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Tensor numerical methods, based on the rank-structured tensor representationof $d$-variate functions and operators, are designed to provide $O(dn)$complexity of numerical calculations on $n^{\otimes d }$ grids contrary to$O(n^d)$ scaling by conventional grid-based methods. However, multiple tensoroperations may lead to enormous increase in the tensor ranks (curse of ranks)of the target data, making calculation intractable. Therefore one of the mostimportant steps in tensor calculations is the robust and efficient rankreduction procedure which should be performed many times in the course ofvarious 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 the development of tensornumerical methods. Here, we briefly survey the essentials of RHOSVD method andthen focus on some new theoretical and computational aspects of the RHOSVDdemonstrating that this rank reduction technique constitutes the basicingredient in tensor computations for real-life problems. In particular, thestability analysis of RHOSVD is presented. We introduce the multilinear algebraof tensors represented in the range-separated (RS) tensor format. This allowsto apply the RHOSVD rank-reduction techniques to non-regular functional datawith many singularities, for example, to the rank-structured computation of thecollective multi-particle interaction potentials in bio-molecular modeling, aswell as to complicated composite radial functions. The new theoretical andnumerical results on application of the RHOSVD in scattered data modeling arepresented. RHOSVD proved to be the efficient rank reduction technique innumerous applications ranging from numerical treatment of multi-particlesystems up to a numerical solution of PDE constrained control problems.

Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing