  We apply the Tensor Train (TT) decomposition to construct the tensor product
Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic
elliptic diffusion PDE with the stochastic Galerkin discretization, and to
compute some quantities of interest (mean, variance, exceedance probabilities).
We assume that the random diffusion coefficient is given as a smooth
transformation of a Gaussian random field. In this case, the PCE is delivered
by a complicated formula, which lacks an analytic TT representation. To
construct its TT approximation numerically, we develop the new block TT cross
algorithm, a method that computes the whole TT decomposition from a few
evaluations of the PCE formula. The new method is conceptually similar to the
adaptive cross approximation in the TT format, but is more efficient when
several tensors must be stored in the same TT representation, which is the case
for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin
matrix and to compute the solution of the elliptic equation and its
post-processing, staying in the TT format.
  We compare our technique with the traditional sparse polynomial chaos and the
Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial
degree is bounded for each random variable independently. This provides higher
accuracy than the sparse polynomial set or the Monte Carlo method, but the
cardinality of the tensor product set grows exponentially with the number of
random variables. However, when the PCE coefficients are implicitly
approximated in the TT format, the computations with the full tensor product
polynomial set become possible. In the numerical experiments, we confirm that
the new methodology is competitive in a wide range of parameters, especially
where high accuracy and high polynomial degrees are required.
