Reduced‐order FFT‐based homogenisation by tensor approximation

Abstract

We enhanced the efficiency of Fast Fourier transform (FFT) based Galerkin methods on numerical homogenisation problems by exploiting low-rank tensor approximations in canonical, Tucker, and tensor train formats. This leads to a significant reduction in computational complexity and memory requirement. The advantages of the approach are demonstrated in a numerical example of a model homogenisation problem with stochastic heterogeneous material coefficients. Homogenisation by Fourier-Galerkin methodsWe consider microscopic homogenisation of a scalar linear elliptic problem on a d-dimensional cell Y = (− 1 2 , 1 2 ) d with bounded, symmetric, and uniformly elliptic material coefficients A : Y → R d×d . The variational formulation of finding effective material properties A H ∈ R d×d for any macroscopic gradient fieldis discretised with the Fourier-Galerkin method [1,2] that incorporates trigonometric polynomials as basis functions.The space H 1 0 (Y) consists of the Y-periodic scalar function v : Y → R with a square integrable gradient and zero mean. The Fourier coefficientsû of the minimisers can be identified by solving the following linear system of equations [2] in discretised Fourier domainThe tensorP is an inverse Laplacian preconditioner [3], N ∈ N d is the discretization size, F N stands for Fourier transform, ∇ N and ∇ * N for tensors of differentiation and divergence operators in Fourier domain.Ã ∈ R d×d×N ×N is a block diagonal tensor obtained by a weighted projection of A onto the dicretised space, and E a constant vector.We speed up the numerical solution by exploiting tensor decomposition techniques [4]. In the low-rank tensor format the multidimensional Fast Fourier transform (FFT) can be implemented at the cost of a series of 1-dimensional FFTs. For tensors with moderate representation rank this alleviates the "curse of dimension". The system is solved by a Minimal Residual iteration method. Low-rank tensor approximationLow rank tensor approximation techniques is applied to handle the huge number of degrees of freedom in the solution of (2). We use three formats, namely canonical, Tucker, and Tensor Train (TT) format of low-rank tensors. As an example, a canonical approximation a tensor v ∈ K N1×···×N d (K is R or C) is a sum of r rank-1 tensors in the form:where c ∈ R r stores the coefficients with respect to basis vectors b (j) ∈ K r×Nj in spatial direction j; operator ⊗ denotes tensor product.The linearity of Fourier transform facilitates to express d-dimensional FFT of a tensor as the sum of tensor products of 1-dimensional FFTs, i.e.,