Approximation by sum of products (of uni-variate functions) is a widely used, e.g. in statistics and physics, concept to approximate high dimensional functions, and high-dimensional problems, e.g. PDE's are cast into optimzation problems. However, the non-linear parametrization leads to problems, where the most simplest are shown to be NP hard (Hillard-Lim 2012). For a particular class of multi-linear parametrizations, namely hierarchical tensor representations (Hackbusch et al. 2009) or tree tensor networks, these difficulties can be resolved to a fairely wide extent, and provide a mathematically sound concept for tensor product approximation. Hierarchcial tensors will be introduced by a hierarchy of optimal subspace approximations, and demostrated by tensor trains (TT), resp. matrix products states (MPS).
We treat different applications ranging from quantum chemistry, Langevin dynamics of biomolecules, uncertainty quantification to tensor completion by variant of DMRG (density matrix renormalization group) and improved algorithm.
S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, \"O. Legeza, Tensor product methods and entanglement optimization for ab initio quantum chemistry, International Journal of Quantum Chemistry 115 (19), 1342-1391 (2015)
M. Bachmayr, R. Schneider, A. Uschmajew, Tensor Networks and Hierarchical Tensors for the Solution of High-dimensional Partial Differential Equations, http://www3.math.tu-berlin.de/preprints/files/Preprint-28-2015.pdf, revised version to appear in FOCM