Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

Eric Chung,Ming-Fai Lam,Hyea-Hyun Kim,Lina Zhao

doi:10.3390/mca26020044

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 realization 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–Loève 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.

Highlights

Coefficient functions are often one of the main difficulties in modeling a real life problem
Supporting numerical results are presented to show the performance of our proposed learning adaptive BDDC algorithm
After we obtain an accurate neural network from the proposed algorithm, besides the normalized root mean squared error (NRMSE) of testing set samples, we present some characteristics of the pre-conditioner based on the approximate eigenvectors from the learning adaptive BDDC algorithm, which include the number of iterations required, minimum and maximum eigenvalues of the pre-conditioner

Summary

Introduction

Coefficient functions are often one of the main difficulties in modeling a real life problem. Numerical methods like discontinuous Galerkin methods could give a robust scheme, as in [1,2]. A stochastic partial differential equation (SPDE) is considered for the modeling of these stochastic coefficient functions. There are several popular approaches for solving SPDE numerically, for example, the Monte Carlo simulation, stochastic Galerkin method, and the stochastic collocation method in [3,4,5]. All these methods require a high computational power in realistic simulations and cannot be generalized to similar situations. Instead of directly solving the SPDE by machine learning, we choose a moderate step that involves the advantage of machine learning and, the accuracy from the adaptive

Methods

Results

Conclusion