A meshless method to compute the proper orthogonal decomposition and its variants from scattered data

TL;DR

This paper introduces a meshless, physics-constrained radial basis function method to perform proper orthogonal decomposition directly on scattered data, eliminating the need for costly interpolation and improving accuracy in analyzing complex spatiotemporal phenomena.

Contribution

The proposed method enables direct decomposition of scattered data into principal components without interpolation, offering higher accuracy and mesh independence compared to traditional techniques.

Findings

Higher accuracy in component extraction.

Mesh-independent decomposition.

Elimination of interpolation step.

Abstract

Complex phenomena can be better understood when broken down into a limited number of simpler "components". Linear statistical methods such as the principal component analysis and its variants are widely used across various fields of applied science to identify and rank these components based on the variance they represent in the data. These methods can be seen as factorisations of the matrix collecting all the data, assuming it consists of time series sampled from fixed points in space. However, when data sampling locations vary over time, as with mobile monitoring stations in meteorology and oceanography or with particle tracking velocimetry in experimental fluid dynamics, advanced interpolation techniques are required to project the data onto a fixed grid before the factorisation. This interpolation is often expensive and inaccurate. This work proposes a method to decompose scattered data without interpolating. The approach employs physics-constrained radial basis function regression to compute inner products in space and time. The method provides an analytical and mesh-independent decomposition in space and time, demonstrating higher accuracy. Our approach allows distilling the most relevant "components" even for measurements whose natural output is a distribution of data scattered in space and time, maintaining high accuracy and mesh independence.