Adaptive discontinuous Galerkin methods for convection dominated optimal control problems

Download

index.pdf

Date

2012

Author

Yücel, Hamdullah

Metadata

Show full item record

Item Usage Stats

195
views

135
downloads

Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection di usion partial di erential equations (PDEs). When convection dominates di usion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall e ciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite di erence, finite volume and continuous finite element methods for solving wave dominated problems like convection di usion equations since they possess higher accuracy. This thesis will focus on analysis and application of DG methods for linear-quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection di usion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraints

Subject Keywords

Mathematical optimization., Galerkin methods., Numerical analysis.

URI

http://etd.lib.metu.edu.tr/upload/12614523/index.pdf
https://hdl.handle.net/11511/21698

Collections

Graduate School of Applied Mathematics, Thesis

Suggestions

OpenMETU
Core

Adaptive discontinuous Galerkin (DG) methods for state constrained optimal control problems governed by convection dominated equations Yücel, Hamdullah; Benner, Peter (2013-07-19) Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection diffusion partial differential equations (PDEs). When convection dominates diffusion, the solutions of these PDEs typically exhibit layers, localized regions where the derivative of the solution is large. Hence, it requires special numerical techniques, which take into account...
Solving optimal control time-dependent diffusion-convection-reaction equations by space time discretizations Seymen, Zahire; Karasözen, Bülent; Department of Mathematics (2013) Optimal control problems (OCPs) governed by convection dominated diffusion-convection-reaction equations arise in many science and engineering applications such as shape optimization of the technological devices, identification of parameters in environmental processes and flow control problems. A characteristic feature of convection dominated optimization problems is the presence of sharp layers. In this case, the Galerkin finite element method performs poorly and leads to oscillatory solutions. Hence, thes...
Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations Yücel, Hamdullah (2015-09-01) We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau-Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a poste...
Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints Yücel, Hamdullah; Karasözen, Bülent (2014-01-02) In this paper, we study a posteriori error estimates of the upwind symmetric interior penalty Galerkin (SIPG) method for the control constrained optimal control problems governed by linear diffusion-convection-reaction partial differential equations. Residual based error estimators are used for the state, the adjoint and the control. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented for convection dominated problems to illustrate the theoretic...
Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations Stoll, Martin; Yücel, Hamdullah; Benner, Peter (2018-01-01) In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. ...

Citation Formats

H. Yücel, “Adaptive discontinuous Galerkin methods for convection dominated optimal control problems,” M.S. - Master of Science, Middle East Technical University, 2012.