  We consider the approximate solution of parametric PDEs using the low-rank
Tensor Train (TT) decomposition. Such parametric PDEs arise for example in
uncertainty quantification problems in engineering applications. We propose an
algorithm that is a hybrid of the alternating least squares and the TT cross
methods. It computes a TT approximation of the whole solution, which is
beneficial when multiple quantities of interest are sought. This might be
needed, for example, for the computation of the probability density function
(PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm.,
1986]. The new algorithm exploits and preserves the block diagonal structure of
the discretized operator in stochastic collocation schemes. This disentangles
computations of the spatial and parametric degrees of freedom in the TT
representation. In particular, it only requires solving independent PDEs at a
few parameter values, thus allowing the use of existing high performance PDE
solvers. In our numerical experiments, we apply the new algorithm to the
stochastic diffusion equation and compare it with preconditioned steepest
descent in the TT format, as well as with (multilevel) quasi-Monte Carlo and
dimension-adaptive sparse grids methods. For sufficiently smooth random fields
the new approach is orders of magnitude faster.
