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 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 CUR decomposition is computed at each iteration of the time stepping scheme, enabling a near-optimal approximation at a fraction of the cost. 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：时变基降阶模型已成功用于逼近非线性随机偏微分方程的解。对于许多实际感兴趣的问题，离散化这些偏微分方程会生成庞大的矩阵微分方程，使用传统方法求解成本过高。尽管时变基降阶模型有潜力显著降低计算负担，但仍面临以下挑战：(i) 对一般非线性问题效率低下，(ii) 实现具有侵入性，(iii) 存在小奇异值时病态，以及(iv) 固定秩导致误差累积。为此，我们提出了一种可扩展的方法来求解时变基降阶模型，该方法具有计算高效、最小侵入性、在小奇异值下鲁棒、秩自适应且高度可并行化的特性。这些优良性质通过在时间离散的矩阵微分方程上进行低秩逼近实现。利用离散经验插值方法，在时间步进方案的每次迭代中计算低秩CUR分解，从而以极小的成本实现近乎最优的逼近。我们还提出了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一系列不同问题上的准确性、效率和鲁棒性。