  Transport maps have become a popular mechanic to express complicated
probability densities using sample propagation through an optimized
push-forward. Beside their broad applicability and well-known success,
transport maps suffer from several drawbacks such as numerical inaccuracies
induced by the optimization process and the fact that sampling schemes have to
be employed when quantities of interest, e.g. moments are to compute. This
paper presents a novel method for the accurate functional approximation of
probability density functions (PDF) that copes with those issues. By
interpreting the pull-back result of a target PDF through an inexact transport
map as a perturbed reference density, a subsequent functional representation in
a more accessible format allows for efficient and more accurate computation of
the desired quantities. We introduce a layer-based approximation of the
perturbed reference density in an appropriate coordinate system to split the
high-dimensional representation problem into a set of independent
approximations for which separately chosen orthonormal basis functions are
available. This effectively motivates the notion of h- and p-refinement (i.e.
``mesh size'' and polynomial degree) for the approximation of high-dimensional
PDFs. To circumvent the curse of dimensionality and enable sampling-free access
to certain quantities of interest, a low-rank reconstruction in the tensor
train format is employed via the Variational Monte Carlo method. An a priori
convergence analysis of the developed approach is derived in terms of Hellinger
distance and the Kullback-Leibler divergence. Applications comprising Bayesian
inverse problems and several degrees of concentrated densities illuminate the
(superior) convergence in comparison to Monte Carlo and Markov-Chain Monte
Carlo methods.
