A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space-time ROM for linear dynamical problems has been developed, which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 0.00001 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space-time Petrov-Galerkin projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space-time ROM method with two model problems: 2D diffusion and 2D convection diffusion, with and without a linear source term. For each problem, we demonstrate the entire process of generating the full order model (FOM) data, constructing the space-time ROM, and predicting the reduced-order solutions, all in less than 120 lines of Python code. We compare our Petrov-Galerkin method with the traditional Galerkin method and show that the space-time ROMs can achieve O(100) speed-ups with O(0.001) to O(0.0001) relative errors for these problems. Finally, we present an error analysis for the space-time Petrov-Galerkin projection and derive an error bound, which shows an improvement compared to traditional spatial Galerkin ROM methods.

中文翻译：

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使用少于120行代码的Python线性动力学系统的高效时空缩减阶模型

用于动态问题的经典降阶模型（ROM）通常仅涉及给定问题的空间缩小。近来，已经开发出了用于线性动力学问题的新颖的时空ROM，其通过在不减小精度的情况下在空间减小的同时引入时间上的减小来进一步减小问题的大小。对于大规模的玻尔兹曼输运问题，作者显示出加速了数千个级别，相对误差小于0.00001。在这项工作中，我们首次提出了线性动力学系统及其相应块结构的时空彼得罗夫-加勒金投影的推导。利用这些块结构，我们展示了具有两个模型问题的时空ROM方法的构造简便性：2D扩散和2D对流扩散，有和没有线性源项。对于每个问题，我们都演示了生成完整订单模型（FOM）数据，构建时空ROM以及预测降阶解决方案的全过程，所有这些过程均少于120行Python代码。我们将Petrov-Galerkin方法与传统的Galerkin方法进行了比较，结果表明，时空ROM可以实现O（100）加速，相对误差为O（0.001）至O（0.0001）。最后，我们提出了时空彼得罗夫-加勒金投影的误差分析，并得出了误差范围，与传统的空间伽勒金ROM方法相比，它显示出一种改进。全部都用不到120行Python代码。我们将Petrov-Galerkin方法与传统的Galerkin方法进行了比较，结果表明，时空ROM可以实现O（100）加速，相对误差为O（0.001）至O（0.0001）。最后，我们提出了时空彼得罗夫-加勒金投影的误差分析，并得出了误差范围，与传统的空间伽勒金ROM方法相比，它显示出一种改进。全部都用不到120行Python代码。我们将Petrov-Galerkin方法与传统的Galerkin方法进行了比较，结果表明，时空ROM可以实现O（100）加速，相对误差为O（0.001）至O（0.0001）。最后，我们提出了时空彼得罗夫-加勒金投影的误差分析，并得出了误差范围，与传统的空间伽勒金ROM方法相比，它显示出一种改进。