L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations
L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations
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DOI：
10.1007/s10915-021-01416-z
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发表时间：
2020-09
影响因子：
2.5
通讯作者：
Yanlai Chen;Lijie Ji;A. Narayan;Zhenli Xu
中科院分区：
文献类型：
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作者：
Yanlai Chen;Lijie Ji;A. Narayan;Zhenli Xu
在优化、控制或交互式应用中重复求解参数化偏微分方程 (pPDE) 的任务使得设计高效且同样准确的替代模型势在必行。降基法（RBM）就是这样的一种选择。伴随着数学上严格的误差估计器，RBM 仔细构建了参数引起的高保真解流形的低维子空间，并在其上计算近似解。利用离线在线分解过程，它可以将效率提高几个数量级。然而，这种分解通常借助非线性和/或参数非仿射偏微分方程的经验插值法 (EIM) 来实现，但实施起来可能具有挑战性，或者导致在线效率严重下降。在本文中，我们将 EIM 方法增强和扩展为直接求解器（而不是助手），用于在简化水平上求解非线性 pPDE。由此产生的方法称为减少过度配置方法 (ROC)，它是稳定的并且能够避免 EIM 传统应用中出现的效率下降。该方案的两个关键要素是在约两倍于缩减近似空间维度的位置上进行配置，以及一个有效的基于 L1 范数的误差指示器，用于策略性选择其快照跨越缩减近似空间的参数值。这两个要素共同确保了所提出的 L1-ROC 方案同时具有离线和在线效率。一个显着特征是，在离线和在线阶段都避免了利用 EIM 解决非线性和非仿射问题的替代 RBM 方法中出现的效率下降。对不同族瞬态和稳态非线性问题的数值测试证明了L1-ROC的高效率和准确性及其优越的稳定性能。
The task of repeatedly solving parametrized partial differential equations (pPDEs) in optimization, control, or interactive applications makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold on which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However this decomposition, usually implemented with aid from the empirical interpolation method (EIM) for nonlinear and/or parametric-nonaffine PDEs, can be challenging to implement, or results in severely degraded online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding efficiency degradation exhibited in traditional applications of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced approximation space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values whose snapshots span the reduced approximation space. Together, these two ingredients ensure that the proposed L1-ROC scheme is both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.