Stochastic discontinuous Galerkin methods for robust deterministic control of convection-diffusion equations with uncertain coefficients

We investigate a numerical behavior of robust deterministic optimal control problem subject to a convection-diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems,...

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Published in	Advances in computational mathematics Vol. 49; no. 2
Main Authors	Çi̇loğlu, Pelin, Yücel, Hamdullah
Format	Journal Article
Language	English
Published	New York Springer US 01.04.2023 Springer Nature B.V
Subjects	Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Convection-diffusion equation Diffusion Error analysis Galerkin method Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Optimal control Optimization Robust control Visualization Stochastic discontinuous Galerkin Error estimates 35R60 Low-rank approximation 49J20 PDE-constrained optimization Uncertainty quantification 60H15 60H35
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ISSN	1019-7168 1572-9044
DOI	10.1007/s10444-023-10015-5

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Abstract	We investigate a numerical behavior of robust deterministic optimal control problem subject to a convection-diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behavior for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in the energy norm, while the estimate of deterministic control is obtained in the L 2 -norm. Large matrix system emerging from the stochastic Galerkin method is addressed by the low-rank version of GMRES method, which reduces both the computational complexity and the memory requirements by employing Kronecker-product structure of the obtained linear system. Benchmark examples with and without control constraints are presented to illustrate the efficiency of the proposed methodology.
AbstractList	We investigate a numerical behavior of robust deterministic optimal control problem subject to a convection-diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behavior for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in the energy norm, while the estimate of deterministic control is obtained in the L 2 -norm. Large matrix system emerging from the stochastic Galerkin method is addressed by the low-rank version of GMRES method, which reduces both the computational complexity and the memory requirements by employing Kronecker-product structure of the obtained linear system. Benchmark examples with and without control constraints are presented to illustrate the efficiency of the proposed methodology. We investigate a numerical behavior of robust deterministic optimal control problem subject to a convection-diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behavior for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in the energy norm, while the estimate of deterministic control is obtained in the L2-norm. Large matrix system emerging from the stochastic Galerkin method is addressed by the low-rank version of GMRES method, which reduces both the computational complexity and the memory requirements by employing Kronecker-product structure of the obtained linear system. Benchmark examples with and without control constraints are presented to illustrate the efficiency of the proposed methodology.
ArticleNumber	16
Author	Çi̇loğlu, Pelin Yücel, Hamdullah
Author_xml	– sequence: 1 givenname: Pelin surname: Çi̇loğlu fullname: Çi̇loğlu, Pelin organization: Institute of Applied Mathematics, Middle East Technical University – sequence: 2 givenname: Hamdullah orcidid: 0000-0002-0313-9767 surname: Yücel fullname: Yücel, Hamdullah email: yucelh@metu.edu.tr organization: Institute of Applied Mathematics, Middle East Technical University
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Snippet	We investigate a numerical behavior of robust deterministic optimal control problem subject to a convection-diffusion equation containing uncertain inputs....
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SubjectTerms	Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Convection-diffusion equation Diffusion Error analysis Galerkin method Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Optimal control Optimization Robust control Visualization
Title	Stochastic discontinuous Galerkin methods for robust deterministic control of convection-diffusion equations with uncertain coefficients
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