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Master thesis project: Numerical Solutions of Semilinear Elliptic PDEs and Optimal Control Problems

Felix Heinermann

Abstract:
This thesis addresses the topic of semilinear partial differential equations (PDEs) with homogeneous Dirichlet boundary conditions. At first, the existence and uniqueness of a solution to these kind of PDEs are being proven for a wide range of superposition operators. Afterwards, a number of numerical solution algorithms are analysed and a discretised version of them is subsequently employed on the PDE. A semi-smooth Newton method, a pure fixed-point method, a gradient descent, a first discretize, then optimize procedure, a generalised Newton method, and path-following methods are utilised to compute a numerical solution. All of the aforementioned methods are set up in the Sobolev space H_0^1(Ω) and discretised using linear finite elements. Afterwards, the associated tracking-type optimal control problem (OCP) is analysed in terms of the existence and uniqueness of a solution. In order to solve the OCP, a gradient descent and a first discretize, then optimize procedure are employed on the OCP. The first order necessary optimality conditions of the OCP are also solved for with a semi-smooth Newton method, a pure fixed-point method, and a Uzawa like iterative method. Finally, the efficiency, speed of convergence, and mesh dependence of all methods are analysed and compared, which shows that the considered semi-smooth Newton method in a potential combination with pathfollowing methods on the PDE yields the best result. On the OCP the considered semi-smooth Newton method and the Uzawa like iterative method lead to the most favourable results.

Time: Fri 2024-12-13 10.00 - 10.30

Location: Seminar room 3721

Language: English

Participating: Felix Heinermann

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The master thesis has been performed at Technische Universität München, under the supervision of Julia Kowalczyk and Constantin Christof

Johan Karlsson has been the examiner at KTH