AUTHOR=Khoromskaia Venera , Khoromskij Boris N. 

TITLE=Ubiquitous Nature of the Reduced Higher Order SVD in Tensor-Based Scientific Computing

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 8 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.826988

DOI=10.3389/fams.2022.826988

ISSN=2297-4687

ABSTRACT=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.