期刊
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
卷 13, 期 4, 页码 1716-1732出版社
SIAM PUBLICATIONS
DOI: 10.1137/130949282
关键词
dynamical systems; bifurcations; classification; compressive sensing; sparse representation; proper orthogonal decomposition
资金
- U.S. Department of Energy (DOE) Office of Advanced Scientific Computing Research
- National Science Foundation [DMS-1007621]
- U.S. Air Force Office of Scientific Research [FA9550-09-0174]
We show that for complex nonlinear systems, model reduction and compressive sensing strategies can be combined to great advantage for classifying, projecting, and reconstructing the relevant low-dimensional dynamics. L-2-based dimensionality reduction methods such as the proper orthogonal decomposition are used to construct separate modal libraries and Galerkin models based on data from a number of bifurcation regimes. These libraries are then concatenated into an overcomplete library, and L-1-sparse representation in this library from a few noisy measurements results in correct identification of the bifurcation regime. This technique provides an objective and general framework for classifying the bifurcation parameters and, therefore, the underlying dynamics and stability. After classifying the bifurcation regime, it is possible to employ a low-dimensional Galerkin model, only on modes relevant to that bifurcation value. These methods are demonstrated on the complex Ginzburg-Landau equation using sparse, noisy measurements. In particular, three noisy measurements are used to accurately classify and reconstruct the dynamics associated with six distinct bifurcation regimes; in contrast, classification based on least-squares fitting (L-2) fails consistently.
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