A Randomized GMsFEM with Data-Driven Predictors for Parametric Flow Problems in Multiscale Heterogeneous Media

TL;DR

This paper introduces a randomized GMsFEM that uses data-driven predictors to efficiently solve parametric flow problems in complex heterogeneous media, with proven convergence and superior numerical performance.

Contribution

It develops a novel randomized GMsFEM framework with a data-driven predictor for multiscale basis construction, improving efficiency and accuracy over traditional methods.

Findings

Achieves accurate approximations for high-contrast heterogeneous media.

Demonstrates superior performance in numerical experiments.

Provides rigorous convergence analysis for the proposed method.

Abstract

In this paper, we propose a randomized generalized multiscale finite element method (Randomized GMsFEM) for flow problems with parameterized inputs and high-contrast heterogeneous media. The method employs a data-driven predictor to construct multiscale basis functions in two stages: offline and online. In the offline stage, a snapshot space is generated via spectral decompositions, and a reduced matrix is obtained using SVD to predict eigenfunctions. In the online stage, these eigenfunctions are evaluated for new parameter realizations to construct the multiscale space. Furthermore, our approach addresses the complexity of multiple permeability fields with random inputs and multiple multiscale information, providing accurate and efficient approximations. Moreover, we conduct a rigorous convergence analysis for our Randomized GMsFEM. Finally, we present extensive numerical examples, demonstrating its superior performance compared to the traditional GMsFEM.