Tensor methods in multi-dimensional spectral problems with particular application in electronic structure calculations

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Many problems from modern physics and other fields are posed on high-dimensional tensor spaces in a natural way. Numerical approximation of solutions suffers from the curse of dimensionality, i.e. the computational complexity scales exponentially with the dimension of the space. Their numerical treatment therefore always involves methods of approximation by data-sparse representation ontensor spaces. There has very recently been remarkable progress in tensor product approximations. Newly introduced multiscale tensorisation techniques are provided by novel tensor formats. We aim to continue and enhance these developments and to use it for the design and critical assessment of algorithms for the new tensor formats treating high dimensional spectral problems. A particular emphasis is on the systematic examination of the potential which these methods may have in connection with various well-known algorithms used for the treatment of the electronic Schrödinger equation. In particular, our goal is to combine those novel techniques with the requirements and existing wellestablished algorithms for the electronic Schrödinger equation, in particular with one-particle models as the Hartree-Fock and Kohn-Sham method. In this respect, the proposal aims at the continuation of our project within the SPP 1324, concerned with the development of novel, mathematically sound tensor product methods for the numerical treatment of high-dimensional eigenvalue problems.