Speaker
Description
The Dynamic Mode Decomposition (DMD) is a tool of the trade in computational data driven analysis of complex dynamical systems, e.g. fluid flows, where it can be used to reveal coherent structures by decomposing the flow field into component fluid structures, called DMD modes, that describe the evolution of the flow. The theoretical underpinning of the DMD is the Koopman composition operator that can be used for a spectral analysis of nonlinear dynamical system.
One of the computational/numerical challenges in the Koopman/DMD framework is the case of non-normal operator, when the eigenvectors of the discretized problem become severely ill-conditioned. To alleviate the potential problem of ill-conditioned eigenvectors in the existing implementations of the Dynamic Mode Decomposition (DMD) and the Extended Dynamic Mode Decomposition (EDMD), we introduce a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics.
The new approach introduces a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator compression is replaced with a modal decomposition in terms of a flag of invariant subspaces that correspond to selected eigenvalues.
The main computational tool from the numerical linear algebra is the partial ordered Schur decomposition that provides convenient orthonormal bases for these subspaces. The new computational scheme is presented in the framework of the Extended DMD, with the same functionalities (snapshot reconstruction and forecasting) and the kernel trick is used.
