Solving large-scale Bayesian inverse problems governed by complex models suffers from the twin difficulties of the high dimensionality of the uncertain parameters and computationally expensive forward models. In this talk, we focus on 1. reducing the computational cost when solving these problems (via joint parameter and state dimension reduction) and 2. accounting for the error due to using a reduced order forward model (via Bayesian Approximation Error (BAE)). To reduce the parameter dimension, we exploit the underlying problem structure (e.g., local sensitivity of the data to parameters, the smoothing properties of the forward model, the fact that the data contain limited information about the (infinite-dimensional) parameter field, and the covariance structure of the prior) and identify a likelihood-informed parameter subspace that shows where the change from prior to posterior is most significant. For the state dimension reduction, we employ a proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM) to approximate the nonlinear term in the forward model. We illustrate our approach with a model ice sheet inverse problem governed by the nonlinear Stokes equation for which the basal sliding coefficient field (a parameter that appears in a Robin boundary condition at the base of the geometry) is inferred from the surface ice flow velocity. The results show the potential to make the exploration of the full posterior distribution of the parameter or subsequent predictions more tractable.
This is joint work with Ki-Tae Kim (UC Merced), Benjamin Peherstorfer (NYU) and Tiangang Cui (Monash University).