[Paper Review] An Evolve-Then-Filter Regularized Reduced Order Model

David Wells, Zhu Wang+2 more|arXiv (Cornell University)|Jun 24, 2015

Model Reduction and Neural Networks34 references3 citations

TL;DR

This paper proposes a novel evolve-then-filter regularized reduced order model (EF-ROM) for convection-dominated flows, using explicit spatial filtering via POD projection (Proj) and a new POD differential filter (DF) to stabilize unstable standard ROMs. The EF-ROM-DF combination achieves the highest accuracy, significantly outperforming the standard G-ROM and other Reg-ROM/filter variants, with CPU times orders of magnitude lower than DNS.

ABSTRACT

In this paper, we propose a new evolve-then-filter reduced order model (EF-ROM). This is a regularized ROM (Reg-ROM), which aims at the numerical stabilization of proper orthogonal decomposition (POD) ROMs for convection-dominated flows. We also consider the Leray ROM (L-ROM). These two Reg-ROMs use explicit ROM spatial filtering to smooth (regularize) various terms in the ROMs. Two spatial filters are used: a POD projection onto a POD subspace (Proj) and a new POD differential filter (DF). The four Reg-ROM/filter combinations are tested in the numerical simulation of the three-dimensional flow past a circular cylinder at a Reynolds number $Re=1000$. Overall, the most accurate Reg-ROM/filter combination is EF-ROM-DF. Furthermore, the spatial filter has a higher impact on the Reg-ROM than the regularization used. Indeed, the DF generally yields better results than Proj for both the EF-ROM and L-ROM. Finally, the CPU times of the four Reg-ROM/filter combinations are orders of magnitude lower than the CPU time of the DNS.

Motivation & Objective

• Address the numerical instability of standard proper orthogonal decomposition (POD) reduced order models (ROMs) in convection-dominated flows.
• Develop a new regularization strategy using explicit spatial filtering applied directly to ROM terms, rather than preprocessing snapshots.
• Investigate the impact of two spatial filters—POD projection (Proj) and POD differential filter (DF)—on ROM accuracy and stability.
• Compare the new evolve-then-filter ROM (EF-ROM) with the established Leray ROM (L-ROM) across multiple filtering strategies.
• Demonstrate the computational efficiency and accuracy of the proposed Reg-ROM/filter combinations in a 3D flow past a cylinder at Re=1000.

Proposed method

• Propose the evolve-then-filter (EF-ROM) Reg-ROM, which applies spatial filtering after time evolution to stabilize the solution.
• Apply two explicit ROM spatial filters: (1) POD projection (Proj), projecting variables onto a POD subspace, and (2) POD differential filter (DF), a regularization via a differential filtering operation.
• Integrate the filters into the Galerkin ROM formulation by modifying the nonlinear terms and viscous terms using the filtered variables.
• Use the same time discretization (explicit Euler) across all ROMs to ensure fair comparison, with parameters (δ for DF, r₁ for Proj) optimized on a short interval.
• Perform numerical simulations of 3D flow past a circular cylinder at Re=1000 using DNS as a benchmark.
• Assess results using multiple metrics: kinetic energy spectrum, mean velocity, Reynolds stresses, RMS velocity fluctuations, POD coefficient evolution, and Strouhal number.

Experimental results

Research questions

• RQ1 How does the new evolve-then-filter (EF-ROM) Reg-ROM compare in accuracy and stability to the standard G-ROM and existing Reg-ROMs like the L-ROM?
• RQ2 Which spatial filter—POD projection (Proj) or POD differential filter (DF)—yields better accuracy and stability in Reg-ROMs for convection-dominated flows?
• RQ3 What is the relative impact of the regularization method (EF-ROM vs. L-ROM) versus the choice of spatial filter on ROM performance?
• RQ4 Can the EF-ROM with DF filtering achieve high accuracy while maintaining computational efficiency orders of magnitude below DNS?
• RQ5 How do the four Reg-ROM/filter combinations (EF-ROM-Proj, EF-ROM-DF, L-ROM-Proj, L-ROM-DF) perform across key flow metrics like energy spectrum and Reynolds stresses?

Key findings

• The EF-ROM-DF combination was the most accurate across all metrics, significantly outperforming the standard G-ROM and other Reg-ROM/filter combinations.
• The EF-ROM-Proj was the least accurate among the four combinations, indicating that the choice of spatial filter has a greater impact than the regularization method.
• The POD differential filter (DF) consistently yielded better results than the POD projection (Proj) for both EF-ROM and L-ROM, demonstrating the superiority of the DF in stabilizing convection-dominated flows.
• The L-ROM-DF and L-ROM-Proj performed significantly better than the standard G-ROM, confirming the value of regularization in ROMs.
• All four Reg-ROM/filter combinations achieved CPU times that were orders of magnitude lower than the direct numerical simulation (DNS), confirming their computational efficiency.
• The EF-ROM-DF accurately captured key flow features such as the Strouhal number, mean velocity profiles, Reynolds stresses, and kinetic energy spectrum, closely matching the DNS benchmark.