  This paper is concerned with the development and analysis of an iterative
solver for high-dimensional second-order elliptic problems based on
subspace-based low-rank tensor formats. Both the subspaces giving rise to
low-rank approximations and corresponding sparse approximations of
lower-dimensional tensor components are determined adaptively. A principal
obstruction to a simultaneous control of rank growth and accuracy turns out to
be the fact that the underlying elliptic operator is an isomorphism only
between spaces that are not endowed with cross norms. Therefore, as central
part of this scheme, we devise a method for preconditioning low-rank tensor
representations of operators. Under standard assumptions on the data, we
establish convergence to the solution of the continuous problem with a
guaranteed error reduction. Moreover, for the case that the solution exhibits a
certain low-rank structure and representation sparsity, we derive bounds on the
computational complexity, including in particular bounds on the tensor ranks
that can arise during the iteration. We emphasize that such assumptions on the
solution do not enter in the formulation of the scheme, which in fact is shown
to detect them automatically. Our findings are illustrated by numerical
experiments that demonstrate the practical efficiency of the method in high
spatial dimensions.
