A Hybrid Chebyshev–Tucker Tensor Format for Approximation of Multivariate Functions

Abstract.

We introduce and analyze a mesh-free two-level hybrid Chebyshev–Tucker tensor representation for approximating multivariate functions, which combines tensor-product Chebyshev interpolation with the low-rank Tucker decomposition of the tensor of Chebyshev coefficients. This construction allows us 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 moderate-sized core tensor of Chebyshev coefficients. Thus, we can compute the nearly optimal Tucker decomposition of the 3D function with controllable accuracy [Formula: see text] without discretizing the function on a full fine grid in the domain, but only using its values at a small set of Chebyshev nodes computed either from the explicit analytic expression of the target function or from its data-sparse representation in a rank-structured tensor format with moderate rank parameter. Finally, we can represent the function in the algebraic Tucker format with optimal [Formula: see text]-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 nonlinear Tucker-ALS decomposition of the coefficient tensor can be much smaller than the polynomial degrees of the initial Chebyshev linear interpolation in the 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 multiparticle systems represented on a fine grid in the range-separated tensor format. The resulting low-rank tensor can be used in applications to numerical PDE, in computational quantum chemistry, and for calculation of various physical quantities of multiparticle systems. We provide error and complexity estimates and demonstrate the computational efficiency of the proposed techniques on challenging examples, including the collective electrostatic potential for large biomolecular systems and lattice-type compounds.

Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/bonans/ChebTuck and in the supplementary materials ( ChebTuck-main.zip [113MB]). [Formula: see text]