  We present a flexible method for computing Bayesian optimal experimental
designs (BOEDs) for inverse problems with intractable posteriors. The approach
is applicable to a wide range of BOED problems and can accommodate various
optimality criteria, prior distributions and noise models. The key to our
approach is the construction of a transport-map-based surrogate to the joint
probability law of the design, observational and inference random variables.
This order-preserving transport map is constructed using tensor trains and can
be used to efficiently sample from (and evaluate approximate densities of)
conditional distributions that are required in the evaluation of many
commonly-used optimality criteria. The algorithm is also extended to sequential
data acquisition problems, where experiments can be performed in sequence to
update the state of knowledge about the unknown parameters. The sequential BOED
problem is made computationally feasible by preconditioning the approximation
of the joint density at the current stage using transport maps constructed at
previous stages. The flexibility of our approach in finding optimal designs is
illustrated with some numerical examples inspired by disease modeling and the
reconstruction of subsurface structures in aquifers.
