A direct method for the numerical solution of optimization problems with time-periodic PDE constraints. In this thesis we develop a numerical method based on direct multiple shooting for optimal control problems (OCPs) constrained by time-periodic partial differential equations (PDEs). The proposed method features asymptotically optimal scale-up of the numerical effort with the number of spatial discretization points. It consists of a linear iterative splitting approach (LISA) within a Newton-type iteration with globalization on the basis of natural level functions. We investigate the LISA-Newton method in the framework of Bock’s kappa-theory and develop reliable a-posteriori kappa-estimators. Moreover we extend the inexact Newton method to an inexact sequential quadratic programming (SQP) method for inequality constrained problems and provide local convergence theory. In addition we develop a classical and a two-grid Newton-Picard preconditioner for LISA and prove grid independent convergence of the classical variant for a model problem. Based on numerical results we can claim that the two-grid version is even more efficient than the classical version for typical application problems. Moreover we develop a two-grid approximation for the Lagrangian Hessian which fits well in the two-grid Newton-Picard framework and yields a reduction of 68% in runtime for a nonlinear benchmark problem compared to the use of the exact Lagrangian Hessian. We show that the quality of the fine grid controls the accuracy of the solution while the quality of the coarse grid determines the asymptotic linear convergence rate, i.e., Bock’s kappa. Based on reliable kappa-estimators we facilitate automatic coarse grid refinement to guarantee fast convergence. For the solution of the occurring large-scale quadratic programming problems (QPs) we develop a structure exploiting two-stage approach. In the first stage we exploit the multiple shooting and Newton-Picard structure to reduce the large-scale QP to an equivalent QP whose size is independent of the number of spatial discretization points. For the second stage we develop extensions for a parametric active set method (PASM) to achieve a reliable and efficient solver for the resulting, possibly nonconvex QP. Furthermore we construct three illustrative, counter-intuitive toy examples which show that convergence of a one-shot one-step optimization method is neither necessary nor sufficient for the convergence of the forward problem method. For three regularization approaches to recover convergence our analysis shows that de-facto loss of convergence cannot be avoided with these approaches. We have further implemented the proposed methods within a code called MUSCOP which features automatic derivative generation for the model functions and dynamic system solutions of first and second order, parallelization on the multiple shooting structure, and a hybrid language programming paradigm to minimize setup and solution time for new application problems. We demonstrate the applicability, reliability, and efficiency of MUSCOP and thus the proposed numerical methods and techniques on a sequence of PDE OCPs of growing difficulty ranging from linear academic problems, over highly nonlinear academic problems of mathematical biology to a highly nonlinear real-world chemical engineering problem in preparative chromatography: The simulated moving bed process.
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References in zbMATH (referenced in 7 articles , 1 standard article )
Showing results 1 to 7 of 7.
- Potschka, Andreas; Bock, Hans Georg: A sequential homotopy method for mathematical programming problems (2021)
- Potschka, Andreas: Backward step control for Hilbert space problems (2019)
- Potschka, Andreas: Backward step control for global Newton-type methods (2016)
- Hante, F. M.; Mommer, M. S.; Potschka, A.: Newton-Picard preconditioners for time-periodic parabolic optimal control problems (2015)
- Potschka, Andreas: Direct multiple shooting for parabolic PDE constrained optimization (2015)
- Potschka, Andreas: A direct method for parabolic PDE constrained optimization problems (2014)
- Potschka, Andreas: A direct method for the numerical solution of optimization problems with time-periodic PDE constraints. (2001)