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Projekte

Algorithms for Optimization Problems in Banach Spaces with Non-smooth Structure

Collaboration partners

The project is a joint research project with Christian Kanzow.

Project description

The aim of this project is to develop and analyse algorithms for the numerical solution of some highly difficult optimization problems in Banach spaces. This includes mathematical programs with complementarity constraints,  switching constraints, or problems involving a sparsity term either in the objective function or the constraints. By exploiting the special structure of these problems, the goal is to derive solution methods with strong global and local convergence properties under realistic, problem-tailored assumptions. All methods will be implemented and tested extensively on several relevant examples.

Funding

This project is funded by the German Research Foundation DFG under project grant Wa 3626/3-2 within SPP 1962.

  • Iterative hard-thresholding applied to optimal control problems with L0(Ω) control cost Wachsmuth, D. SIAM J. Control Optim. 57, 854--879 (2019).
     

Algorithms for Quasi-Variational Inequalities in Infinite-dimensional Spaces

Collaboration partners

The project is a joint research project with Christian Kanzow.

Project description

The aim of this project is to develop and analyse algorithms for the numerical solution of some classes of quasi-variational inequalities. Such inequalities occur, e.g., in connection with generalized Nash equilibria in multi-player control problems. Moreover, they are widely used to describe the value function in stochastic control problems. Our goal is twofold: (a) Transfer existing solution methods from finite-dimensional to infinite-dimensional problems. (b) Develop problem-tailored solution methods by taking into account the particular structure of certain quasi-variational inequalities. All methods should have a strong theoretical background and will be tested extensively on suitable examples.

Funding

This project is funded by the German Research Foundation DFG under project grant Wa 3626/3-1 within SPP 1962.

  • Safeguarded augmented Lagrangian methods in Banach spaces Karl, V., Kanzow, C., Steck, D., Wachsmuth, D. (2019).
     
  • The multiplier-penalty method for generalized Nash equilibrium problems in Banach spaces Kanzow, C., Karl, V., Steck, D., Wachsmuth, D. SIAM J. Optim. 29, 767--793 (2019).
     
  • On non-reducible multi-player control problems and their numerical computation Karl, V., Pörner, F. https://arxiv.org/abs/1805.03009v2 (2018).
     
  • A joint Tikhonov regularization and augmented Lagrange approach for ill-posed state constrained control problems with sparse controls Karl, V., Pörner, F. Numer. Funct. Anal. Optim. 39, 1543--1573 (2018).
     
  • An augmented Lagrangian method for optimization problems in Banach spaces Steck, D., Kanzow, C., Wachsmuth, D. SIAM J. Control Optim. 56, 272-291 (2018).
     
  • A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems Karl, V., Neitzel, I., Wachsmuth, D. http://arxiv.org/abs/1806.08124 (2018).
     
  • An augmented Lagrange method for elliptic state constrained optimal control problems Karl, V., Wachsmuth, D. Comp. Opt. Appl. 69, 857--880 (2018).
     

Regularization and Discretization of Inverse Problems for PDEs in Banach Spaces

Collaboration partners

The project is a joint research project with Frank Pörner, Christian Clason (Duisburg-Essen) and Barbara Kaltenbacher (Klagenfurt).

Project description

The aim of this project is a combined analysis of regularization and discretization of ill-posed problems in Banach spaces specifically in the context of partial differential equations. Such problems play a crucial role in numerous applications ranging from medical imaging via nondestructive testing to geophysical prospecting, with the Banach space setting mandated by the inherent regularity of the sought coefficients as well as structural features such as sparsity.

Funding

This project is funded by the German Research Foundation DFG under project grant Wa 3626/1-1.

  • Iterative hard-thresholding applied to optimal control problems with L0(Ω) control cost Wachsmuth, D. SIAM J. Control Optim. 57, 854--879 (2019).
     
  • On non-reducible multi-player control problems and their numerical computation Karl, V., Pörner, F. https://arxiv.org/abs/1805.03009v2 (2018).
     
  • A joint Tikhonov regularization and augmented Lagrange approach for ill-posed state constrained control problems with sparse controls Karl, V., Pörner, F. Numer. Funct. Anal. Optim. 39, 1543--1573 (2018).
     
  • Second-order analysis and numerical approximation for bang-bang bilinear control problems Casas, E., Wachsmuth, D., Wachsmuth, G. SIAM J. Control Optim. 56, 4203--4227 (2018).
     
  • Regularization Methods for Ill-Posed Optimal Control Problems Pörner, F. http://nbn-resolving.org/urn:nbn:de:bvb:20-opus-163153 (2018).
     
  • A priori stopping rule for an iterative Bregman method for optimal control problems Pörner, F. Optimization Methods and Software 33, 249-267 (2018).
     
  • Inexact Iterative Bregman Method for Optimal Control Problems Pörner, F. Numerical Functional Analysis and Optimization 39, 491-516 (2018).
     
  • Stability for bang-bang control problems of partial differential equations Qui, N. T., Wachsmuth, D. Optimization 67, 2157-2177 (2018).
     
  • Error estimates for the approximation of a discrete-valued optimal control problem Clason, C., Do, T. B. T., Pörner, F. Comp. Opt. Appl. 71, 857-878 (2018).
     
  • On the switching behavior of sparse optimal controls for the one-dimensional heat equation Tröltzsch, F., Wachsmuth, D. Mathematical Control & Related Fields 8, 135--153 (2017).
     
  • Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations Pörner, F., Wachsmuth, D. Mathematical Control & Related Fields 8, 315--335 (2017).
     
  • Sufficient second-order conditions for bang-bang control problems Casas, E., Wachsmuth, D., Wachsmuth, G. SIAM J. Control Optim. 55, 3066--3090 (2017).
     
  • Functional error estimators for the adaptive discretization of inverse problems Clason, C., Kaltenbacher, B., Wachsmuth, D. Inverse Problems 32, 104004 (2016).
     
  • An iterative Bregman regularization method for optimal control problems with inequality constraints Pörner, F., Wachsmuth, D. Optimization 65, 2195--2215 (2016).
     
  • A sharp regularization error estimate for bang-bang solutions for an iterative Bregman regularization method for optimal control problems Pörner, F. In: Proceedings in Applied Mathematics and Mechanics. p. 787--788 (2016).
     

Higher-order finite elements for optimal control problems

Collaboration partners

The project is a joint research project with Sven Beuchler (Bonn).

Project description

The mathematical models of many technical processes contain partial differential equations. Here, it is important to optimize these processes. Often the optimization variables are subject to constraints that have to be taken into account. As model problem, consider the minimization of a functional

g(y)+j(u)
subject to the elliptic equation
-Δy =u on Ω,   y=0 on Γ
and pointwise control
ua ≤ u ≤ ub
and state constraints
ya ≤ y ≤ yb

The numerical solution of this problem offers many challenges. One of them is the low regularity of Lagrange multipliers associated to the state constraints, which are measures. The global regularity of the solution of the optimal control problem is limited by quantities that have small support.
The project will exploit this property and develop methods for an adaptive hp-discretization of the optimal control problem, where the solution is approximated by high-order polynomials on large triangles where it is smooth, whereas the solution is approximated by low-order polynomials on small triangles in regions, where it is non-smooth.

Funding

This project was funded by the Austrian Research Fund FWF under project grant P23484.

Numerical verification of optimality and optimality conditions for optimal control problems

Project description

Many technical processes are described by partial differential equations. Here, it is important to optimize these processes. This leads to optimization problems in an infinite-dimensional setting.
As model problem, consider the minimization of a functional

g(y)+j(u)
subject to the elliptic equation
-Δy + d(y)=u on Ω,   y=0 on Γ
and pointwise control constraints
ua ≤ u ≤ ub

Despite its simple structure, this problem offers many difficulties and challenges. Due to the non-linear elliptic equation this optimisation problem becomes non-convex.
If one has computed solutions yh and uh of discretized versions of this problem, the question arises

Are yh and uh indeed an approximation of a solution of the infinite-dimensional problem?

Due to the inherent non-convexity of the optimization problem, this question is by far non-trivial. The project wants to give answers to this question with information that is computable from the numerical solution. The methods that will be applied are based on techniques from optimal control, finite element methods, and eigenvalue computations.

Funding

This project was funded by the Austrian Research Fund FWF under project grant P21564.