Tensor-based numerical method for stochastic homogenisation

TL;DR

This paper introduces a tensor-based numerical method to efficiently approximate stochastic homogenisation in random materials, reducing computational costs while analyzing accuracy and limitations across different material structures.

Contribution

The paper presents a novel tensor-based approach exploiting quasi-periodicity for cost-effective homogenisation, and proposes using homogenised quantities as control variates for variance reduction in Monte Carlo simulations.

Findings

The method achieves significant cost reduction in homogenisation calculations.

It performs well for materials with quasi-periodic or mesoscopic structures.

Limitations are identified for materials lacking periodicity or mesoscopic features.

Abstract

This paper addresses the complexity reduction of stochastic homogenisation of a class of random materials for a stationary diffusion equation. A cost-efficient approximation of the correctors is built using a method designed to exploit quasi-periodicity. Accuracy and cost reduction are investigated for local perturbations or small transformations of periodic materials as well as for materials with no periodicity but a mesoscopic structure, for which the limitations of the method are shown. Finally, for materials outside the scope of this method, we propose to use the approximation of homogenised quantities as control variates for variance reduction of a more accurate and costly Monte Carlo estimator (using a multi-fidelity Monte Carlo method). The resulting cost reduction is illustrated in a numerical experiment with a control variate from weakly stochastic homogenisation for comparison, and the limits of this variance reduction technique are tested on materials without periodicity or mesoscopic structure.