A Model Reduction Method for Multiscale Elliptic Pdes with Random Coefficients Using an Optimization Approach
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 an 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 simulations.
© 2019 Society for Industrial and Applied Mathematics. Received by the editors August 16, 2018; accepted for publication (in revised form) April 15, 2019; published electronically June 20, 2019. The research of the first author is partially supported by the NSF grant DMS 1613861. The research of the third author is supported by the Hong Kong RGC grants (projects 27300616, 17300817, and 17300318), National Natural Science Foundation of China (project 11601457), Seed Funding Programme for Basic Research (HKU), and an RAE Improvement Fund from the Faculty of Science (HKU). This research was made possible by a donation to the Big Data Project Fund, HKU, from Dr. Patrick Poon, whose generosity is gratefully acknowledged. We would like to thank Professor Lei Zhang for stimulating discussions.
Submitted - 1807.02394.pdf
Published - 18m1205844.pdf