# An adaptive dynamically low-dimensional approximation method for multiscale stochastic diffusion equations

## TL;DR

This paper introduces an adaptive low-dimensional approximation method combining multiscale basis functions and generalized polynomial chaos to efficiently solve complex multiscale stochastic diffusion equations.

## Contribution

It develops a novel adaptive DyBO method with multiscale basis functions and online adaptation for improved efficiency and accuracy in multiscale stochastic PDEs.

## Key findings

- Significant reduction in computational cost demonstrated.
- High accuracy maintained with multiscale and online basis functions.
- Method effectively handles multiscale and stochastic features.

## Abstract

In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within each coarse grid block for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical results are presented to demonstrate the efficiency of the proposed method in solving time-dependent PDEs with multiscale and random features.