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A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu \cite{xu1994novel}, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh $\mathcal{T}_H$ with a low level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{\boldsymbol{P}}$) and solve linearized equations on a fine mesh $\mathcal{T}_h$ using high level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{\boldsymbol{p}}$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $\mathcal{T}_h$ and $\mathcal{P}_{\boldsymbol{p}}$. The two-level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients. Numerical experiments are provided to verify the theoretical results.