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Adaptive Dimensionality-Reduction for Time-Stepping in Differential and Partial Differential Equations

Part of: Infinite-dimensional dissipative dynamical systems Numerical methods Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 September 2017

Xing Fu  and
J. Nathan Kutz
Xing Fu
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
J. Nathan Kutz*
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
*
*Corresponding author. Email address:kutz@uw.edu (J. N. Kutz)
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Abstract

A numerical time-stepping algorithm for differential or partial differential equations is proposed that adaptively modifies the dimensionality of the underlying modal basis expansion. Specifically, the method takes advantage of any underlying low-dimensional manifolds or subspaces in the system by using dimensionality-reduction techniques, such as the proper orthogonal decomposition, in order to adaptively represent the solution in the optimal basis modes. The method can provide significant computational savings for systems where low-dimensional manifolds are present since the reduction can lower the dimensionality of the underlying high-dimensional system by orders of magnitude. A comparison of the computational efficiency and error for this method are given showing the algorithm to be potentially of great value for high-dimensional dynamical systems simulations, especially where slow-manifold dynamics are known to arise. The method is envisioned to automatically take advantage of any potential computational saving associated with dimensionality-reduction, much as adaptive time-steppers automatically take advantage of large step sizes whenever possible.

Keywords

Time-steppingdimensionality reductionproper orthogonal decomposition

MSC classification

Secondary: 74S25: Spectral and related methods 65M70: Spectral, collocation and related methods 37L65: Special approximation methods (nonlinear Galerkin, etc.)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 872 - 894
Copyright
Copyright © Global-Science Press 2017 

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