Structured rank-\((r_1, \dots , r_d)\) decomposition of function-related tensors in \(\mathbb R^d\). (English) Zbl 1120.65052

Khoromskij, B. N.

Comput. Methods Appl. Math. 6, No. 2, 194-220 (2006).

This paper is concerned with the tensor-product approximation of multidimensional nonlocal operators. Nested multi-level variations of the Tucker model and the CP decomposition are proposed. This includes a two-level rank-\((r_1,\dots,r_d)\) decomposition, where the core tensor of a Tucker model is represented by a CP decomposition. While a survey of existing iterative methods to compute such CP and Tucker decompositions for a given general tensor is given, the emphasis is clearly on the approximation of function-generated tensors.
It is demonstrated how sinc methods applied to the generating univariate function can be used to prove the existence of low two-level rank approximations in a constructive way. Several examples are given, including the multivariate functions \(1/\sqrt{x_1^2+\cdots+x_d^2}\) and \(e^{-\alpha \sqrt{x_1^2+\cdots+x_d^2}}\). The paper nicely demonstrates the importance of taking the analytic background of tensor-product approximations into account.

Reviewer: Daniel Kressner (Berlin)

MSC:

65F30	Other matrix algorithms (MSC2010)
65F50	Computational methods for sparse matrices
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems
41A63	Multidimensional problems

Keywords:

tensor products; multivariate function approximation; H-matrices; multidimensional integral operators; Newton potential; block Toeplitz algorithm; Tucker model; CP decomposition; sinc methods; numerical examples