Performance comparisons of preconditioned iterative methods for problems arising in PDE-constrained optimization

by Shiraz Farouq

Institution: Uppsala University
Year: 2015
Keywords: Engineering and Technology; Teknik och teknologier; Masterprogram i tillämpad beräkningsvetenskap; Master Programme in Computational Science
Record ID: 1329129
Full text PDF: http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-253688


The governing dynamics of simple and complex processes, whether physical, biological, social, economic, engineering, or even rather a mere figment of imagination, can be studied via numerical simulations of mathematical models. These models in many cases can be thought to consist of one, or frequently, several coupled partial differential equations (PDEs). In many applications, the aim of such simulations is not only to study the behavior of the underlying processes, but also to optimize or control those in some optimal way. These are referred to as optimal control problems constrained by PDEs and are stated in the form of a constrained minimization problem. The general framework under which such problems are studied is referred to as PDE-constrained optimization. In this thesis, we aim to solve three benchmark optimal control problems, namely, the optimal control of the Poisson equation, the optimal control of the convection-diffusion equation and the optimal control of the Stokes system. Numerically tackling these problems lead to a large optimality system with a saddle point structure. Systems with a saddle point structure are indefinite and in general, ill-conditioned, thus posing great challenges for iterative solvers seeking to find their solution. Preconditioning the optimality system is a possible strategy to deal with the issue. The main focus of the thesis is therefore to solve the resulting optimality systems with various preconditioners available in literature and compare their efficiency. Moreover, additional challenges arise when dealing with convection-diffusion control problems which we effectively deal by employing the local projection stabilization (LPS) scheme. Furthermore, Axelsson and Neytcheva in [40] proposed a preconditioner for efficiently solving large nonlinear coupled multi-physics problems. We successfully apply this preconditioner to the first two benchmark problems with promising results.