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

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.