Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction
Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction
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DOI：
10.1016/j.jcp.2016.10.033
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发表时间：
2017-02-01
影响因子：
4.1
通讯作者：
Antil, Harbir
中科院分区：
文献类型：
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作者：
Carlberg, Kevin;Barone, Matthew;Antil, Harbir
最小二乘彼得罗夫-伽辽金（LSPG）模型降阶技术，如高斯牛顿近似张量（GNAT）方法已显示出希望，因为它们已经产生了稳定，准确的解决方案，大规模的湍流，可压缩流问题的标准伽辽金技术失败。然而，对这两种方法的比较分析有限。这部分是由于困难所产生的事实，Galerkin技术进行最佳投影与残留最小化在时间连续的水平，而LSPG技术这样做在时间离散level.This工作提供了一个详细的理论和计算比较两种技术的两个常见的时间积分：线性多步计划和Runge-Kutta计划。我们提出了一些新的发现，包括条件下，LSPG ROM具有时间连续表示，条件下，这两种技术是等效的，和时间离散误差界的两种方法。也许最令人惊讶的是，我们从理论上和计算上证明，减少时间步长并不一定会减少LSPG ROM的误差，相反，时间步长应该“匹配”的频谱内容的减少的基础。在一个超过一百万个未知数的湍流可压缩流问题进行的数值实验中，我们表明，增加时间步长到一个中间值减少了一个数量级的误差和模拟时间的LSPG降阶模型。(C)2016 Elsevier Inc. All rights reserved.
Least-squares Petrov-Galerkin (LSPG) model-reduction techniques such as the Gauss Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time discrete level.This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge-Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be 'matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude. (C) 2016 Elsevier Inc. All rights reserved.