Randomized methods to characterize large-scale vortical flow network

TL;DR

This paper introduces randomized linear algebra techniques, specifically the Nyström method, to efficiently analyze large-scale vortical flow networks, enabling faster computation of key network properties in high-dimensional turbulent flows.

Contribution

The paper presents a novel application of randomized methods, like the Nyström method, for approximating network eigenvalues and eigenvectors in vortical flow analysis, reducing computational costs.

Findings

Randomized methods effectively approximate network eigenvalues in high-dimensional flows.

Quasi-uniform column sampling outperforms uniform sampling in accuracy.

Significant computational savings achieved with minimal loss of accuracy.

Abstract

We demonstrate the effective use of randomized methods for linear algebra to perform network-based analysis of complex vortical flows. Network theoretic approaches can reveal the connectivity structures among a set of vortical elements and analyze their collective dynamics. These approaches have recently been generalized to analyze high-dimensional turbulent flows, for which network computations can become prohibitively expensive. In this work, we propose efficient methods to approximate network quantities, such as the leading eigendecomposition of the adjacency matrix, using randomized methods. Specifically, we use the Nystr\"om method to approximate the leading eigenvalues and eigenvectors, achieving significant computational savings and reduced memory requirements. The effectiveness of the proposed technique is demonstrated on two high-dimensional flow fields: two-dimensional flow past an airfoil and two-dimensional turbulence. We find that quasi-uniform column sampling outperforms uniform column sampling, while both feature the same computational complexity.