Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation

Grid-based lattice summation of electrostatic potentials by low-rank tensor approximation

We introduce and study the grid-based rank-structured tensor method for fast and accurate calculation of the lattice sums of Coulomb interactions on large 3D periodicstructured compounds. The approach is based on the low-rank canonical tensor representation of the Newton kernels discretized in a computational box using fine N×N×N 3D Cartesian grid. This reduces the 3D summation to a sequence of...

Tensor Low Multilinear Rank Approximation by Structured Matrix Low-Rank Approximation

We present a new connection between higherorder tensors and affinely structured matrices, in the context of low-rank approximation. In particular, we show that the tensor low multilinear rank approximation problem can be reformulated as a structured matrix low-rank approximation, the latter being an extensively studied and well understood problem. We first consider symmetric tensors. Although t...

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely ...

Low-rank Tensor Approximation

Approximating a tensor by another of lower rank is in general an ill posed problem. Yet, this kind of approximation is mandatory in the presence of measurement errors or noise. We show how tools recently developed in compressed sensing can be used to solve this problem. More precisely, a minimal angle between the columns of loading matrices allows to restore both existence and uniqueness of the...

Structured Lower Rank Approximation