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IGPM385.pdf March 2014 
TITLE 
Hierarchical Tensor Approximation of Output Quantities of ParameterDependent PDEs 
AUTHORS 
Jonas Ballani, Lars Grasedyck 
ABSTRACT 
Parametric PDEs appear in a large number of applications, as, e.g., in uncertainty quantification or optimisation. In many cases, one is interested in scalar output quantities induced by the parameterdependent solution. The output can be interpreted as a tensor living on a highdimensional parameter space. Our aim is to adaptively construct an approximation of this tensor in a datasparse 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. 
KEYWORDS 
hierarchical Tucker, low rank tensor, SPDE, cross approximation, sampling 