Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected Systems

Broomhead, D. S. and Kirby, M. J. (2005) Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected Systems. Nonlinear Dynamics, 41 (1-3). pp. 47-67. ISSN 1573-269X

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Abstract

In previous papers we have developed an approach to the data reduction problem which is based on a well-known, constructive proof of Whitney’s embedding theorem [Broomhead, D. S. and Kirby, M., SIAM Journal of Applied Mathematics 60(6), 2000, 2114–2142; Broomhead, D. S. and Kirby, M., Neural Computation 13, 2001, 2595–2616]. This approach involves picking projections of the high-dimensional system which are optimised in the sense that they are easy to invert. This is done by considering the effect of the projections on the set of unit secants constructed from the data. In the present paper we discuss the implications of this idea in the case that the high-dimensional data is generated by a dynamical system. We ask if the existence of an easily invertible projection leads to practical methods for the construction of an equivalent, low-dimensional dynamical system. The paper consists of a review of the secant-based projection method and simple methods for finding good representations of the (nonlinear) inverse of the projections. We then discuss two variants of a way to find the dynamical system induced by a projection which lead to quite distinct numerical approximations. One of these is developed further as we describe various ways in which knowledge of the full dynamical system can be incorporated into the approximate projected system. The ideas of the paper are illustrated in some more or less simple examples, which range from a simple system of nonlinear ODEs which have an attracting limit cycle, to low-dimensional solutions of the Kuramoto–Sivashinsky equation which need many Galerkin modes for their description.

Item Type:	Article
Additional Information:	The original publication is available at www.springerlink.com.
Uncontrolled Keywords:	reduced dynamical systems, secant projections
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory MSC 2010, the AMS's Mathematics Subject Classification > 53 Differential geometry MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Dr Mark Muldoon
Date Deposited:	02 Mar 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/168

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