  In this paper, we introduce and analyze a new low-rank multilevel strategy
for the solution of random diffusion problems. Using a standard stochastic
collocation scheme, we first approximate the infinite dimensional random
problem by a deterministic parameter-dependent problem on a high-dimensional
parameter domain. Given a hierarchy of finite element discretizations for the
spatial approximation, we make use of a multilevel framework in which we
consider the differences of the solution on two consecutive finite element
levels in the collocation points. We then address the approximation of these
high-dimensional differences by adaptive low-rank tensor techniques. This
allows to equilibrate the error on all levels by exploiting analytic and
algebraic properties of the solution at the same time. We arrive at an explicit
representation in a low-rank tensor format of the approximate solution on the
entire parameter domain, which can be used for, e.g., the direct and cheap
computation of statistics. Numerical results are provided in order to
illustrate the approach.
