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Model Free Techniques for Reduction of High-Dimensional Dynamics

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dc.contributor.advisor Sauer, Timothy Berry, Tyrus
dc.creator Berry, Tyrus en_US 2013-08-09T15:39:28Z 2013-08-09T15:39:28Z 2013 en_US
dc.description.abstract There is a growing need in science and engineering to extract information about complex phenomena from large data sets. A rapidly developing approach to building a model from data is manifold learning, and analysis of such a model may allow isolation of the desired features of the data. By introducing an additional geometric structure, the techniques of differential geometry become available for analyzing the model. In this dissertation we extend previous methods of analyzing the geometry of data. Our key contribution is the theory of local kernels, which generalizes previous nonparametric techniques such as Laplacian eigenmaps and diffusion maps. We show that every geometry can be represented by a local kernel in the limit of large data. Moreover, using the discrete exterior calculus (DEC) we show that a local kernel can be used to introduce a discrete Hodge star operator on a data set. This shows that local kernels introduce a discrete geometry on a data set without the need for an explicit simplicial complex.
dc.format.extent 150 pages en_US
dc.language.iso en en_US
dc.rights Copyright 2013 Tyrus Berry en_US
dc.subject Mathematics en_US
dc.subject diffusion maps en_US
dc.subject dynamical systems en_US
dc.subject geometry of data en_US
dc.subject kalman filtering en_US
dc.subject local kernels en_US
dc.subject manifold learning en_US
dc.title Model Free Techniques for Reduction of High-Dimensional Dynamics en_US
dc.type Dissertation en Doctoral en Mathematics en George Mason University en

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