A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction
This work addresses computational challenges in solving complex PDEs for applications in physics and engineering, representing an incremental improvement through a novel method for a known bottleneck.
The authors tackled the problem of solving multiscale elliptic PDEs with random coefficients by proposing a data-driven approach that leverages intrinsic low-dimensional structures, achieving significant dimension reduction and demonstrating accuracy and efficiency in numerical examples.
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the underlying Green's functions. Different online construction methods are proposed depending on the problem setup. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method.