Numerical approximation of multi-dimensional PDEs in quantized tensor spaces

Modern methods of tensor-product approximation by separation of variables allow an efficient lowparametric calculus of functions and operators in higher dimensions. Most common separable representations combine the Tucker, canonical, tensor train (TT) and the more general MPS-type decompositions. The idea of data quantization makes it possible to represent (approximate) the multivariate functions, operators and dynamical systems in the quantized tensor spaces (QTT format) with the log-volume complexity scaling with respect to the grid size [2]. This opens the way to the profound numerical simulation of high-dimensional PDEs getting rid of the “curse of dimensionality” and rigorous restrictions on the grid size. However, the approach may be limited by the “curse of ranks” characterizing the essential entanglement in the system of interest.