Prof. Dr. Daniel Wachsmuth

Optimale Steuerung



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.

Related publications

  • A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
    V. Karl, I. Neitzel, D. Wachsmuth
    arXiv:1806.08124. (2018)
  • On non-reducible multi-player control problems and their numerical computation
    V. Karl, F. Pörner
    arXiv:1805.03009v2. (2018)
  • An augmented Lagrange method for ill-posed elliptic state constrained optimal control problems with sparse controls
    V. Karl, F. Pörner
    arXiv:1707.08460. (2017)
  • The multiplier-penalty method for generalized Nash equilibrium problems in Banach spaces
    C. Kanzow, V. Karl, D. Steck, D. Wachsmuth
    Preprint SPP1962-028. (2017)
  • An augmented Lagrange method for elliptic state constrained optimal control problems
    V. Karl, D. Wachsmuth
    Comp. Opt. Appl. 69(3), 857-880 (2018).
  • An augmented Lagrangian method for optimization problems in Banach spaces
    D. Steck, C. Kanzow, D. Wachsmuth
    SIAM J. Control Optim. 56(1), 272-291 (2018).
    Preprint version.


Regularization and Discretization of Inverse Problems for PDEs in Banach Spaces

Collaboration partners

The project is a joint research project with 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.

Related publications



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.

Related publications



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.

Related publications



Institut für Mathematik    Am Hubland    97074 Würzburg    Tel. 0931/31-85091



Institut für Mathematik  >  Mitarbeiter  > Daniel Wachsmuth > Projekte