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Volume 10, Issue 3
A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations

M. M. Butt & Y. Yuan

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 639-655.

Published online: 2017-10

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  • Abstract

A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

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@Article{NMTMA-10-639, author = {}, title = {A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2017}, volume = {10}, number = {3}, pages = {639--655}, abstract = {

A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017.m1637}, url = {http://global-sci.org/intro/article_detail/nmtma/12362.html} }
TY - JOUR T1 - A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 639 EP - 655 PY - 2017 DA - 2017/10 SN - 10 DO - http://doi.org/10.4208/nmtma.2017.m1637 UR - https://global-sci.org/intro/article_detail/nmtma/12362.html KW - AB -

A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

M. M. Butt & Y. Yuan. (2020). A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations. Numerical Mathematics: Theory, Methods and Applications. 10 (3). 639-655. doi:10.4208/nmtma.2017.m1637
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