Structure Preserving Model Reduction of Parametric Hamiltonian Systems
Abstract
While reduced-order models (ROMs) have been popular for efficiently solving large systems of differential equations, the stability of reduced models over long-time integration presents challenges. We present a greedy approach for a ROM generation of parametric Hamiltonian systems that captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis. We use the error in the Hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors. Under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters, we show that the greedy algorithm converges at an exponential rate. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure. This enables the reduction of the computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy, and stability of this model reduction technique is illustrated through simulations of the parametric wave equation and the parametric Schrödinger equation.
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Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A2616 - A2644
DOI: 10.1137/17M1111991
ISSN (online): 1095-7197
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 17 January 2017
Accepted: 8 September 2017
Published online: 16 November 2017
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Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-17-1-0241
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