MATLAB Implementation of Proper Orthogonal Decomposition (POD) with Code Description

Proper Orthogonal Decomposition (POD) is a dimensionality reduction technique widely used in fluid dynamics, structural dynamics, and other fields, capable of extracting dominant modal features from high-dimensional data. Implementing POD in MATLAB typically involves the following key steps: Data Preparation: The first step requires acquiring time-series or spatial distribution data, typically stored in matrix format where each column represents observations at a time step or spatial sampling points. In MATLAB implementation, this involves organizing data into a matrix X with dimensions [n_points × n_snapshots]. Mean Centering: To improve decomposition accuracy, data typically undergoes mean-centering by subtracting temporal or spatial averages. This can be implemented using MATLAB's mean function: X_centered = X - mean(X,2) for spatial averaging or mean(X,1) for temporal averaging. Covariance Matrix Computation: The core POD implementation involves either computing the data covariance matrix C = X_centered' * X_centered or directly performing Singular Value Decomposition (SVD) on the data matrix. The SVD approach using MATLAB's built-in svd function is more numerically stable: [U,S,V] = svd(X_centered,'econ'). Modal Extraction: Utilizing the left and right singular vectors obtained from SVD decomposition to identify dominant data modes. The left singular vectors (U) correspond to spatial modes, while the right singular vectors (V) represent temporal coefficients. The singular values (S) indicate the energy distribution among modes. Energy Contribution Analysis: Evaluating each mode's contribution to the data through singular values' magnitude. Typically, low-energy modes can be truncated for dimensionality reduction by selecting modes based on energy threshold: cumulative_energy = cumsum(diag(S).^2)/sum(diag(S).^2). POD's core advantage lies in efficiently reconstructing original data using few modes, making it suitable for flow field analysis, vibration mode extraction, and other applications. In MATLAB, the built-in svd function enables efficient decomposition computation, while additional functionality can be implemented for mode sorting, reconstruction error analysis, and dynamic mode interpretation.