Projects
Algorithms for Optimization Problems in Banach Spaces with Nonsmooth 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, problemtailored 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 3626/32 within SPP 1962.
Algorithms for QuasiVariational Inequalities in Infinitedimensional 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 quasivariational inequalities. Such inequalities occur, e.g., in connection with generalized Nash equilibria in multiplayer 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 finitedimensional to infinitedimensional problems. (b) Develop problemtailored solution methods by taking into account the particular structure of certain quasivariational 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 3626/31 within SPP 1962.

Safeguarded augmented Lagrangian methods in Banach spaces (2019).

The multiplierpenalty method for generalized Nash equilibrium problems in Banach spaces SIAM J. Optim. 29, 767793 (2019).[ DOI ]

On nonreducible multiplayer control problems and their numerical computation https://arxiv.org/abs/1805.03009v2 (2018).[arxiv]

A joint Tikhonov regularization and augmented Lagrange approach for illposed state constrained control problems with sparse controls Numer. Funct. Anal. Optim. 39, 15431573 (2018).

An augmented Lagrangian method for optimization problems in Banach spaces SIAM J. Control Optim. 56, 272291 (2018).

A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems http://arxiv.org/abs/1806.08124 (2018).[arxiv]

An augmented Lagrange method for elliptic state constrained optimal control problems Comp. Opt. Appl. 69, 857880 (2018).[ DOI ]
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 (DuisburgEssen) and Barbara Kaltenbacher (Klagenfurt).
Project description
The aim of this project is a combined analysis of regularization and discretization of illposed 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 3626/11.

Iterative hardthresholding applied to optimal control problems with L^{0}(Ω) control cost SIAM J. Control Optim. 57, 854879 (2019).

On nonreducible multiplayer control problems and their numerical computation https://arxiv.org/abs/1805.03009v2 (2018).[arxiv]

A joint Tikhonov regularization and augmented Lagrange approach for illposed state constrained control problems with sparse controls Numer. Funct. Anal. Optim. 39, 15431573 (2018).

Secondorder analysis and numerical approximation for bangbang bilinear control problems SIAM J. Control Optim. 56, 42034227 (2018).

Regularization Methods for IllPosed Optimal Control Problems http://nbnresolving.org/urn:nbn:de:bvb:20opus163153 (2018).

A priori stopping rule for an iterative Bregman method for optimal control problems Optimization Methods and Software 33, 249267 (2018).

Inexact Iterative Bregman Method for Optimal Control Problems Numerical Functional Analysis and Optimization 39, 491516 (2018).

Stability for bangbang control problems of partial differential equations Optimization 67, 21572177 (2018).

Error estimates for the approximation of a discretevalued optimal control problem Comp. Opt. Appl. 71, 857878 (2018).

On the switching behavior of sparse optimal controls for the onedimensional heat equation Mathematical Control & Related Fields 8, 135153 (2017).

Tikhonov regularization of optimal control problems governed by semilinear partial differential equations Mathematical Control & Related Fields 8, 315335 (2017).

Sufficient secondorder conditions for bangbang control problems SIAM J. Control Optim. 55, 30663090 (2017).[ DOI ]

Functional error estimators for the adaptive discretization of inverse problems Inverse Problems 32, 104004 (2016).

An iterative Bregman regularization method for optimal control problems with inequality constraints Optimization 65, 21952215 (2016).

A sharp regularization error estimate for bangbang solutions for an iterative Bregman regularization method for optimal control problems In: Proceedings in Applied Mathematics and Mechanics. p. 787788 (2016).
Higherorder 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
u_{a} ≤ u ≤ u_{b}
and state constraints
y_{a} ≤ y ≤ y_{b}
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 hpdiscretization of the optimal control problem, where the solution is approximated by highorder polynomials on large triangles where it is smooth, whereas the solution is approximated by loworder polynomials on small triangles in regions, where it is nonsmooth.
Funding
This project was funded by the Austrian Research Fund FWF under project grant P23484.
 Exponential convergence of hpfinite element discretization of optimal boundary control problems with elliptic partial differential equations
D. Wachsmuth, J.E. Wurst
SIAM J. Control Optim. 54(5), 25262552 (2016).
 Optimal control of interface problems with hpfinite elements
D. Wachsmuth, J.E. Wurst
Numerical Functional Analysis and Optimization 37(3), 363390 (2016).
Final version. The Version of Record of this manuscript has been published and is available in Numerical functional analysis and optimization, 09.02.2016, at http://www.tandfonline.com.
 An interior point method designed for solving linear quadratic optimal control problems with hp finite elements
D. Wachsmuth, J.E. Wurst
Optimization methods and software 30(6), 12761302 (2015).
Preprint version.
 Boundary concentrated finite elements for optimal control problems with distributed observation
S. Beuchler, K. Hofer, D. Wachsmuth, J.E. Wurst
Comp. Opt. Appl. 62, 3165 (2015).
Final version. The final publication is available at link.springer.com.
 Robust error estimates for regularization and discretization of bangbang control problems
D. Wachsmuth
Comp. Opt. Appl. 62, 271289 (2014).
Final version. The final publication is available at link.springer.com.
 Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs
S. Beuchler, C. Pechstein, D. Wachsmuth
Comp. Opt. Appl. 51(2), 883908 (2012).
 Schwarz type solvers for hpFEM discretizations of mixed problems
S. Beuchler, M. Purrucker
Comput. Methods Appl. Math. 12(4), 369390 (2012).
Preprint version.
 Inexact additive Schwarz solvers for hpFEM discretizations in three dimensions
S. Beuchler
In: Advanced finite element methods and applications, Lect. Notes Appl. Comput. Mech. Vol. 66, 91108, Springer (2013).
Preprint version.
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 infinitedimensional 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
u_{a} ≤ u ≤ u_{b}
Despite its simple structure, this problem offers many difficulties and challenges. Due to the nonlinear elliptic equation this optimisation problem becomes nonconvex.
If one has computed solutions y_{h} and u_{h} of discretized versions of this problem, the question arises
Are y_{h} and u_{h} indeed an approximation of a solution of the infinitedimensional problem?
Due to the inherent nonconvexity of the optimization problem, this question is by far nontrivial. 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.
 Aposteriori error estimates for optimal control problems with state and control constraints
A. Rösch, D. Wachsmuth
Numerische Mathematik 120(4), 733762 (2012).
Final version. The final publication is available at link.springer.com.
 Aposteriori verification of optimality conditions for control problems with finitedimensional control space
S. Akindeinde, D. Wachsmuth
Numerical Functional Analysis and Optimization 33(5), 473523 (2012).
Preprint version.
 Numerical verification of optimality conditions
A. Rösch, D. Wachsmuth
SIAM J. Control Optim. 47(5), 25572581 (2008).
 Adaptive methods for control problems with finitedimensional control space
S. Akindeinde, D. Wachsmuth
In: System Modelling and Optimization, 5969, Springer (2013).
Final version. The final publication is available at link.springer.com.
 How to check numerically the sufficient optimality conditions for infinitedimensional optimization problems
A. Rösch, D. Wachsmuth
In: Control of Coupled Partial Differential Equations, Internat. Ser. Numer. Math. Vol. 158, 297317, Birkhäuser (2009).