Speaker
Description
The Dynamic Mode Decomposition (DMD) is a powerful and versatile numerical method for data driven analysis of nonlinear dynamical systems, with a wide spectrum of applications. It can be used for model order reduction, analysis of latent structures in the dynamics, and e.g. for forecasting and control. The theoretical bedrock upon which the more general Extended DMD (EDMD) framework is built is the Koopman composition operator. The EDMD can be described as a data driven Rayleigh-Ritz extraction of spectral information of a Koopman operator associated with the underlying dynamical system. The nonlinear data snapshots are represented using the eigenvectors of the operator resulting in a modal decomposition KMD (Koopman Mode Decomposition). This becomes a model order reduction tool that represents the nonlinear dynamics using selected eigenpairs. It can be used to reveal coherent states and for forecasting.
The numerical realization of the Koopman operator framework (i.e. DMD and EDMD) for computational analysis of nonlinear dynamics is an excellent illustration of the importance and power of numerical linear algebra, and an instructive case study of the software development based on the design principles introduced in the state of the art software packages such as e.g. LAPACK. The main ingredients of all variants of the DMD are the SVD and low rank approximations,the QR factorization and the orthogonal projections, structured least squares approximations,
and approximations of eigenvalues and eigenvectors from subspaces.
The problem of illconditioned eigenvectors is solved using a Koopman-Schur decomposition, based on unitary transformations. The analysis in terms of the eigenvectors is replaced with a modal decomposition in terms of flag of invariant subspaces that correspond to selected eigenvalues. Accuracy is controlled using computable residuals.
We will discuss these and other fine details that are built in robust software solutions.