Hierarchical Tensor Approximation of Output Quantities of Parameter-Dependent PDEs

Parametric PDEs appear in a large number of applications such as, e.g., uncertainty quantification and optimization. In many cases, one is interested in scalar output quantities induced by the parameter-dependent solution. The output can be interpreted as a tensor living on a high-dimensional parameter space. Our aim is to adaptively construct an approximation of this tensor in a data-sparse hierarchical tensor format. Once this approximation from an offline computation is available, the evaluation of the output for any parameter value becomes a cheap online task. Moreover, the explicit tensor representation can be used to compute stochastic properties of the output in a straightforward way. The potential of this approach is illustrated by numerical examples.