Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

Abstract

We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unfied framework. We build on the Bayesian approximation error (BAE) framework, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

Hyper-differential sensitivity analysis for nonlinear Bayesian inverse problems

“…Suitable approximations of the posterior, such as a Laplace approximation, can be considered, to mitigate the high cost of HDSA of such quantities in large-scale nonlinear inverse problems. Another interesting line of inquiry is to use HDSA within the context of optimal experimental design (OED) under uncertainty [30,31]. HDSA can reveal model uncertainties that the OED criterion is most sensitive to and thus must be accounted for in the optimal design process.…”

A Greedy Sensor Selection Algorithm for Hyperparameterized Linear Bayesian Inverse Problems

“…Our proposed algorithm is directly related to the orthogonal matching pursuit (OMP) algorithm [8,9] for the parameterizedbackground data-weak (PBDW) method and the empirical interpolation method (EIM) ( [10,11]). Closely related OED methods for linear Bayesian inverse problems over partial differential equations (PDEs) include [12,13,14,15,16,17], mostly for A-and D-OED and uncorrelated noise. In recent years, these methods have also been extended to non-linear Bayesian inverse problems, e.g., [18,19,20,21,22], while an advance to correlated noise has been made in [23].…”

Edge-Promoting Adaptive Bayesian Experimental Design for X-ray Imaging

“…In addition to the works mentioned above, let us list [8,11,22,23,24,25,30,32,36,37,9] to name a few papers on this topic. In particular, there is an interesting line of research developing Bayesian OED for infinite-dimensional inverse problems [5,4,6,7]. Here, we test our novel ideas in a sequential optimization strategy, which has previously been formalized for large-scale problems in [29] based on ideas from dynamical programming.…”