08:30
MS10: PDE constrained optimization and variational inequalities (Part 1)
Chair: Daniel Walter
08:30
25 mins
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Fenchel duality for convex optimization on Riemannian manifolds
Jose Vidal-Nunez, Ronny Bergmann, Roland Herzog, Daniel Tenbrinck
Abstract: This talk introduces a new duality theory that generalizes the classical Fenchel-Legendre conjugation to functions defined on Riemannian manifolds. We present that results from convex analysis also hold for this novel duality theory on manifolds. Especially the Fenchel--Moreau theorem and properties involving the Riemannian subdifferential can be stated in this setting. A main application of this theory is that a specific class of optimization problems can be rewritten into a primal-dual saddle-point formulation. This is a first step towards efficient algorithms.
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08:55
25 mins
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Optimal Control Problems and Algebraic Flux Correction Schemes
Jens Baumgartner, Arnd Rösch
Abstract: Solutions of convection-diffusion-reaction equations may possess layers, i.e., narrow regions where the solution has a large gradient (in particular for convection dominated equations). Standard finite element methods lead to discrete solutions which are polluted by spurious oscillations.
The main motivation for the construction of the so-called algebraic flux correction (AFC) schemes is the satisfaction of the DMP to avoid spurious oscillations in the discrete solutions.
We apply an AFC scheme to an optimal control problem governed by a convection-diffusion-reaction equation. Due to the fact that the AFC schemes are nonlinear and usually non-differentiable the approaches "optimize-then-discretize" and "discretize-then-optimize" do not commute. We use the "optimize-then-discretize" approach, i.e., we discretize the state equation and besides the adjoint equation with the AFC method.
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09:20
25 mins
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Numerical analysis of optimal control of fracture propagation
Masoumeh Mohammadi, Winnifried Wollner
Abstract: An optimal control problem governed by a time-discrete fracture propagation process is considered. The nonlinear fracture model is treated once as a linearized one, while the original nonlinear model is dealt with afterwards. The discretization of the problem in both cases is done using a conforming finite element method. Regarding the linearized case, in contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, fracture equation. Based on this, a priori error estimates for the control, state, and adjoint variables will be derived. The discretized nonlinear fracture model will be analyzed as well, which leads to a quantitative error estimate, while we avoid unrealistic regularity assumptions.
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09:45
25 mins
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A Semi smooth Newton Method for Regularized L^q–quasinorm Sparse Optimal Control Problems
Pedro Merino
Abstract: We present a numerical method for control problems with non--convex integral costs of the form $\displaystyle\int_\Omega |u|^{\frac1p}dx$, with $p\in(0,1)$. By introducing a Huber like regularization of the original non--convex term we are able to approximate the original optimal control problem by a difference--of--convex function representation. In this way we obtain a DC-algorithm that useful to solve this kind of problems.
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