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  Solving Optimal Control Problems governed by Random Navier-Stokes Equations using Low-Rank Methods

Benner, P., Dolgov, S., Onwunta, A., & Stoll, M. (in preparation). Solving Optimal Control Problems governed by Random Navier-Stokes Equations using Low-Rank Methods.

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Item Permalink: http://hdl.handle.net/21.11116/0000-0000-2E27-2 Version Permalink: http://hdl.handle.net/21.11116/0000-0001-E73D-7
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1703.06097.pdf (Preprint), 2MB
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 Creators:
Benner, Peter1, Author              
Dolgov, Sergey2, Author
Onwunta, Akwum1, Author              
Stoll, Martin3, Author              
Affiliations:
1Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society, ou_1738141              
2University of Bath, ou_persistent22              
3Numerical Linear Algebra for Dynamical Systems, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society, ou_1832293              

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Free keywords: Mathematics, Numerical Analysis, math.NA
 Abstract: Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

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 Dates: 2017-03-17
 Publication Status: Not specified
 Pages: 29 pages
 Publishing info: -
 Table of Contents: -
 Rev. Method: -
 Identifiers: arXiv: 1703.06097
URI: http://arxiv.org/abs/1703.06097
 Degree: -

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