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In this paper, we propose a model reduction method for solving multiscaleelliptic PDEs with random coefficients in the multiquery setting using anoptimization approach. The optimization approach enables us to construct a setof localized multiscale data-driven stochastic basis functions that giveoptimal approximation property of the solution operator. Our method consists ofthe offline and online stages. In the offline stage, we construct the localizedmultiscale data-driven stochastic basis functions by solving an optimizationproblem. In the online stage, using our basis functions, we can efficientlysolve multiscale elliptic PDEs with random coefficients with relatively smallcomputational costs. Therefore, our method is very efficient in solving targetproblems with many different force functions. The convergence analysis of theproposed method is also presented and has been verified by the numericalsimulation.

A model reduction method for multiscale elliptic PDEs with random coefficients using an optimization approach