1,714 research outputs found
Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization
We study the extension of the Chambolle--Pock primal-dual algorithm to
nonsmooth optimization problems involving nonlinear operators between function
spaces. Local convergence is shown under technical conditions including metric
regularity of the corresponding primal-dual optimality conditions. We also show
convergence for a Nesterov-type accelerated variant provided one part of the
functional is strongly convex.
We show the applicability of the accelerated algorithm to examples of inverse
problems with - and -fitting terms as well as of
state-constrained optimal control problems, where convergence can be guaranteed
after introducing an (arbitrary small, still nonsmooth) Moreau--Yosida
regularization. This is verified in numerical examples
A cubic C\u3csup\u3e0\u3c/sup\u3e interior penalty method for elliptic distributed optimal control problems with pointwise state and control constraints
We design and analyze a cubic C interior penalty method for linearāquadratic elliptic distributed optimal control problems with pointwise state and control constraints. Numerical results that corroborate the theoretical error estimates are also presented.
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Numerical Techniques for Optimization Problems with PDE Constraints
The development, analysis and implementation of eļ¬cient and robust numerical techniques for optimization problems associated with partial diļ¬erential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, signiļ¬cant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm āOptimize ļ¬rst, then discretizeā and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reļ¬ected the progress made in the ļ¬eld. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled āall-at-onceā approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identiļ¬cation of parameters in multi-scale physical and physiological processes
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