in High-Dimensional Numerical Modeling

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high dimensional applications. We describe the multi-folding or quantics based tensor approximation method of O(d logN)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1, ..., N}⊗d, or briefly N -d tensors of size N, and to the respective discretised differential-integral operators in R. As the basic approximation result, we prove that complex exponential sampled on equispaced grid has quantics rank 1. Moreover, the Chebyshev polynomial sampled over Chebyshev Gauss-Lobatto grid, has separation rank 2 in quantics tensor format, while for the polynomial of degree m the respective quantics rank is at most m + 1. For N -d tensors generated by certain analytic functions, we give the constructive proof on the O(d logN log ε)-complexity bound for their approximation by low rank 2-(d logN) quantics tensors up to the accuracy ε > 0. In the case ε = O(N), α > 0, our approach leads to the quantics tensor numerical method in dimension d, with the nearly optimal asymptotic complexity O(d/α log ε). ¿From numerics presented, we observe that the quantics tensor method has proved its value in application to various function related tensors/matrices arising in computational quantum chemistry and in the traditional FEM/BEM—the tool apparently works. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10