  We propose a guaranteed and fully computable upper bound on the energy norm
of the error in low-rank Tensor Train (TT) approximate solutions of (possibly)
high dimensional reaction-diffusion problems. The error bound is obtained from
Euler-Lagrange equations for a complementary flux reconstruction problem, which
are solved in the low-rank TT representation using the block Alternating Linear
Scheme. This bound is guaranteed to be above the energy norm of the total
error, including the discretization error, the tensor approximation error, and
the error in the solver of linear algebraic equations, although quadrature
errors, in general, can pollute its evaluation. Numerical examples with the
Poisson equation and the Schroedinger equation with the Henon-Heiles potential
in up to 40 dimensions are presented to illustrate the efficiency of this
approach.
