An Efficient Intrusive Uncertainty Propagation Method For Multi-Physics System With Random Inputs

TL;DR

This paper introduces a modular intrusive spectral projection method for efficient uncertainty propagation in multi-physics PDE systems, significantly reducing computational costs by localizing stochastic parameters and using reduced chaos approximations.

Contribution

The paper presents a novel module-based intrusive spectral projection approach that enhances efficiency in uncertainty propagation for coupled PDE systems by localizing stochastic analysis.

Findings

Significant computational savings over standard ISP methods.

Effective handling of high-dimensional stochastic parameters.

Successful numerical demonstrations of the proposed method.

Abstract

Coupled partial differential equation (PDE) systems, which often represent multi-physics models, are naturally suited for modular numerical solution methods. However, several challenges yet remain in extending the benefits of modularization practices to the task of uncertainty propagation. Since the cost of each deterministic PDE solve can be usually expected to be quite significant, statistical sampling based methods like Monte-Carlo (MC) are inefficient because they do not take advantage of the mathematical structure of the problem, and suffer for poor convergence properties. On the other hand, even if each module contains a moderate number of uncertain parameters, implementing spectral methods on the combined high-dimensional parameter space can be prohibitively expensive due to the curse of dimensionality. In this work, we present a module-based and efficient intrusive spectral projection (ISP) method for uncertainty propagation. In our proposed method, each subproblem is separated and modularized via block Gauss-Seidel (BGS) techniques, such that each module only needs to tackle the local stochastic parameter space. Moreover, the computational costs are significantly mitigated by constructing reduced chaos approximations of the input data that enter each module. We demonstrate implementations of our proposed method and its computational gains over the standard ISP method using numerical examples.