  Obtaining samples from the posterior distribution of inverse problems with
expensive forward operators is challenging especially when the unknowns involve
the strongly heterogeneous Earth. To meet these challenges, we propose a
preconditioning scheme involving a conditional normalizing flow (NF) capable of
sampling from a low-fidelity posterior distribution directly. This conditional
NF is used to speed up the training of the high-fidelity objective involving
minimization of the Kullback-Leibler divergence between the predicted and the
desired high-fidelity posterior density for indirect measurements at hand. To
minimize costs associated with the forward operator, we initialize the
high-fidelity NF with the weights of the pretrained low-fidelity NF, which is
trained beforehand on available model and data pairs. Our numerical
experiments, including a 2D toy and a seismic compressed sensing example,
demonstrate that thanks to the preconditioning considerable speed-ups are
achievable compared to training NFs from scratch.
