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

Proper orthogonal decomposition (POD) combined with hierarchical tensor approximation (HTA) in the context of uncertain parameters