D. Peláez, Université de Lillle
We present the new Singular-Value Decomposition Multigrid POTFIT (SVD-MGPF), a grid-based tensor decomposition algorithm for large dimensional systems with particular focus on Quantum Dynamical problems.
SVD-MGPF constitutes a generalisation of our previous (eigenvalue-decomposition based) EVD-MGPF method . Both MGPF methods provide the Tucker decomposition of a target tensor (e.g. a potential energy surface). For such, a fine grid, the one needed to accurately represent the system, and a coarse one, a subset of the former, are defined. Tucker factor matrices are obtained from a series of POTFIT  decompositions carried out on grids which are fine for some degrees of freedom and coarse for the rest. The core tensor is obtained by overlapping these factor matrices with the values of the original tensor on the coarse grid.
The difference between SVD-MGPF and EVD-MGPF (or POTFIT) lies in the fact that the Tucker factor matrices are obtained through singular value decompositions of one-particle potential density matrices instead of eigenvalue ones. This is shown to remove the numerical instabilities present in MGPF. Additionally, a black-box method for the choice of the coarse grid choice is proposed. The novelty of the latter is that it leads to a fully-relaxed non-product coarse grid.