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首页> 外文期刊>Journal of Computational and Applied Mathematics >Model reduction of dynamical systems by proper orthogonal decomposition: Error bounds and comparison of methods using snapshots from the solution and the time derivatives
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Model reduction of dynamical systems by proper orthogonal decomposition: Error bounds and comparison of methods using snapshots from the solution and the time derivatives

机译：适当正交分解的动态系统模型减少：误差界限和使用解决方案快照的方法和时间衍生物的比较

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摘要

We consider two proper orthogonal decomposition (POD) methods for dimension reduction of dynamical systems. The first method (M1) uses only time snapshots of the solution, while the second method (M2) augments the snapshot set with time-derivative snapshots. The goal of the paper is to analyze and compare the approximation errors resulting from the two methods by using error bounds. We derive several new bounds of the error from POD model reduction by each of the two methods. The new error bounds involve a multiplicative factor depending on the time steps between the snapshots. For method M1 the factor depends on the second power of the time step, while for method 2 the dependence is on the fourth power of the time step, suggesting that method M2 can be more accurate for small between-snapshot intervals. However, three other factors also affect the size of the error bounds. These include (i) the norm of the second (for M1) and fourth derivatives (M2); (ii) the first neglected singular value and (iii) the spectral properties of the projection of the system's Jacobian in the reduced space. Because of the interplay of these factors neither method is more accurate than the other in all cases. Finally, we present numerical examples demonstrating that when the number of collected snapshots is small and the first neglected singular value has a value of zero, method M2 results in a better approximation. (C) 2017 Elsevier B.V. All rights reserved.
机译：我们考虑两个适当的正交分解（POD）动力系统尺寸减少的方法。第一种方法（M1）仅使用解决方案的时间快照，而第二种方法（M2）增加了与时间导数快照设置的快照。本文的目标是通过使用错误界限来分析和比较这两种方法产生的近似误差。我们通过两种方法中的每一种从POD模型减少源的几个新界限。根据快照之间的时间步长，新的错误界限涉及乘法因子。对于方法M1，因子取决于时间步骤的第二功率，而对于方法2，依赖性是时间步骤的第四个电源，暗示方法M2可以更准确地在快照间隔中较小。但是，其他三个因素也会影响错误界限的大小。这些包括（i）第二种（对于M1）和第四衍生物（M2）的标准; （ii）第一种被忽略的奇异值和（iii）系统雅加诺在减少空间中的投影的光谱特性。由于这些因素的相互作用，在所有情况下，任何方法都没有比其他方式更准确。最后，我们存在证明当收集的快照的数量小并且第一被忽略的奇异值具有零的值时，方法M2导致更好的近似。 （c）2017年Elsevier B.V.保留所有权利。

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