CUR-Based Implicit Integration of Random Partial Differential Equations on Low-Rank Manifolds*

This work presents a cost-effective approach for the implicit time integration of parametric partial differential equations (PDEs) on low-rank matrix manifolds using time-dependent bases (TDBs). This enables the low-rank approximation of stiff PDEs, providing a computationally efficient solution. The key to the cost-effectiveness of the proposed methodology is the utilization of a CUR matrix decomposition for low-rank approximation, which avoids costly nonlinear solves. By evaluating the PDE only at selected points, the method achieves substantial speedups compared to full-order model integration. Moreover, the presented framework incorporates rank adaptivity, allowing for the adjustment of the approximation rank over time. This rank adaptivity facilitates error control and ensures that only the necessary modes are retained to achieve the desired accuracy. The efficacy of the implicit TDB method is demonstrated through several analytical and PDE examples, including the stochastic Burgers' and Gray-Scott equations.