A Hierarchical Flow Solver for Optimisation with PDE Constraints
Creators: name: Köster, Michael
Publisher: name: Technische Universität Dortmund
Year: 2011
Subjects: subject: Block-Glätter
subject: Block smoother
subject: CFD
subject: Crank-Nicolson
subject: Crystal growth
subject: Czochralski
subject: Distributed Control
subject: Edge-oriented stabilisation
subject: Elliptic
subject: Elliptisch
subject: EOJ stabilisation
subject: EOJ Stabilisierung
subject: FEAT
subject: FEATFLOW
subject: Finite Elemente
subject: Finite Elements
subject: First discretise then optimise
subject: First discretize then optimize
subject: First optimise then discretise
subject: First optimize then discretize
subject: Flow-Around-Cylinder
subject: Full Newton-SAND
subject: Heat equation
subject: Hierarchical
subject: Hierarchical solution concept
subject: Hierarchisch
subject: Hierarchisches Lösungskonzept
subject: Inexact Newton
subject: Inexaktes Newton-Verfahren
subject: Instationär
subject: Inverse Probleme
subject: Inverse Problems
subject: Kantenbasierte Stabilisierung
subject: KKT system
subject: Kristallwachstum
subject: Krylov
subject: Large-Scale
subject: linear complexity
subject: lineare Komplexität
subject: Mehrgitter
subject: Mehrgitter-Krylov
subject: Monolithic
subject: Monolithisch
subject: Multigrid
subject: Multigrid-Krylov
subject: Multilevel
subject: Navier-Stokes
subject: Nichtparametrische Finite Elemente
subject: Nonparametric finite elements
subject: Nonstationary
subject: OPTFLOW
subject: Optimierung
subject: Optimisation
subject: Optimization
subject: PDE Constraints
subject: Raum-Zeit
subject: saddle point
subject: SAND
subject: Sattelpunkt
subject: Schur complement preconditioning
subject: Schurkomplement-Vorkonditionierer
subject: Space-time
subject: SQP
subject: Stokes
subject: Theta schema
subject: Theta scheme
subject: Time-dependent
subject: Transient
subject: Unstructured Grids
subject: Unstrukturierte Gitter
subject: Vanka
subject: Verteilte Kontrolle
subject: Wärmeleitung
subject: Wärmeleitungsgleichung
subject: 510

description: Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations. The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes-- and Navier--Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level. Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement.