Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities

Description

Quasi-variational inequalities (QVls) often arise in applications where non-smooth and nonlinear phenomena lead to complex state-dependent constraints. They can be used to describe, for instance, the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, or the chemotactic behavior of bacteria in competition. Mathematically, the solutions to these problems are not unique and their dependence on input quantities (data, controls, etc.) is non-smooth.

This project is devoted to analyzing and numerically solving optimal control problems associated with elliptic and parabolic QVis. The research work is organized as follows: (a) It starts with the development of function-space based solution algorithms for QVIs tailored to constraints of obstacle- or gradient-type. In particular, we aim at path-following semi smooth Newton schemes which exhibit fast local mesh-independent convergence. (b) Then it focuses on an enhanced solution theory for the underlying QVls. More specifically, properties of the minimal and maximal solutions will be studied along with associated (differential) stability and numerical approximation schemes. (c) Then, in a two progressively more demanding research steps, stationary conditions for optimal control problems for the QVls of interest will be derived. These optimization problems fall into the realm of mathematical programs with equilibrium constraints (MPECs) in function space. In technical terms, in our stationarity considerations two smoothing approaches will be pursued, one utilizing a Moreau-Yosida technique and the the other one relying on a technique modifying the underlying differential operators. (d) Finally, bundle-free implicit programming methods for the numerical solution of the MPEC under consideration are pursued. These also involve relaxation and path-following techniques, and advanced discretization schemes.

The analytical as well as numerical advance in the project work will be validated against prototypical applications. These involve in particular the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, and the chemotactic behavior of bacteria in competition.

Publications

Amal Alphonse, Carlos N. Rautenberg, Michael Hintermüller: Existence, Iteration Procedures and Directional Differentiability for Parabolic QVIs, Calc. Var. 59, 95, 2020 (SPP1962-114r).

Amal Alphonse, Carlos N. Rautenberg, Michael Hintermüller: Stability of the solution set of quasi-variational inequalities and optimal control, SIAM J. Control Optim., 58(6), 3508–3532, 2020 (SPP1962-107r).

Amal Alphonse, Carlos N. Rautenberg, Michael Hintermüller: Recent Trends and Views on Elliptic Quasi-variational Inequalities, In: Hintermüller M., Rodrigues J. (eds) Topics in Applied Analysis and Optimisation. CIM Series in Mathematical Sciences. Springer, Cham. , 2019 (SPP1962-071).

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Directional Differentiability for Elliptic Quasi-variational Inequalities of Obstacle Type, Calc. Var. 58: 39. , 2019 (SPP1962-049).

Andrea N. Ceretani, Carlos N. Rautenberg: The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow, Z. Angew. Math. Phys. 70: 14, 2019.

Michael Hintermüller, Carlos N. Rautenberg, Nikolai Strogies: Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities With Gradient Constraints, Set-Valued Var. Anal, 2018 (SPP1962-036).

Michael Hintermüller, Simon Rösel: Duality Results and Regularization Schemes for Prandtl-Reuss Perfect Plasticity, ESAIM: COCV, 2018 (SPP1962-006).

Harbir Antil, Carlos N. Rautenberg: A Fractional Elliptic Quasi-Variational Inequalities: Theory and Numerics , Interfaces Free Bound. 20, 1-24., 2018.

Michael Hintermüller, Carlos N. Rautenberg, Simon Rösel: Density of Convex Intersections and Applications, Proc. R. Soc. A 2017 473 20160919, 2017 (SPP1962-005).

Michael Hintermüller, Carlos N. Rautenberg, Simon Rösel: Density of Convex Intersections and Applications, Proc. R. Soc. A 2017 473 20160919, 2017 (SPP1962-005).

Preprints

Amal Alphonse, Michael Hintermüller, Carlos N.: Stability of the solution set of quasi-variational inequalities and optimal control (SPP1962-107r, 08/2020, [bib])

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Existence, Iteration Procedures and Directional Differentiability for Parabolic QVIs (SPP1962-114r, 05/2020, [bib])

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Directional Differentiability for Elliptic Quasi-variational Inequalities of Obstacle Type (SPP1962-049r, 11/2019, [bib])

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Existence, Iteration Procedures and Directional Differentiability for Parabolic QVIs (SPP1962-114, 06/2019, [bib])

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Stability of the Solution Set of Quasi-variational Inequalities and Optimal Control (SPP1962-107, 03/2019, [bib])

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Recent Trends and Views on Elliptic Quasi-variational Inequalities (SPP1962-071, 09/2018, [bib])

Amal Alphonse, Michael Hintermüller, Carlos N. Rautenberg: Directional Differentiability for Elliptic Quasi-variational Inequalities of Obstacle Type (SPP1962-049, 02/2018, [bib])

Michael Hintermüller, Carlos N. Rautenberg, Nikolai Strogies: Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities With Gradient Constraints (SPP1962-036, 10/2017, [bib])

Michael Hintermüller, Simon Rösel: Duality Results and Regularization Schemes for Prandtl-Reuss Perfect Plasticity (SPP1962-006, 02/2017, [bib])

Michael Hintermüller, Carlos N. Rautenberg, Simon Rösel: Density of Convex Intersections and Applications (SPP1962-005, 01/2017, [bib])

Research Area

Modeling, problem analysis, algorithm design and convergence analysis

The focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.

Realization of algorithms, adaptive discretization and model reduction

As the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.

Members

Project Related News

  • Aug 24, 2020 : New revised preprint submitted

    Amal Alphonse submitted the revised preprint SPP1962-107r, Stability of the solution set of quasi-variational inequalities and optimal control

  • May 27, 2020 : New revised preprint submitted

    Amal Alphonse submitted the revised preprint SPP1962-114r, Existence, Iteration Procedures and Directional Differentiability for Parabolic QVIs

  • Nov 26, 2019 : New revised preprint submitted

    Amal Alphonse submitted the revised preprint SPP1962-049r, Directional Differentiability for Elliptic Quasi-variational Inequalities of Obstacle Type

  • Jun 21, 2019 : New preprint submitted

    Amal Alphonse submitted the preprint SPP1962-114, Existence, Iteration Procedures and Directional Differentiability for Parabolic QVIs

  • Mar 05, 2019 : New preprint submitted

    Amal Alphonse submitted the preprint SPP1962-107, Stability of the Solution Set of Quasi-variational Inequalities and Optimal Control

  • Sep 13, 2018 : New preprint submitted

    Amal Alphonse submitted the preprint SPP1962-071, Recent Trends and Views on Elliptic Quasi-variational Inequalities

  • Feb 10, 2018 : New preprint submitted

    Amal Alphonse submitted the Preprint SPP1962-049, Directional Differentiability for Elliptic Quasi-variational Inequalities of Obstacle Type

  • Sep 20, 2016 : Welcome to our new project member

    Amal Alphonse joins project 11.