Generalized and Multiscale Modal Analysis

TL;DR

This chapter presents a unified framework for various modal decompositions, introduces the multiscale POD (mPOD), and provides algorithms and Python tools for advanced spectral and multiresolution analysis.

Contribution

It generalizes modal decompositions within a matrix factorization framework and introduces the multiscale POD, combining MRA and POD for enhanced analysis.

Findings

Unified framework for modal decompositions

Introduction of multiscale POD (mPOD)

Comparison of DFT, POD, DMD, EF on example

Abstract

This chapter describes modal decompositions in the framework of matrix factorizations. We highlight the differences between classic space-time decompositions and 2D discrete transforms and discuss the general architecture underpinning \emph{any} decomposition. This setting is then used to derive simple algorithms that complete \emph{any} linear decomposition from its spatial or temporal structures (bases). Discrete Fourier Transform, Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), and Eigenfunction Expansions (EF) are formulated in this framework and compared on a simple exercise. Finally, this generalization is used to analyze the impact of spectral constraints on the classical POD, and to derive the Multiscale Proper Orthogonal Decomposition (mPOD). This decomposition combines Multiresolution Analysis (MRA) and POD. This chapter contains four exercises and two tutorial test cases. The \textsc{Python} scripts associated to these are provided on the book's website.