Learning adaptive coarse spaces of BDDC algorithms for stochastic elliptic problems with oscillatory and high contrast coefficients

TL;DR

This paper introduces a machine learning approach using deep neural networks to efficiently construct adaptive coarse spaces in BDDC algorithms for stochastic elliptic problems with complex coefficients, reducing computational costs.

Contribution

The paper proposes a novel DNN-based method to approximate coarse spaces in BDDC algorithms, addressing the computational challenge of local spectral problem solutions for stochastic coefficients.

Findings

The DNN effectively predicts coarse spaces for stochastic coefficients.

The method demonstrates robustness with oscillatory and high contrast coefficients.

Numerical results confirm efficiency and accuracy of the approach.

Abstract

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realisation of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen-Lo\`eve expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme.