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 of the mu-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB (R)/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach.

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

Marco Caliari
;
Fabio Cassini;Franco Zivcovich
2023-01-01

Abstract

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 of the mu-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB (R)/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach.
2023
mu-mode product
Tensor-structured problems
Exponential of Kronecker sum
ADI preconditioners
Multivariate interpolation
Multidimensional spectral transforms
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/1089487
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