The conventional numerical methods when applied to multidimensional problems suffer from the
so-called curse of dimensionality that cannot be eliminated by using parallel architectures
and high performance computing. The novel tensor numerical methods are based on a smart
rank-structured tensor representation of the multivariate functions and operators discretized
on Cartesian grids thus reducing solution of the multidimensional integral-differential
equations to 1D calculations. We explain basic tensor formats and algorithms and show how the
orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in
numerical analysis relying on rigorous results from approximation theory. Benefits of tensor
approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D
convolution integrals for functions with multiple singularities is replaced by a sequence of 1D
operations thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor
grids of the size up to 1015. Fast tensor-based Hartree-Fock solver incorporating the
grid-based low-rank factorization of the two-electron integrals serves as a prerequisite for
economical calculation of the excitation energies of molecules. Tensor approach suggests
efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D
finite lattices with defects.The novel range-separated tensor format applies to interaction
potentials of multi-particle systems of general type opening the new prospects for tensor
methods in scientific computing. This research monograph presenting the modern tensor
techniques applied to problems in quantum chemistry may be interesting for a wide audience of
students and scientists working in computational chemistry material science and scientific
computing.
