A model reduction method for multiscale elliptic PDEs with random coefficients using an optimization approach
For computational scientists solving multiscale PDEs with random coefficients, this method offers an efficient approach for repeated solves with different force functions, though it is an incremental improvement over existing model reduction techniques.
The paper proposes a model reduction method for multiscale elliptic PDEs with random coefficients using an optimization approach to construct localized multiscale data-driven stochastic basis functions, enabling efficient solution in multiquery settings with reduced computational costs. Numerical simulations verify the convergence analysis.
In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized multiscale data-driven stochastic basis functions that give optimal approximation property of the solution operator. Our method consists of the offline and online stages. In the offline stage, we construct the localized multiscale data-driven stochastic basis functions by solving an optimization problem. In the online stage, using our basis functions, we can efficiently solve multiscale elliptic PDEs with random coefficients with relatively small computational costs. Therefore, our method is very efficient in solving target problems with many different force functions. The convergence analysis of the proposed method is also presented and has been verified by the numerical simulation.