网格数据与散点数据的统计方法及模态分解：无网格统计与无网格数据驱动模态分析
Statistical Methods and Modal Decompositions for Gridded and Scattered Data: Meshless Statistics and Meshless Data Driven Modal Analysis
AI总结：
针对湍流研究中散点与网格流场数据处理需求，本研究提供教程讲解无网格统计计算及基于RBF的无网格数据驱动模态分解方法（含POD、DMD等），并开放相关代码。
摘要：
统计工具对于湍流研究和建模至关重要，湍流中混沌的速度波动跨越广泛的时空尺度。图像测速技术的进展，尤其是基于跟踪的方法，如今可实现高速、高密度粒子图像处理，能收集详细的3D流场数据。本讲座提供一套教程，讲解如何处理这类数据集以提取平均值、二阶矩（湍流应力）等关键量，以及利用本征正交分解（POD）等模态分解方法提取相干模式。在简要回顾统计与模态分析基础后，讲座深入探讨跟踪测速散点数据处理的挑战，并与传统网格数据方法对比。还涵盖研究主题，包括用于无网格湍流统计计算的物理基径向基函数（RBF）回归，以及支持无网格数据驱动分解计算的RBF内积定义，涉及POD、动态模态分解（DMD）等传统方法，以及谱POD（SPOD）、多尺度POD（mPOD）等高级变体。我们将此方法称为“无网格数据驱动分解”框架，所有代码可在https://github.com/mendezVKI/PIV_LS_2024_Signal获取。
英文摘要：
Statistical tools are crucial for studying and modeling turbulent flows, where chaotic velocity fluctuations span a wide range of spatial and temporal scales. Advances in image velocimetry, especially in tracking-based methods, now allow for high-speed, high-density particle image processing, enabling the collection of detailed 3D flow fields. This lecture provides a set of tutorials on processing such datasets to extract essential quantities like averages, second-order moments (turbulent stresses) and coherent patterns using modal decompositions such as the Proper Orthogonal Decomposition (POD). After a brief review of the fundamentals of statistical and modal analysis, the lecture delves into the challenges of processing scattered data from tracking velocimetry, comparing it to traditional gridded-data approaches. It also covers research topics, including physics-based Radial Basis Function (RBF) regression for meshless computation of turbulent statistics and the definition of an RBF inner product, which enables meshless computation of data-driven decompositions. These include traditional methods like Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD), as well as advanced variants such as Spectral POD (SPOD) and Multiscale POD (mPOD). We refer to this approach as the "Meshless Data-Driven Decomposition" framework. All codes are available at https://github.com/mendezVKI/PIV_LS_2024_Signal
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
