Ubiquitous Nature of the Reduced Higher Order SVD in Tensor-Based Scientific Computing - Khoromskaia, Venera; Khoromskij, Boris N. (Frontiers in Applied Mathematics and Statistics, 2022)
Tensor numerical methods based on the representation of $d$-variate functions and operators on large $n^{\otimes d }$ grids in the rank-structured tensor formats provide $O(dn)$ complexity of numerical calculations instead of $O(n^d)$ by conventional methods. However, tensor operations may lead to enormous increase in the ranks of the tensor data, making calculation intractable. Therefore one of the most important steps in tensor calculations is the robust and efficient rank reduction procedure which should be performed many times in the course of various tensor transforms in multidimensional operator and function calculus. The rank reduction scheme based on the Reduced Higher Order SVD (RHOSVD) introduced in [33] played a significant role in development of tensor numerical methods. Here, we analyze some new theoretical and computational aspects of the RHOSVD and show that this rank reduction technique constitutes the main ingredient in tensor computations for real-life problems. In particular, we recall the performance of the RHOSVD in tensor-based calculation of the Hartree potential in computational quantum chemistry and of the collective multi-particle interaction potentials. The new results on application of the RHOSVD in scattered data analysis are presented. RHOSVD is also the basis of rank reduction techniques for recent tensor-structured solvers for the 3D elliptic equations.