  The evaluation of robustness and reliability of realistic structures in the
presence of uncertainty involves costly numerical simulations with a very high
number of evaluations. This motivates model order reduction techniques like the
proper orthogonal decomposition. When only a few quantities are of interest an
approximative mapping from the high-dimensional parameter space onto each
quantity of interest is sufficient. Appropriate methods for this task are for
instance the polynomial chaos expansion or low-rank tensor approximations.
  In this work we focus on a non-linear neo-hookean deformation problem with
the maximal deformation as our quantity of interest. POD and adaptive POD
models of this problem are constructed and compared with respect to
approximation quality and construction cost. Additionally, the adapative proper
orthogonal decomposition (APOD) is introduced and compared to the regular POD.
Building upon that, several hierarchical Tucker approximations (HTAs) are
constructed from the reduced and unreduced models. A simple Monte Carlo method
in combination with HTA and (A)POD is used to estimate the mean and variance of
our quantity of interest. Furthermore, the HTA of the unreduced model is
employed to find feasible snapshots for (A)POD.
