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

TL;DR

This paper introduces an optimization-based model reduction technique for efficiently solving multiscale elliptic PDEs with random coefficients, utilizing localized stochastic basis functions for rapid multi-query solutions.

Contribution

The paper presents a novel offline-online framework that constructs localized data-driven stochastic basis functions via optimization, enhancing efficiency in solving multiscale PDEs with randomness.

Findings

Constructed basis functions via optimization improve approximation accuracy.

Method significantly reduces computational costs in multi-query scenarios.

Numerical simulations verify convergence and efficiency.

Abstract

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.