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Tensor-based computation of metastable and coherent sets
lib:9aad8de8659bb2b6 (v1.0.0)
Authors: Feliks Nüske,Patrick Gelß,Stefan Klus,Cecilia Clementi
ArXiv: 1908.04741
Document: PDF DOI
Artifact development version: GitHub
Abstract URL: https://arxiv.org/abs/1908.04741v2
Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory -- with extended dynamic mode decomposition (EDMD) being a cornerstone of the field. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT) format -- have become a valuable tool for the solution of large-scale problems in a number of fields. In this work, we combine EDMD and the TT format, enabling the application of EDMD to high-dimensional problems in conjunction with a large set of features. We derive efficient algorithms to solve the EDMD eigenvalue problem based on tensor representations of the data, and to project the data into a low-dimensional representation defined by the eigenvectors. We extend this method to perform canonical correlation analysis (CCA) of non-reversible or time-dependent systems. We prove that there is a physical interpretation of the procedure and demonstrate its capabilities by applying the method to several benchmark data sets.
ArXiv: 1908.04741
Document: PDF DOI
Artifact development version: GitHub
Abstract URL: https://arxiv.org/abs/1908.04741v2
Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory -- with extended dynamic mode decomposition (EDMD) being a cornerstone of the field. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT) format -- have become a valuable tool for the solution of large-scale problems in a number of fields. In this work, we combine EDMD and the TT format, enabling the application of EDMD to high-dimensional problems in conjunction with a large set of features. We derive efficient algorithms to solve the EDMD eigenvalue problem based on tensor representations of the data, and to project the data into a low-dimensional representation defined by the eigenvectors. We extend this method to perform canonical correlation analysis (CCA) of non-reversible or time-dependent systems. We prove that there is a physical interpretation of the procedure and demonstrate its capabilities by applying the method to several benchmark data sets.
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