Real-Time Reduced-Order Modeling of Stochastic Partial Differential Equations via Time-Dependent Subspaces

TL;DR

This paper introduces the DBO method for real-time reduced-order modeling of stochastic PDEs, offering improved accuracy, stability, and efficiency over existing methods like DO and BO by utilizing time-dependent orthonormal modes.

Contribution

The paper develops the DBO decomposition, a novel approach that enhances stochastic PDE modeling by addressing limitations of prior methods and providing better numerical stability and accuracy.

Findings

DBO outperforms DO and BO in ill-conditioned covariance cases.

DBO avoids eigenvalue crossing issues present in BO.

DBO achieves higher accuracy in numerical experiments.

Abstract

We present a new methodology for the real-time reduced-order modeling of stochastic partial differential equations called the dynamically/bi-orthonormal (DBO) decomposition. In this method, the stochastic fields are approximated by a low-rank decomposition to spatial and stochastic subspaces. Each of these subspaces is represented by a set of orthonormal time-dependent modes. We derive exact evolution equations of these time-dependent modes and the evolution of the factorization of the reduced covariance matrix. We show that DBO is equivalent to the dynamically orthogonal (DO) and bi-orthogonal (BO) decompositions via linear and invertible transformation matrices that connect DBO to DO and BO. However, DBO shows several improvements compared to DO and BO: (i) DBO performs better than DO and BO for cases with ill-conditioned covariance matrix; (ii) In contrast to BO, the issue of eigenvalue crossing is not present in the DBO formulation; (iii) In contrast to DO, the stochastic modes are orthonormal, which leads to more accurate representation of the stochastic subspace. We study the convergence properties of the method and compare it to the DO and BO methods. For demonstration, we consider three cases: (i) stochastic linear advection equation, (ii) stochastic Burgers' equation, and (iii) stochastic incompressible flow over a bump in a channel. Overall we observe improvements in the numerical accuracy of DBO compared against DO and BO.