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Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty
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.Comment: 26 Page
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