A Domain-Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

TL;DR

This paper introduces a domain-decomposition model reduction technique for linear convection-diffusion equations with random coefficients, effectively managing high-dimensional stochastic inputs and irregular solution behaviors to reduce computational costs.

Contribution

The paper presents a novel domain-decomposition model reduction method that efficiently handles high-dimensional randomness and irregularities in convection-diffusion equations.

Findings

Significant reduction in PDE solves through domain decomposition.

Effective operator approximation for non-affine, high-dimensional random fields.

Robust handling of irregular behaviors like sharp solution transitions.

Abstract

We develop a domain-decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equations with random diffusivities, and the convection-dominated transport equations with random velocities. We investigate the equations with two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence. The motivation is to use domain decomposition to exploit low-dimensional structures of local problems in the sub-domains, such that the total number of expensive PDE solves can be greatly reduced. Our objective is to develop an efficient model reduction method to simultaneously handle high-dimensionality and irregular behaviors of the stochastic PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) operator approximation for handling non-affine and high-dimensional random fields; (iii) effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solution. Two numerical examples will be provided to demonstrate the advantageous performance of our method.