Institut für Mathematik
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Institute
A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
(2020)
In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.
ADMM-Type Methods for Optimization and Generalized Nash Equilibrium Problems in Hilbert Spaces
(2020)
This thesis is concerned with a certain class of algorithms for the solution of constrained optimization problems and generalized Nash equilibrium problems in Hilbert spaces. This class of algorithms is inspired by the alternating direction method of multipliers (ADMM) and eliminates the constraints using an augmented Lagrangian approach. The alternating direction method consists of splitting the augmented Lagrangian subproblem into smaller and more easily manageable parts.
Before the algorithms are discussed, a substantial amount of background material, including the theory of Banach and Hilbert spaces, fixed-point iterations as well as convex and monotone set-valued analysis, is presented. Thereafter, certain optimization problems and generalized Nash equilibrium problems are reformulated and analyzed using variational inequalities and set-valued mappings. The analysis of the algorithms developed in the course of this thesis is rooted in these reformulations as variational inequalities and set-valued mappings.
The first algorithms discussed and analyzed are one weakly and one strongly convergent ADMM-type algorithm for convex, linearly constrained optimization. By equipping the associated Hilbert space with the correct weighted scalar product, the analysis of these two methods is accomplished using the proximal point method and the Halpern method.
The rest of the thesis is concerned with the development and analysis of ADMM-type algorithms for generalized Nash equilibrium problems that jointly share a linear equality constraint. The first class of these algorithms is completely parallelizable and uses a forward-backward idea for the analysis, whereas the second class of algorithms can be interpreted as a direct extension of the classical ADMM-method to generalized Nash equilibrium problems.
At the end of this thesis, the numerical behavior of the discussed algorithms is demonstrated on a collection of examples.
This thesis is concerned with the solution of control and state constrained optimal control problems, which are governed by elliptic partial differential equations. Problems of this type are challenging since they suffer from the low regularity of the multiplier corresponding to the state constraint. Applying an augmented Lagrangian method we overcome these difficulties by working with multiplier approximations in $L^2(\Omega)$. For each problem class, we introduce the solution algorithm, carry out a thoroughly convergence analysis and illustrate our theoretical findings with numerical examples.
The thesis is divided into two parts. The first part focuses on classical PDE constrained optimal control problems. We start by studying linear-quadratic objective functionals, which include the standard tracking type term and an additional regularization term as well as the case, where the regularization term is replaced by an $L^1(\Omega)$-norm term, which makes the problem ill-posed. We deepen our study of the augmented Lagrangian algorithm by examining the more complicated class of optimal control problems that are governed by a semilinear partial differential equation.
The second part investigates the broader class of multi-player control problems. While the examination of jointly convex generalized Nash equilibrium problems (GNEP) is a simple extension of the linear elliptic optimal control case, the complexity is increased significantly for pure GNEPs. The existence of solutions of jointly convex GNEPs is well-studied. However, solution algorithms may suffer from non-uniqueness of solutions. Therefore, the last part of this thesis is devoted to the analysis of the uniqueness of normalized equilibria.
This thesis covers a wide range of results for when a random vector is in the max-domain of attraction of max-stable random vector. It states some new theoretical results in D-norm terminology, but also gives an explaination why most approaches to multivariate extremes are equivalent to this specific approach. Then it covers new methods to deal with high-dimensional extremes, ranging from dimension reduction to exploratory methods and explaining why the Huessler-Reiss model is a powerful parametric model in multivariate extremes on par with the multivariate Gaussian distribution in multivariate regular statistics. It also gives new results for estimating and inferring the multivariate extremal dependence structure, strategies for choosing thresholds and compares the behavior of local and global threshold approaches. The methods are demonstrated in an artifical simulation study, but also on German weather data.
In this paper we introduce a theoretical framework concerned with fostering functional thinking in Grade 8 students by utilizing digital technologies. This framework is meant to be used to guide the systematic variation of tasks for implementation in the classroom while using digital technologies. Examples of problems and tasks illustrate this process. Additionally, results of an empirical investigation with Grade 8 students, which focusses on the students’ skills with digital technologies, how they utilize these tools when engaging with the developed tasks, and how they influence their functional thinking, are presented. The research aim is to investigate in which way tasks designed according to the theoretical framework could promote functional thinking while using digital technologies in the sense of the operative principle. The results show that the developed framework — Function-Operation-Matrix — is a sound basis for initiating students’ actions in the sense of the operative principle, to foster the development of functional thinking in its three aspects, namely, assignment, co-variation and object, and that digital technologies can support this process in a meaningful way.
In this paper we consider the class (θA, B) of parameter-dependent linear systems given by matrices A ∈ ℂ\(^{nxn}\) and B ∈ ℂ\(^{nxm}\). This class is of interest for several applications and the frequently met task for such systems is to steer the origin toward a given target family f(θ) by using an input that is independent from the parameter. This paper provides a collection of necessary and sufficient conditions for ensemble reachability for these systems.
We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.
This dissertation investigates the application of multivariate Chebyshev polynomials in the algebraic signal processing theory for the development of FFT-like algorithms for discrete cosine transforms on weight lattices of compact Lie groups. After an introduction of the algebraic signal processing theory, a multivariate Gauss-Jacobi procedure for the development of orthogonal transforms is proven. Two theorems on fast algorithms in algebraic signal processing, one based on a decomposition property of certain polynomials and the other based on induced modules, are proven as multivariate generalizations of prior theorems. The definition of multivariate Chebyshev polynomials based on the theory of root systems is recalled. It is shown how to use these polynomials to define discrete cosine transforms on weight lattices of compact Lie groups. Furthermore it is shown how to develop FFT-like algorithms for these transforms. Then the theory of matrix-valued, multivariate Chebyshev polynomials is developed based on prior ideas. Under an existence assumption a formula for generating functions of these matrix-valued Chebyshev polynomials is deduced.
We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural L1-topology and show that its complement is rather large, specifically it is an open dense set.