This article presents two extensions of the empirical interpolation method (EIM) designed to deal with vector interpolation problems. These reduced-order modeling techniques are aimed at exploiting pointwise (vector) measurements to obtain the unknown field reconstruction and they are preferred to other, more efficient, techniques as the proper orthogonal decomposition (POD) because of their intrinsic capability to identify measurement positions and to perform field reconstruction. The “EIM-roto” method implements rotation matrix coefficients and should be intended as a composition of rotations and dilatations of the vector basis functions. The “EIM-orto” implements diagonal matrices coefficients and can be intended as the interpolation, component by component, of the unknown vector field, projected on a fixed reference system. The two techniques are tested over the lid-driven cavity benchmark, in laminar conditions. The results obtained on this study case highlight how the “EIM-orto” interpolation does not allow a reliable reconstructions, while the “EIM-roto” interpolation allows reconstruction performances close to the POD ones (here used as reference method). In particular, the worst reconstruction error, that is, the maximum L² − norm of the residuals, decreases exponentially, reaching 5% with 25 basis functions. This result can be consider satisfactory, considering the nature of the problem.

中文翻译：

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二维矢量场和矢量测量的经验插值方法

本文介绍了旨在处理矢量插值问题的经验插值方法 (EIM) 的两个扩展。这些降阶建模技术旨在利用逐点（矢量）测量来获得未知场重建，并且它们优于其他更有效的技术作为适当的正交分解（POD），因为它们具有识别测量位置和进行场重建。“EIM-roto”方法实现了旋转矩阵系数，应该作为矢量基函数的旋转和膨胀的组合。“EIM-orto”实现对角矩阵系数，可以用作投影到固定参考系统上的未知矢量场的逐个分量插值。这两种技术在层流条件下在盖子驱动的腔基准上进行了测试。在本研究案例中获得的结果突出显示了“EIM-orto”插值如何不允许可靠的重建，而“EIM-roto”插值允许重建性能接近 POD 的性能（此处用作参考方法）。特别是最坏的重构误差，即最大L ²  - 残差范数呈指数下降，在 25 个基函数下达到 5%。考虑到问题的性质，这个结果可以认为是令人满意的。