A $\mu$-mode BLAS approach for multidimensional tensor-structured problems

TL;DR

This paper introduces a tensor framework utilizing the $0$-mode product for efficient multidimensional numerical computations, demonstrated through MATLAB implementations and experiments up to six dimensions.

Contribution

It generalizes 1D numerical tasks to higher dimensions using tensor product formulas and introduces an efficient BLAS formulation based on $0$-mode tensor operations.

Findings

Effective tensor-based approach up to six dimensions

MATLAB implementations in the KronPACK package

Numerical experiments demonstrate efficiency and versatility

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/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.