Stochastic BDDC algorithms

Stochastic balancing domain decomposition by constraints (BDDC) algorithms are developed and analyzed for the sampling of the solutions of linear stochastic elliptic equations with random coefficients. Different from the deterministic BDDC algorithms, the stochastic BDDC algorithms have online and offline stages. At the offline stage, the Polynomial Chaos (PC) expansions of different components of the BDDC algorithms are constructed based on the subdomain local parametrization of the stochastic coefficients. During the online stage, the sample-dependent BDDC algorithm can be implemented with a small cost. Under some assumptions, the condition number of the stochastic BDDC preconditioned operator is estimated. Numerical experiments confirm the theory and show that the stochastic BDDC algorithm outperforms the BDDC preconditioner constructed using the mean value of the stochastic coefficients.