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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Main Authors: Eric Chung, Hyea-Hyun Kim, Ming-Fai Lam, Lina Zhao
Format: Article
Language:English
Published: MDPI AG 2021-06-01
Series:Mathematical and Computational Applications
Subjects:
Online Access:https://www.mdpi.com/2297-8747/26/2/44
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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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AT mingfailam learningadaptivecoarsespacesofbddcalgorithmsforstochasticellipticproblemswithoscillatoryandhighcontrastcoefficients
AT linazhao learningadaptivecoarsespacesofbddcalgorithmsforstochasticellipticproblemswithoscillatoryandhighcontrastcoefficients