Abstract of Aram Khachatryan

Data-sparse approximation of singular functions
A data-sparse structure for representing function-related tensors with a finite number of singularities is described. As it is well known, singularities (or fast growth of the derivatives) can prevent the existence of a low-rank approximation. A strategy for representing such a tensor (matrix) in a data-sparse form could be to approximate the blocks far from singularities (admissible blocks) and continue hierarchically with the non-admissible blocks (e.g. H-matrix technique). This will however lead to a complicated structure if there are several singular points, and the number of blocks will increase exponentially w.r.t. the dimension. We propose to approximate not the admissible blocks but the whole tensor without the non-admissible ones. The approximation problem in this case goes beyond the standard theory of linear algebra already for d=2. Some successful algorithms and numerical examples from electronic structure calculations are presented.


Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)