  We consider optimal design of infinite-dimensional Bayesian linear inverse
problems governed by partial differential equations that contain secondary
reducible model uncertainties, in addition to the uncertainty in the inversion
parameters. By reducible uncertainties we refer to parametric uncertainties
that can be reduced through parameter inference. We seek experimental designs
that minimize the posterior uncertainty in the primary parameters, while
accounting for the uncertainty in secondary parameters. We accomplish this by
deriving a marginalized A-optimality criterion and developing an efficient
computational approach for its optimization. We illustrate our approach for
estimating an uncertain time-dependent source in a contaminant transport model
with an uncertain initial state as secondary uncertainty. Our results indicate
that accounting for additional model uncertainty in the experimental design
process is crucial.
