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MiS Preprint

$O(d \log N)$-Quantics Approximation of $N$-$d$ Tensors in High-Dimensional Numerical Modeling

Boris N. Khoromskij


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 $O(d \log N)$-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set $\{1,...,N \}^{\otimes d}$, or briefly $N$-$d$ tensors of size $N^d$, and to the respective discretised differential-integral operators in $\mathbb{R}^d$. 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 $O(d \log N \log \varepsilon^{-1})$-complexity bound for their approximation by low rank $2$-$(d \log N)$ quantics tensors up to the accuracy $\varepsilon >0$. In the case $\varepsilon = O(N^{-\alpha})$, $\alpha>0$, our approach leads to the quantics tensor numerical method in dimension $d$, with the nearly optimal asymptotic complexity $O(d/\alpha \log^2 \varepsilon^{-1})$. >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.

MSC Codes:
65F30, 65F50, 65N35, 65F10
high-dimensional problems, quantics folding of vectors, rank structured tensor approximation, fem, matrix valued functions, material sciences, stochastic modeling

Related publications

2011 Repository Open Access
Boris N. Khoromskij

\(O(d łog N) \)-quantics approximation of \(N\)-\(d\) tensors in high-dimensional numerical modeling

In: Constructive approximation, 34 (2011) 2, pp. 257-280