Structured Rank-(r1, . . . , rd) Decomposition of Function-related Tensors in R_D

By Boris N. Khoromskij. 2006

72 citations recorded by the source corpus.

Abstract The structured tensor-product approximation of multidimensional nonlocal operators by a two-level rank-(r1, . . . , rd) decomposition of related higher-order tensors is proposed and analysed. In this approach, the construction of the desired approximant to a target tensor is a reminiscence of the Tucker-type model, where the canonical components are represented in a fixed (uniform) basis, while the core tensor is given in the canonical format. As an alternative, the multilevel nested canonical decomposition is presented. The complexity analysis of the corresponding multilinear algebra shows an almost linear cost in the one-dimensional problem size. The existence of a low Kronecker rank two-level representation is proven for a class of function-related tensors.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Signal Processing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “One popular combination is the Tucker format combined with the CP format for the approximation of the core tensor [148, 156, 157, 170], see [168, Sec.”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Multigrid Accelerated Tensor Approximation of Function Related Multi-dimensional Arrays Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays * (unknown): “Under the natural compatibility conditions (2.10) r ℓ ≤ ¯ r ℓ := r 1 ...r ℓ−1 r ℓ+1 ...r d ,ℓ =1,...,d, the dual maximization problem (2.8) can be proven to have a global maximum (see [16]).”
Mathematik in den Naturwissenschaften Leipzig Tensor-Product Approximation to Operators and Functions in High Dimensions (2007, methodology): “Approximations via the Nyström and Galerkin methods have been considered in [14, 13, 19].”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians.”
Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd (2009, methodology): “To our best knowledge, the low rank tensor-structured approximation for oscillating kernels have not been yet addressed in the literature (cf. the asymptotically optimal method in the volume size based on the fast multipole type representations [5, 6]).”