Tractable optimal experimental design using transport maps

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.