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In this work, we propose a novel two-level discretization for solvingsemilinear elliptic equations with random coefficients. Motivated by thetwo-grid method for deterministic partial differential equations (PDEs)introduced by Xu \cite{xu1994novel}, our two-level stochastic collocationmethod utilizes a two-grid finite element discretization in the physical spaceand a two-level collocation method in the random domain. In particular, wesolve semilinear equations on a coarse mesh $\mathcal{T}_H$ with a low levelstochastic 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 thepolynomial space $\mathcal{P}_{\boldsymbol{p}}$). We prove that theapproximated solution obtained from this method achieves the same order ofaccuracy as that from solving the original semilinear problem directly bystochastic collocation method with $\mathcal{T}_h$ and$\mathcal{P}_{\boldsymbol{p}}$. The two-level method is computationally moreefficient than the standard stochastic collocation method for solving nonlinearproblems with random coefficients. Numerical experiments are provided to verifythe theoretical results.

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