Martin P. Neuenhofen

Martin P. Neuenhofen, Eric C. Kerrigan

We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. We prove convergence of our numerical method under reasonably mild assumptions: boundedness and Lipschitz-continuity of the problem-defining functions. We do not require uniqueness, differentiability or constraint qualifications to hold and we avoid the use of Lagrange multipliers. Our approach differs fundamentally from well-known methods based on collocation; we follow a penalty-barrier approach, where we compute integral quadratic penalties on the equality path constraints and point constraints, and integral log-barriers on the inequality path constraints. The resulting penalty-barrier functional can be minimized numerically using finite elements and penalty-barrier interior-point nonlinear programming solvers. Order of convergence results are derived, even if components of the solution are discontinuous. We also present numerical results to compare our method against collocation methods. The numerical results show that for the same degree and mesh, the computational cost is similar, but that the new method can achieve a smaller error and converges in cases where collocation methods fail.

Martin P. Neuenhofen

In this thesis we propose a novel implementation of IDRstab that avoids several unlucky breakdowns of current IDRstab implementations and is further capable of benefiting from a particular lucky breakdown scenario. IDRstab is a very efficient short-recurrence Krylov subspace method for the numerical solution of linear systems. Current IDRstab implementations suffer from slowdowns in the rate of convergence when the basis vectors of their oblique projectors become linearly dependent. We propose a novel implementation of IDRstab that is based on a successively restarted GMRES method. Whereas the collinearity of basis vectors in current IDRstab implementations would lead to an unlucky breakdown, our novel IDRstab implementation can strike a benefit from it in that it terminates with the exact solution whenever a new basis vector lives in the span of the formerly computed basis vectors. Numerical experiments demonstrate the superior robustness of our novel implementation with regards to convergence maintenance and the achievable accuracy of the numerical solution.