An Adaptive CUR Algorithm and its Application to Reduced-Order Modeling of Random PDEs

TL;DR

This paper addresses the challenge of robust, data-efficient low-rank approximations for nonlinear matrix differential equations arising from reduced-order modeling of random PDEs. It introduces CUR-CR-OS, an adaptive cross oversampling strategy that samples only cross entries to improve stability while controlling data access, and combines this with DLRA for time integration on low-rank matrix manifolds. A deterministic error bound is derived, and an adaptive oversampling and rank-adaptation framework is proposed to balance accuracy and computational cost. Numerical demonstrations on nonlinear stochastic PDEs (Burgers, Allen-Cahn, KdV) and a heat-conduction-radiation problem show that CUR-CR-OS achieves accuracy comparable to SVD-based approaches, with enhanced robustness and scalability, especially for high-dimensional problems.

Abstract

Certain classes of CUR algorithms, also referred to as cross or pseudoskeleton algorithms, are widely used for low-rank matrix approximation when direct access to all matrix entries is costly. Their key advantage lies in constructing a rank-r approximation by sampling only r columns and r rows of the target matrix. This property makes them particularly attractive for reduced-order modeling of nonlinear matrix differential equations, where nonlinear operations on low-rank matrices can otherwise produce high-rank or even full-rank intermediates that must subsequently be truncated to rank $r$. CUR cross algorithms bypass the intermediate step and directly form the rank-$r$ matrix. However, standard cross algorithms may suffer from loss of accuracy in some settings, limiting their robustness and broad applicability. In this work, we propose a cross oversampling algorithm that augments the intersection with additional sampled columns and rows. We provide an error analysis demonstrating that the proposed oversampling improves robustness. We also present an algorithm that adaptively selects the number of oversampling entries based on efficiently computable indicators. We demonstrate the performance of the proposed CUR algorithm for time integration of several nonlinear stochastic PDEs on low-rank matrix manifolds.

Figures (9)

• Figure 1: Schematic of CUR oversampling methodology using cross oversampling (CR-OS), to oversampling $m_r$ rows and $m_c$ columns of matrix $\mathbf A \in \mathbb R^{n\times s}$. We observe the only extra cost resulting from oversampling is $m_rm_c$, which are the total number of cross points used for oversampling.
• Figure 2: Comparative analysis of fast and slow decay for matrix low-rank approximation: (a), (b) relative error $\mathcal{E}$ versus reduction order $r$.
• Figure 3: Comparative analysis of fast and slow decay for matrix low-rank approximation: (a), (b) condition number for row $\eta_p$ and column $\eta_s$ for CUR-CR ($m=0$) i.e. without oversampling and CUR-CR-OS i.e. with adaptive column and row oversampling versus reduction order $r$. (c),(d) adaptive row ($m_r$) and column ($m_c$) oversampling used for CUR-CR-OS versus reduction order $r$.
• Figure 4: Stochastic Burgers: (a) Analysis of the normalized TDB-CUR-CR-OS error ($\mathcal{E}$) to the rank ($r = 18, 40, 100$) without oversampling ($m=0$, solid line) and with adaptive oversampling ($m_r, m_c$ adaptive, dotted line). (b) Sensitivity study of the effect of chosen oversampling bound $(\epsilon_{os})$ on the normalized error, with constant rank $r=18$.
• Figure 5: Mean flow solution for Burgers, Allen-Cahn, and Korteweg-De Vries equations with the rank adaptive QDEIM points (black dots).

Theorems & Definitions (7)

• definition 1: Low-rank matrix manifold
• lemma 1
• proof
• lemma 2
• proof
• theorem 1
• proof