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

• 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 )
• Multiobjective Optimal Control of Partial Differential Equations Using Reduced-Order Modeling (Applicants Dellnitz, Michael ;  Peitz, Sebastian ;  Volkwein, Stefan )
• Multiscale control concepts for transport-dominated problems (Applicants Göttlich, Simone ;  Herty, Michael )
• 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 )
• 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 )
• Optimizing Fracture Propagation Using a Phase-Field Approach (Applicants Neitzel, Ira ;  Wick, Thomas ;  Wollner, Winnifried )
• Optimizing Variational Inequalities on Shape Manifolds (Applicant Schulz, Volker )
• 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 )
• 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 )

• Spokesperson: Professor Dr. Michael Hintermüller
• Participating Person: Jana Segmehl