RECONSTRUCTION OF TURBULENT DATA WITH GAPPY POD METHOD
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摘要: Gappy POD 是一种基于本征正交分解(proper orthogonal decomposition, POD)的数据重构方法. 本文研究了gappy POD在湍流数据重构中的应用, 主要关注了以下两个因素的影响: 第一, 数据本身的复杂程度, 即构成流场的POD模态数量; 第二, 破损区域的面积大小和几何形状. 考虑到上述因素, 本文重新严格地表述了gappy POD的重构过程, 并推导出gappy POD重构误差的公式. 论文选取旋转湍流数据为案例进行了gappy POD重构的研究, 并解释了构成gappy POD重构误差的两个部分. 第一部分来自流场POD展开的截断误差, 该截断误差会被POD基函数在已知点上的值组成的矩阵的最小特征值放大. 这部分误差主要取决于流场的复杂程度, 当流场复杂程度较低时, 相应误差随采用的POD模态数目增大而减小. 当流场复杂程度较高时, 很小的POD截断误差也会导致很大的重构误差, 此时需要采用流场所有的POD模态进行重构以消除截断误差. 重构误差的第二部分来自POD基函数在已知点上的值组成的矩阵的非列满秩性, 它主要取决于破损区域的面积大小和几何形状. 破损区域的面积越大, 或者破损面积相同时, 破损区域内信息所包含的相关性越大, 第二部分的重构误差越大.Abstract: Gappy POD is a method of data reconstruction based on the proper orthogonal decomposition (POD). We study the applicability of gappy POD to the reconstruction of fluid turbulence configurations and focus mainly on two factors. The first factor is the complexity of the data, which mostly depends on the number of POD modes with non-zero eigenvalues. The second factor is the area and the geometry of the gap. By taking these factors into account, we reformulate the gappy POD reconstruction and derive a formula to compute the reconstruction error. Rotating turbulence data is used as a case study of gappy POD reconstruction, where the reconstruction error can be separated into two parts. The first contribution to the reconstruction error is from the truncation error during the POD expansion and it is amplified by the smallest eigenvalue of the matrix, which consists of POD modes at known indexes. This error mainly depends on the flow complexity, e.g. for flow of moderate complexity, this error decreases with the increase in number of POD modes employed during the reconstruction process. For flow of large complexity, a small POD truncation error can be detrimental and contribute signification to the reconstruction error. Therefore, all POD modes should be considered when utilizing Gappy POD reconstruction to eliminate the truncation error, especially for the turbulent flow field. The second part of the reconstruction error appears when the matrix composed of POD modes at the known points is not of full column rank. This part of error depends on the area and the geometry of the gap. The gap area determines the amount of the lost information. For the same gap area, the gap geometry determines the correlation of the lost information. Gappy POD reconstruction works well when both the amount and the correlation of the lost information are small.
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Key words:
- gappy POD /
- flow reconstruction /
- rotating turbulence
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2 (a)-(e) 破损面积相同但破损区域几何不同的部分缺失流场, (f) 真实的完整流场
2. (a)-(e) Damaged flow fields with gaps of the same area but different geometries. (f) The complete flow field
图 2 (a)-(e) 破损面积相同但破损区域几何不同的部分缺失流场, (f) 真实的完整流场 (续)
Figure 2. (a)-(e) Damaged flow fields with gaps of the same area but different geometries. (f) The complete flow field (continued)
图 3 破损区域中的均方重构误差
$ {\rm{MS}}{{\rm{E}}_{{\rm{gap}}}} $ 关于${{{N_{{\rm{comp}}}}} \mathord{\left/ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} \right. } {{N_{{\rm{flow}}}}}}$ 的变化曲线Figure 3. Normalized mean square error in the gap,
$ {{MS}}{{{E}}_{{\rm{gap}}}} $ , as a function of${{{N_{{\rm{comp}}}}} \mathord{\left/ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} \right. } {{N_{{\rm{flow}}}}}}$ 
图 4
$\boldsymbol{\tilde X}$ 的最小奇异值$ {\sigma _{\min }} $ 关于${{N_{{\rm{comp}}}}}$ /$ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} {{N_{{\rm{flow}}}}} $ 的变化曲线Figure 4. The minimum singular value of
$ \boldsymbol{\tilde X} $ ,$ {\sigma _{\min }} $ , as a function of${{N_{{\rm{comp}}}}}$ /${ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} } {{N_{{\rm{flow}}}}} $ 
图 5 对于不同几何形状的破损, 破损区域中的均方重构误差
$ {{MS}}{{{E}}_{{\rm{gap}}}} $ 关于破损尺寸的变化曲线Figure 5. Normalized mean square error in the gap,
$ {{MS}}{{{E}}_{{\rm{gap}}}} $ , as a function of the gap size for different gap geometries
图 6 对于不同几何形状的破损,
${N_{{\rm{flow}}}} - r\left( {\boldsymbol{\tilde X}} \right)$ 关于破损尺寸的变化曲线Figure 6.
${N_{{\rm{flow}}}} - r\left( {\boldsymbol{\tilde X}} \right)$ as a function of the gap size for different gap geometries
图 7 相同破损面积, 不同破损几何的gappy POD重构结果(对应不同行). 第1列: 破损的流场; 第2列: 重构的流场; 第3列: 原始流场; 第4列: 第一列中红色参考线上重构结果(虚线)与真实结果(实线)的分布; 第5列: 平均重构误差(红色),
$ {\varDelta _u}\left( {{x_1}} \right) $ , 及其在测试集上的平均(黑色),$ \left\langle {{\varDelta _u}\left( {{x_1}} \right)} \right\rangle $ .${x_1}$ 和${x_2}$ 分别代表水平和竖直方向Figure 7. Gappy POD reconstruction results for the same gap area and different gap geometries (one for each row). 1st column: damaged image in input. 2nd column: image generated in output. 3rd column: ground truth. 4th column: generated (dashed) and ground truth (solid) profiles along the vertical line shown in the 1st column. 5th column: mean reconstruction error,
$ {\varDelta _u}\left( {{x_1}} \right) $ , for each image (red line) and the average error,$ \left\langle {{\varDelta _u}\left( {{x_1}} \right)} \right\rangle $ , over the test images (black curve). Note that${x_1}$ and${x_2}$ denotes the horizontal and vertical directions, respectively -
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