A μ-mode BLAS approach for multidimensional tensor-structured problems

@article{Caliari2021AB,
title={A $\mu$-mode BLAS approach for multidimensional tensor-structured problems},
author={Marco Caliari and Fabio Cassini and Franco Zivcovich},
journal={ArXiv},
year={2021},
volume={abs/2112.11238}
}
• Published 21 December 2021
• Computer Science, Mathematics
• ArXiv
In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage…
3 Citations

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References

SHOWING 1-10 OF 39 REFERENCES

Tensor Toolbox for MATLAB, Version 3.2.1

• https://www.tensortoolbox.org (April,
• 2021

Tensor Decompositions and Applications

• Computer Science
SIAM Rev.
• 2009
This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order

A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations

• Mathematics
ArXiv
• 2020
A novel matrix-oriented reduction process is derived leading to an effective, structure aware low order approximation of the original problem, giving rise to a new two-sided version of DEIM.

Approximation of the matrix exponential for matrices with a skinny field of values

• Computer Science
• 2020
A rigorous bound is proposed for the relative backward error of the matrix exponential, which is of particular interest for matrices whose field of values is skinny, such as the discretization of the advection–diffusion or the Schrödinger operators.

Recursive blocked algorithms for linear systems with Kronecker product structure

• Computer Science, Mathematics
Numerical Algorithms
• 2019
This work shows that recursive blocked algorithms extend in a seamless fashion to higher-dimensional variants of generalized Sylvester matrix equations, as they arise from the discretization of PDEs with separable coefficients or the approximation of certain models in macroeconomics.

Multilinear operators for higher-order decompositions

• T. Kolda
• Mathematics, Computer Science
• 2006
Two new multilinear operators are proposed for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions and one of them is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor.