| In this thesis we develop numerical methods for the solution of large-scale PDE-based constrained optimization problems. Overall three studies are presented; the first two are application driven, addressing volume pre-serving image registration and optimal mass transport problems. The third study is more generic and embarks at the development of a new inexact sequential quadratic programming framework. Image registration aims at finding a plausible transformation which aligns images taken at different times, different view-points or by different modalities. This problem is ill-posed and therefore, regularization is required. In that study, elastic regularizer is considered along with volume preserving constraint. A numerical framework based on augmented Lagrangian along with geometrical multi-grid preconditioner was devised. The proposed algorithm was tested with real data. The optimal mass transport seeks for an optimal way to move a pile of soil from one site to another using minimal energy, while preserving the overall mass. In that study, a fluid dynamics formulation was considered. This formulation introduces an artificial time stepping, which on the one hand transforms the non-convex problem to a convex one, but on the other hand increases the dimensionality of the problem. A Schur complement and algebraic multigrid formed a preconditioner within a sequential quadratic programming scheme. Results for both three and four dimensional problems were presented. Inside each step of nonlinear optimization, solution for an ill-conditioned, indefinite linear system, known as a KKT system is required. As problem size increases, linear iterative solvers become the bottleneck of the optimization scheme. In the third study, a new approach for inexact step computation is proposed. The general idea is to reduce the number of linear iteration while still maintaining convergence of the overall scheme. This is done, by the embedment of a filter inside a linear solver. |
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