A Priori Error Bounds for POD-ROMs for Fluids: A Brief Survey

TL;DR

This survey addresses a gap in POD-based reduced order modeling for incompressible flows by consolidating the development of a priori error bounds. It differentiates POD from RBM analyses, presents the canonical a priori bound form $\| \boldsymbol{u} - {\boldsymbol{u}}_r \| \leq C \, f(\Delta t, h, r, \{\lambda_i\}_{i=r+1}^{R}, \{\boldsymbol{\varphi}_i\}_{i=r+1}^{R})$ with asymptotic rates like $\| \boldsymbol{u} - {\boldsymbol{u}}_r \| = \mathcal{O}(h^{p_1}) + \mathcal{O}(\Delta t^{p_2}) + \mathcal{O}\left( (\sum_{i=r+1}^{R} \lambda_i)^{p_3} \right)$, and contrasts it with a posteriori bounds $\| \boldsymbol{u}_h - {\boldsymbol{u}}_r \| \leq s(\Delta t, h) \ d(\Delta t, h, {\boldsymbol{u}}_r)$ that are computable for fixed FOM parameters. It surveys key contributions across error bounds, optimal pointwise rates, FOM-ROM coupling and parameter scalings, and analyses for pressure, stabilization, closure, and data assimilation/control, highlighting how DQs restore optimal convergence and how coupling with FOM discretization yields robust scalings. The work emphasizes practical impact: guiding ROM dimension choices, enabling cross-problem scalability, and informing stabilization and data-driven approaches for realistic turbulent flows, while noting limitations such as dependence on the exact solution for a priori bounds. Overall, the survey frames a path toward robust, certifiable POD-ROMs suitable for high-Reynolds-number and HPC contexts.”

Abstract

Galerkin reduced order models (ROMs), e.g., based on proper orthogonal decomposition (POD) or reduced basis methods, have achieved significant success in the numerical simulation of fluid flows. The ROM numerical analysis, however, is still being developed. In this paper, we take a step in this direction and present a survey of a priori error bounds, with a particular focus on POD-based ROMs. Specifically, we outline the main components of ROM a priori error bounds, emphasize their practical importance, and discuss significant contributions to a priori error bounds for ROMs for fluids.