normal (Gaussian) approximation of the parameter posterior density. To compute the MAP point we apply an inexact Newton conjugate gradient approach based on the adjoint methodology. The construction of the covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. Two numerical experiments are considered: one where the prior covariance on the conductivity is isotropic, and one where the prior covariance on the conductivity is anisotropic. Results are compared to those based on standard error models, with particular emphasis on the feasibility of the posterior uncertainty estimates. We show that the BAE approach is a feasible one in the sense that the predicted posterior uncertainty is consistent with the actual estimation errors, while neglecting the related modelling error yields infeasible estimates for the Robin coefficient. In addition, we demonstrate that the BAE approach is approximately as computationally expensive (measured in the number of PDE solves) as the conventional error approach.

Terrence Alsup, Tucker Hartland, Benjamin Peherstorfer et al.

Multilevel Stein variational gradient descent is a method for particle-based variational inference that leverages hierarchies of surrogate target distributions with varying costs and fidelity to computationally speed up inference. The contribution of this work is twofold. First, an extension of a previous cost complexity analysis is presented that applies even when the exponential convergence rate of single-level Stein variational gradient descent depends on iteration-varying parameters. Second, multilevel Stein variational gradient descent is applied to a large-scale Bayesian inverse problem of inferring discretized basal sliding coefficient fields of the Arolla glacier ice. The numerical experiments demonstrate that the multilevel version achieves orders of magnitude speedups compared to its single-level version.

Tobin Isaac, Noemi Petra, Georg Stadler et al.

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.