Time-dependent basis reduced order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values, and (iv) error accumulation due to fixed rank. To this end, we present a scalable method based on oblique projections for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive, and highly parallelizable. These favorable properties are achieved via low-rank approximation of the time discrete MDE. Using the discrete empirical interpolation method (DEIM), a low-rank decomposition is computed at each iteration of the time stepping scheme, enabling a near-optimal approximation at a fraction of the cost. We coin the new approach TDB-CUR since it is equivalent to a CUR decomposition based on sparse row and column samples of the MDE. We also propose a rank-adaptive procedure to control the error on-the-fly. Numerical results demonstrate the accuracy, efficiency, and robustness of the new method for a diverse set of problems.

翻译：时间依赖基降阶模型（TDB ROM）已成功应用于近似求解非线性随机偏微分方程（PDEs）。对于许多实际感兴趣的问题，离散化这些PDEs会产生规模巨大的矩阵微分方程（MDEs），传统方法难以承受其求解代价。尽管TDB ROMs有潜力显著降低这一计算负担，但仍面临以下挑战：（i）对一般非线性问题效率低下，（ii）侵入式实现，（iii）存在小奇异值时病态化，以及（iv）固定秩导致的误差累积。为此，我们提出一种基于斜投影的可扩展方法，用于求解TDB ROMs，该方法具有计算高效、最小程度侵入、对小奇异值鲁棒、秩自适应且高度并行化的优点。通过时间离散MDE的低秩近似实现了这些优良性质。利用离散经验插值方法（DEIM），在时间步进方案的每次迭代中计算低秩分解，从而以极小代价获得近最优近似。我们将新方法命名为TDB-CUR，因其等价于基于MDE稀疏行列采样的CUR分解。我们还提出一种秩自适应程序以实时控制误差。数值结果展示了新方法在多种问题上的准确性、高效性和鲁棒性。