Manifold Approximations via Transported Subspaces: Model Reduction for Transport-Dominated Problems
Abstract.
This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures, which means that traditional model reduction methods based on linear approximations are inefficient when applied to these problems. In contrast, the approach introduced in this work derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. A time-stepping scheme evolves the nonlinear reduced models by transporting local approximation spaces along the characteristic curves of the governing equations. The proposed computational procedure allows an offline/online decomposition and is online-efficient in the sense that the complexity of accurately time stepping the nonlinear reduced model is independent of that of the full model. Numerical experiments with transport through heterogeneous media and the Burgers equation show orders of magnitude speedups of the proposed nonlinear reduced models based on transported subspaces compared to traditional linear reduced models and full models.
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Acknowledgment.
The numerical experiments were computed with support through the NYU IT High Performance Computing resources, services, and staff expertise.
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Information & Authors
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Published In
SIAM Journal on Scientific Computing
Pages: A170 - A199
DOI: 10.1137/20M1316998
ISSN (online): 1095-7197
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 4 February 2020
Accepted: 12 August 2022
Published online: 23 February 2023
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Authors
Funding Information
Air Force Center of Excellence on Multi-Fidelity Modeling of Rocket Combustor Dynamics: FA9550-17-1-0195
AFOSR MURI: FA9550-15-1-0038
National Science Foundation (NSF): OAC-1735609, DMS-170288
Funding: The work of the first and second authors was supported by the Air Force Center of Excellence on MultiFidelity Modeling of Rocket Combustor Dynamics under Award FA9550-17-1-0195 and AFOSR MURI on multiinformation sources of multiphysics systems under Award FA9550-15-1-0038 (Program Manager Dr. Fariba Fahroo). The work of the third author was supported by the NSF under grants OAC-1735609 and DMS-1720288.
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