AAROC: Reduced Over-Collocation Method with Adaptive Time Partitioning and Adaptive Enrichment for Parametric Time-Dependent Equations

TL;DR

This work tackles the high online cost and instability of reduced-order models for nonlinear, parametric time-dependent PDEs by introducing AAROC, which fuses adaptive time partitioning with adaptive collocation enrichment within a Reduced Residual Reduced Over-Collocation framework. The offline-online greedy process builds a small reduced basis and segment-specific collocation sets, using a residual-based robustness indicator to trigger enrichment or time-domain partitioning, while GEIM/EIM determine collocation points. Numerical tests on the viscous Burgers' equation and a lid-driven cavity problem show that AAROC achieves comparable or better accuracy than AROC with fewer collocation points, improved stability, and enhanced online efficiency. The results highlight the method's potential for robust, efficient ROMs in convection-dominated, nonlinear parametric flows and suggest avenues for extending to nonlinear ROM frameworks and nonuniform time partitioning.

Abstract

Nonlinear and nonaffine terms in parametric partial differential equations can potentially lead to a computational cost of a reduced order model (ROM) that is comparable to the cost of the original full order model (FOM). To address this, the Reduced Residual Reduced Over-Collocation method (R2-ROC) is developed as a hyper-reduction method within the framework of the reduced basis method in the collocation setting. R2-ROC greedily selects two sets of reduced collocation points based on the (generalized) empirical interpolation method for both solution snapshots and residuals, thereby avoiding the computational inefficiency. The vanilla R2-ROC method can face instability when applied to parametric fluid dynamic problems. To address this, an adaptive enrichment strategy has been proposed to stabilize the ROC method. However, this strategy can involve in an excessive number of reduced collocation points, thereby negatively impacting online efficiency. To ensure both efficiency and accuracy, we propose an adaptive time partitioning and adaptive enrichment strategy-based ROC method (AAROC). The adaptive time partitioning dynamically captures the low-rank structure, necessitating fewer reduced collocation points being sampled in each time segment. Numerical experiments on the parametric viscous Burgers' equation and lid-driven cavity problems demonstrate the efficiency, enhanced stability, and accuracy of the proposed AAROC method.

Figures (10)

• Figure 1: Schematic illustration of the adaptive time partitioning technique of AAROC method. Here, $\text{len}={\mathcal{N}_t}/{N_{\text{tpar}}}$, and ${\mathcal{T}}_j$ is the set of uniformly discretized time nodes $t_i$ within $j$-th time segment $I_j$, $j=1, \cdots, N_{\text{tpar}}$.
• Figure 2: Results of the viscous Burgers' equation: High fidelity solution (Top Left) and AAROC approximations $\widehat{u}_n(x,t)$ at $\mu =0.0950$ with $p_{\rm{adap}} =0.2$, $n=20, 40, 80$ at $\mu =0.0950$.
• Figure 3: Results of the viscous Burgers' equation: Error estimators of the offline process on the training set and relative errors of the online process on the testing set for $p_{\text{adap}}=0.1, 0.2$, and $0.3$ (Top to Bottom).
• Figure 4: Results of the viscous Burgers' equation: Error estimators of the offline process on the training set and relative errors of the online process on the testing set of the AAROC method.
• Figure 5: Results of the viscous Burgers' equation: Distribution of selected parameter-time pairs for the AROC and AAROC basis (Left, and Middle), and that of the selected parameter-time pairs for the EIM process of the second set of collocation points of AAROC method (Right).