  Folding grid value vectors of size $2^L$ into $L$th order tensors of mode
sizes $2\times \cdots\times 2$, combined with low-rank representation in the
tensor train format, has been shown to lead to highly efficient approximations
for various classes of functions. These include solutions of elliptic PDEs on
nonsmooth domains or with oscillatory data. This tensor-structured approach is
attractive because it leads to highly compressed, adaptive approximations based
on simple discretizations. Standard choices of the underlying basis, such as
piecewise multilinear finite elements on uniform tensor product grids, entail
the well-known matrix ill-conditioning of discrete operators. We demonstrate
that for low-rank representations, the use of tensor structure itself
additionally introduces representation ill-conditioning, a new effect specific
to computations in tensor networks. We analyze the tensor structure of a BPX
preconditioner for second-order linear elliptic operators and construct an
explicit tensor-structured representation of the preconditioner, with ranks
independent of the number $L$ of discretization levels. The straightforward
application of the preconditioner yields discrete operators whose matrix
conditioning is uniform with respect to the discretization parameter, but in
decompositions that suffer from representation ill-conditioning. By
additionally eliminating certain redundancies in the representations of the
preconditioned discrete operators, we obtain reduced-rank decompositions that
are free of both matrix and representation ill-conditioning. For an iterative
solver based on soft thresholding of low-rank tensors, we obtain convergence
and complexity estimates and demonstrate its reliability and efficiency for
discretizations with up to $2^{50}$ nodes in each dimension.
