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-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.
University of Southern California（南加州大学）；Institute of Thermomechanics, Academy of Sciences of the Czech Republic（捷克科学院热力学研究所）；University of Colorado Denver（科罗拉多大学丹佛分校）
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Department of Mathematical and Statistical Sciences, University of Colorado Denver（科罗拉多大学丹佛分校数学与统计科学系）；Department of Aerospace and Mechanical Engineering, University of Southern California（南加州大学航空航天与机械工程系）；Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University（捷克理工大学机械工程学院数学系）
University of Colorado Denver（科罗拉多大学丹佛分校）；Institute of Thermomechanics, Academy of Sciences of the Czech Republic（捷克科学院热力学研究所）；Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Czech Technical University in Prague（布拉格捷克理工大学）
Application of the parallel BDDC preconditioner to the Stokes flow
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Department of Mathematical and Statistical Sciences, University of Colorado Denver（科罗拉多大学丹佛分校数学与统计科学系）；Institute of Thermomechanics, Academy of Sciences of the Czech Republic（捷克科学院热力学研究所）；Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague（布拉格捷克理工大学机械工程学院数学系）；Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague（布拉格捷克理工大学土木工程学院数学系）
Speeding up an unsteady flow simulation by adaptive BDDC and Krylov subspace recycling
Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；Faculty of Mechanical Engineering, Czech Technical University in Prague（布拉格捷克理工大学机械工程学院）
Face-based Selection of Corners in 3D Substructuring
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Department of Mathematics, Faculty of Mechanical Engineering Czech Technical University in Prague（布拉格捷克理工大学机械工程学院数学系）；Department of Mathematics, Faculty of Civil Engineering Czech Technical University in Prague（布拉格捷克理工大学土木工程学院数学系）
A Hybrid High-Order method for highly oscillatory elliptic problems
Université Paris-Est, CERMICS (ENPC)（巴黎东大学，CERMICS（国立桥梁与道路学院））；Inria Paris（法国国家信息与自动化研究所巴黎分院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Inria Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧分院）
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
School of Mathematics, Jilin University（吉林大学数学学院）；Ocean University of China（中国海洋大学）；Shenzhen Loop Area Institute（深圳河套学院）；King Abdullah University of Science and Technology（卡布斯科学技术大学）
