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Title: Fast iterative solvers for PDE-constrained optimization problems
Author: Pearson, John W.
ISNI:       0000 0004 2747 5997
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2013
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In this thesis, we develop preconditioned iterative methods for the solution of matrix systems arising from PDE-constrained optimization problems. In order to do this, we exploit saddle point theory, as this is the form of the matrix systems we wish to solve. We utilize well-known results on saddle point systems to motivate preconditioners based on effective approximations of the (1,1)-block and Schur complement of the matrices involved. These preconditioners are used in conjunction with suitable iterative solvers, which include MINRES, non-standard Conjugate Gradients, GMRES and BiCG. The solvers we use are selected based on the particular problem and preconditioning strategy employed. We consider the numerical solution of a range of PDE-constrained optimization problems, namely the distributed control, Neumann boundary control and subdomain control of Poisson's equation, convection-diffusion control, Stokes and Navier-Stokes control, the optimal control of the heat equation, and the optimal control of reaction-diffusion problems arising in chemical processes. Each of these problems has a special structure which we make use of when developing our preconditioners, and specific techniques and approximations are required for each problem. In each case, we motivate and derive our preconditioners, obtain eigenvalue bounds for the preconditioners where relevant, and demonstrate the effectiveness of our strategies through numerical experiments. The goal throughout this work is for our iterative solvers to be feasible and reliable, but also robust with respect to the parameters involved in the problems we consider.
Supervisor: Wathen, Andrew Sponsor: EPSRC
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; Numerical analysis ; Calculus of variations and optimal control ; Partial differential equations ; numerical linear algebra ; preconditioning ; PDE-constrained optimization ; saddle point systems