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

  title={A $\mu$-mode BLAS approach for multidimensional tensor-structured problems},
  author={Marco Caliari and Fabio Cassini and Franco Zivcovich},
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… 

Figures and Tables from this paper

Exponential methods for anisotropic diffusion

The anisotropic di ff usion equation is of crucial importance in understandingcosmic ray di ff usion across the Galaxy and its interplay with the Galactic magnetic field. This di ff usion term

Exploiting Kronecker structure in exponential integrators: fast approximation of the action of φ-functions of matrices via quadrature

An algorithm for approximating the action of ϕ − functions of matrices against vectors, which is a key operation in exponential time integrators, is proposed and shows that it is accurate and orders of magnitude faster than the current state-of-the-art.

A $\mu$-mode approach for exponential integrators: actions of $\varphi$-functions of Kronecker sums

The authors' numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion– reaction, Allen–Cahn, and Brusselator equations show the superiority of the 𝜇 -mode approach of PHIKS.



Tensor Toolbox for MATLAB, Version 3.2.1

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

Tensor Decompositions and Applications

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

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

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.

Batched transpose-free ADI-type preconditioners for a Poisson solver on GPGPUs

Recursive blocked algorithms for linear systems with Kronecker product structure

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.

The ubiquitous Kronecker product