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We consider optimal experimental design (OED) for Bayesian nonlinear inverseproblems governed by partial differential equations (PDEs) under modeluncertainty. Specifically, we consider inverse problems in which, in additionto the inversion parameters, the governing PDEs include secondary uncertainparameters. We focus on problems with infinite-dimensional inversion andsecondary parameters and present a scalable computational framework for optimaldesign of such problems. The proposed approach enables Bayesian inversion andOED under uncertainty within a unified framework. We build on the Bayesianapproximation error (BAE) approach, to incorporate modeling uncertainties inthe Bayesian inverse problem, and methods for A-optimal design ofinfinite-dimensional Bayesian nonlinear inverse problems. Specifically, aGaussian approximation to the posterior at the maximum a posteriori probabilitypoint is used to define an uncertainty aware OED objective that is tractable toevaluate and optimize. In particular, the OED objective can be computed at acost, in the number of PDE solves, that does not grow with the dimension of thediscretized inversion and secondary parameters. The OED problem is formulatedas a binary bilevel PDE constrained optimization problem and a greedyalgorithm, which provides a pragmatic approach, is used to find optimaldesigns. We demonstrate the effectiveness of the proposed approach for a modelinverse problem governed by an elliptic PDE on a three-dimensional domain. Ourcomputational results also highlight the pitfalls of ignoring modelinguncertainties in the OED and/or inference stages.

Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty