The breaking through approach to low-parametric representation of multivariate functions and operators is based on the principle of separation of variables which can be realized by using approximation in rank-structured tensor formats . This allows the linear complexity scaling in dimension, hence breaking the ”curse of dimensionality”. The method of quantized tensor train (QTT) approximation is proven to provide the logarithmic data-compression on a wide class of discretized functions and operators .
We discuss how the tensor methods based on the canonical, Tucker, TT and QTT?approximation apply to calculation of electrostatic potential of many-particle systems, to the (post) Hartree-Fock eigenvalue problem for large 3D lattice-structured molecules (electrostatic potential on a lattice in electronic structure calculations) [3,4,5], to elliptic equations with highly-oscillating coefficients (homogenization theory) , and to parametric/stochastic elliptic PDEs.The other direction is related to tensor approaches for parabolic equations in space-time d + 1formulation.
The efficiency of the tensor approach is illustrated by numerical examples.