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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Detail(s)

Original languageEnglish
Article number44
Journal / PublicationMathematical and Computational Applications
Volume26
Issue number2
Online published6 Jun 2021
Publication statusPublished - Jun 2021
Externally publishedYes

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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

Research Area(s)

  • BDDC, stochastic partial differential equation, artificial neural network, coarse space, high contrast

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