Multigrid in Hierarchical Low Rank Tensor Formats

Lars Grasedyck
RWTH Aachen University

In this presentation we give a brief introduction to hierarchial low rank tensor formats for the representation of high-dimensional tensors that fulfil certain rank bounds or singular value decay of corresponding matricizations.
In order to solve systems of equations in these formats there are already many developed and practically useful methods available. However, their convergence and convergence rates are difficult to analyse. On the other hand multigrid methods are well analysed with superior convergence rates. We use the classical multigrid convergence theory by providing a smoother and prolongations / restrictions that comply with low rank tensor formats. Prolongation and restriction are naturally given in a low rank tensor formulation. The smoother is an approximation of Jacobi based on a truncated exponential sum, thereby adapting it to the low rank format.

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