# FFT-based homogenisation accelerated by low-rank tensor approximations

## TL;DR

This paper introduces a method that accelerates FFT-based homogenisation by applying low-rank tensor approximations, significantly reducing computational costs for large-scale heterogeneous material problems.

## Contribution

The paper presents a novel integration of low-rank tensor formats with FFT-based homogenisation, enabling efficient reduced order modeling of complex heterogeneous materials.

## Key findings

- Reduces computational and memory requirements for homogenisation problems.
- Demonstrates effectiveness on 2D and 3D problems with heterogeneous coefficients.
- Enables efficient approximation of global basis functions without full problem solutions.

## Abstract

Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kind of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem. It significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation. The advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients. This approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material.