SPP 1962:
Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization
Subject Area
Mathematics
Mechanical and Industrial Engineering
Term
since 2016
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 274039581
Many of the most challenging problems in the applied sciences involve non-differentiable structures as well as partial differential operators, thus leading to non-smooth distributed parameter systems. The associated non-smoothness typically arises (1) directly in the problem formulation (through non-smooth energies/objectives or system components), (2) through inequality constraints, nonlinear complementarity or switching systems, or (3) as a result of competition and hierarchy, typically leading to multiobjective/hierarchical optimization or to quasi-variational inequality problems. In this context, the transition from smoothing or simulation based approaches to genuinely non-smooth techniques or to multi-objective respectively multi-level optimization are crucial. This motivates the research of the Priority Programme. The goals of the programme are to: • lay the analytical foundations (through, e.g., the advancement of non-smooth and set-valued analysis)• establish a basis for stable numerical approximation through the design of algorithms with mesh independent convergence• address the influence of parameters, which enter the above-mentioned problems and which fall into a specified parameter range (uncertainty set)The overall research of the Priority Programme aims at combining non-smooth (numerical) analysis of non-linear complementarity, quasi-variational inequality and hierarchical optimization problems, the development, analysis and realization of robust solution algorithms, and applications of large-scale and infinite-dimensional problems where non-smoothness/switching occurs in or are due to:• systems governing an optimization problem• lower level problems of bi- or multilevel equilibrium problems• coupled systems of equilibrium problems (in particular (generalized) Nash games)• systems that require robust solutions• quasi-variational inequalitiesThe research of the Priority Programme will be validated against prototypical applications. These include: • multi-physics problems such as frictional elasto-plastic contact problems in a dynamic regime and coupled with thermal effects• motion optimization and optimal system design in robotics and biomechanics• multi-objective control systems such as (generalized) Nash equilibrium problems in technical or life sciences as well as in economics The cross section of each of the envisaged research areas exhibits a spectrum from basic research projects to research addressing specific applications. Clustered around such proto-typical applications, the research is organized in three communicating research areas:Area 1: Modelling, problem analysis, algorithm design and convergence analysisArea 2: Realization of algorithms, adaptive discretization and model reductionArea 3: Incorporation of parameter dependencies and robustnessThe cross section of each of the envisaged research areas exhibits a spectrum from basic research projects to research addressing specific applications.
DFG Programme
Priority Programmes
International Connection
Australia, Italy, Netherlands, Senegal, South Africa, Switzerland, USA
Projects
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A Calculus for Non-Smooth Shape Optimization with Applications to Geometric Inverse Problems
(Applicants
Herzog, Roland
;
Schmidt, Stephan
)
-
A non-smooth phase-field approach to shape optimization with instationary fluid flow
(Applicants
Hintermüller, Michael
;
Hinze, Michael
)
-
A Unified Approach to Optimal Uncertainty Quantification and Risk-Averse Optimization with
Quasi-Variational Inequality Constraints
(Applicant
Hintermüller, Michael
)
-
Approximation of non-smooth optimal convex shapes with applications in optimal insulation and minimal resistance
(Applicants
Bartels, Sören
;
Wachsmuth, Gerd
)
-
Bilevel Optimal Control: Theory, Algorithms, and Applications
(Applicants
Dempe, Stephan
;
Wachsmuth, Gerd
)
-
Bilevel Optimal Transport
(Applicants
Lorenz, Dirk A.
;
Meyer, Christian
)
-
Constrained Mean Field Games: Analysis and Algorithms
(Applicant
Hintermüller, Michael
)
-
Coordination Funds
(Applicant
Hintermüller, Michael
)
-
Coupling hyperbolic PDEs with switched DAEs: Analysis, numerics and application to blood flow models
(Applicants
Borsche, Raul
;
Trenn, Stephan
)
-
Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion
(Applicants
Hintermüller, Michael
;
Surowiec, Thomas Michael
)
-
Identification of Energies from Observations of Evolutions
(Applicant
Fornasier, Massimo
)
-
Identification of Stresses in Heterogeneous Contact Models
(Applicants
Duda, Georg
;
Weiser, Martin
)
-
Multi-Leader-Follower Games in Function Space
(Applicants
Schwartz, Alexandra
;
Steffensen, Sonja
)
-
Multi-Physics Phenomena in High-Temperature Superconductivity: Analysis, Numerics and Optimization
(Applicant
Yousept, Irwin
)
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Multiobjective Optimal Control of Partial Differential Equations Using Reduced-Order Modeling
(Applicants
Dellnitz, Michael
;
Peitz, Sebastian
;
Volkwein, Stefan
)
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Multiscale control concepts for transport-dominated problems
(Applicants
Göttlich, Simone
;
Herty, Michael
)
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Non-smooth Methods for Complementarity Formulations of Switched Advection-Diffusion Processes
(Applicants
Kirches, Christian
;
Sager, Sebastian
)
-
Nonsmooth and nonconvex optimal transport problems
(Applicants
Schmitzer, Bernhard
;
Wirth, Benedikt
)
-
Nonsmooth Multi-Level Optimization Algorithms for Energetic Formulations of Finite-Strain Elastoplasticity
(Applicants
Sander, Oliver
;
Schiela, Anton
)
-
Numerical Methods for Diagnosis and Therapy Design of Cerebral Palsy by Bilevel Optimal Control of Constrained Biomechanical Multi-Body Systems
(Applicants
Bock, Hans Georg
;
Kostina, Ekaterina
)
-
Optimal Control of Dissipative Solids: Viscosity Limits and Non-Smooth Algorithms
(Applicants
Herzog, Roland
;
Knees, Dorothee
;
Meyer, Christian
)
-
Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities
(Applicant
Hintermüller, Michael
)
-
Optimal Control of Static Contact in Finite Strain Elasticity
(Applicant
Schiela, Anton
)
-
Optimal Control of Variational Inequalities of the Second Kind with Application to Yield Stress Fluids
(Applicants
Meyer, Christian
;
Schweizer, Ben
;
Turek, Stefan
)
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Optimization methods for mathematical programs with equilibrium constraints in function spaces based on adaptive error control and reduced order or low rank tensor approximations
(Applicants
Ulbrich, Stefan
;
Ulbrich, Michael
)
-
Optimization Problems in Banach Spaces with Non-smooth Structure
(Applicants
Kanzow, Christian
;
Wachsmuth, Daniel
)
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Optimizing Fracture Propagation Using a Phase-Field Approach
(Applicants
Neitzel, Ira
;
Wick, Thomas
;
Wollner, Winnifried
)
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Optimizing Variational Inequalities on Shape Manifolds
(Applicant
Schulz, Volker
)
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Parameter identification in models with sharp phase transitions
(Applicants
Clason, Christian
;
Rösch, Arnd
)
-
Semi-Smooth Newton Methods on Shape Spaces
(Applicants
Schulz, Volker
;
Welker, Kathrin
)
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Shape Optimization for Maxwell's Equations Including Hysteresis Effects in the Material Laws
(Applicants
Schmidt, Stephan
;
Walther, Andrea
)
-
Shape Optimization for Mitigating Coastal Erosion
(Applicant
Schulz, Volker
)
-
Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System with Variable Fluid Densities
(Applicants
Hintermüller, Michael
;
Hinze, Michael
)
-
Simulation and Optimization of Rate-Independent Systems with Non-Convex Energies
(Applicants
Knees, Dorothee
;
Meyer, Christian
)
-
Stress-Based Methods for Variational Inequalities in Solid Mechanics: Finite Element Discretization and Solution by Hierarchical Optimization
(Applicant
Starke, Gerhard
)
-
Theory and Solution Methods for Generalized Nash Equilibrium Problems Governed by Networks of Nonlinear Hyperbolic Conservation Laws
(Applicants
Ulbrich, Stefan
;
Ulbrich, Michael
)