We consider the problem of optimal experimental design (OED) for infinite-dimensional Bayesian linear inverse problems governed by PDEs that contain model uncertainties, in addition to the uncertainty in the inversion parameters. The focus will be on the case where these additional (secondary) uncertainties are reducible; such parametric uncertainties can, in principle, be reduced through parameter estimation. We seek experimental designs that minimize the posterior uncertainty in the inversion parameters, while accounting for the uncertainty in the secondary parameters. To accomplish this, we derive a marginalized A-optimality criterion and present an efficient computational approach for its optimization. We illustrate our proposed methods in a problem involving inference of an uncertain time-dependent source in a contaminant transport model, with an uncertain initial state as secondary uncertainty. We will also discuss a sensitivity analysis framework that can help reduce the complexity of inversion and design of experiments under additional model uncertainties.
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