Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients
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 coefficie...
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MDPI AG
2021-06-01
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author | Eric Chung Hyea-Hyun Kim Ming-Fai Lam Lina Zhao |
author_facet | Eric Chung Hyea-Hyun Kim Ming-Fai Lam Lina Zhao |
author_sort | Eric Chung |
collection | DOAJ |
description | 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. |
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spelling | doaj.art-54ceaee3005440beaf9f922b6d67e6cc2023-11-21T22:58:41ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472021-06-012624410.3390/mca26020044Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast CoefficientsEric Chung0Hyea-Hyun Kim1Ming-Fai Lam2Lina Zhao3Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, ChinaDepartment of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin, KoreaDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, ChinaDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, ChinaIn 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.https://www.mdpi.com/2297-8747/26/2/44BDDCstochastic partial differential equationartificial neural networkcoarse spacehigh contrast |
spellingShingle | Eric Chung Hyea-Hyun Kim Ming-Fai Lam Lina Zhao Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients Mathematical and Computational Applications BDDC stochastic partial differential equation artificial neural network coarse space high contrast |
title | Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients |
title_full | Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients |
title_fullStr | Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients |
title_full_unstemmed | Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients |
title_short | Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients |
title_sort | learning adaptive coarse spaces of bddc algorithms for stochastic elliptic problems with oscillatory and high contrast coefficients |
topic | BDDC stochastic partial differential equation artificial neural network coarse space high contrast |
url | https://www.mdpi.com/2297-8747/26/2/44 |
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