A mesh-free hybrid Chebyshev-Tucker tensor format with applications to multi-particle modelling

We introduce and analyze a mesh-free two-level hybrid Tucker tensor format for approximating multivariate functions, which combines the product Chebyshev interpolation with the alternating least-squares (ALS) based Tucker decomposition of the tensor of Chebyshev coefficients. This construction allows to avoid the expensive rank-structured grid-based approximation of function-related tensors on large spatial grids, while benefiting from the Tucker decomposition of the rather small core tensor of Chebyshev coefficients. Thus, we can compute the nearly optimal Tucker decomposition of the 3D function with controllable accuracy $\varepsilon >0$ without discretizing the function on the full grid in the domain, but only using its values at small set of Chebyshev nodes. Finally, we can represent the function in the algebraic Tucker format with optimal $\varepsilon$-rank on an arbitrarily large 3D tensor grid in the computational domain by discretizing the Chebyshev polynomials on that grid. The rank parameters of the Tucker-ALS decomposition of the coefficient tensor can be much smaller than the polynomial degrees of the initial Chebyshev interpolant obtained via a function independent polynomial basis set. It is shown that our techniques can be gainfully applied to the long-range part of the singular electrostatic potential of multi-particle systems approximated in the range-separated tensor format. We provide error and complexity estimates and demonstrate the computational efficiency of the proposed techniques on challenging examples, including the multi-particle electrostatic potential for large bio-molecular systems and lattice-type compounds.