2019/07/01 by Sijing Li, Zhiwen Zhang, Li, Sijing +3
Computer Science · Decision Sciences · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Computational Physics (physics.comp-ph) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1907.00806
openalex publication_date 2019/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.