Optimizing Variational Inequalities on Shape Manifolds


Shape optimization is of importance in many fields of applications similarly to variational inequalities. However, shape optimization problems with constraints consisting of variational inequalities have not yet been considered much in the existing literature. This proposal aims at a novel approach to shape optimization problems in terms of shape manifolds and the resulting framework from infinite dimensional Riemannian geometry, which has been developed recently by the applicant. This approach enables a theoretical connection of shape optimization with optimal control problems in vector bundles, which will be the guiding principle for the analytical and numerical investigations within this project. Thus, the goals of this project are investigations in the area of shape optimization for variational inequalities regarding appropriate Riemannian shape manifold formulations, existence and well-posedness of solutions, semi-smoth Newton methods on shape vector bundles, mesh independent algorithmic approaches, robust treatment of uncertainties and solution approaches to application problems from the field of (thermo-)mechanics. Besides that, the shape manifold approach together with its novel shape metrics enhancing discretization and algorithmic robustness provides a basis for cooperation with other projects addressing shape based problem formulations.


Volker Schulz, Kathrin Welker: On Optimization Transfer Operators in Shape Spaces, Shape Optimization, Homogenization and Optimal Control, International Series of Numerical Mathematics 169, 2018.

Björn Führ, Volker Schulz, Kathrin Welker: Shape Optimization for Interface Identification with Obstacle Problems, Vietnam Journal of Mathematics, 2018.

Athina Panotopoulou, Kathrin Welker, Elissa Ross, Evelyne Hubert, Geraldine Morin: Scaffolding a Skeleton, Research in Shape Analysis, Association for Woman in Mathematics, Springer, 2018.

Martin Siebenborn, Kathrin Welker: Computational aspects of multigrid methods for optimization in shape spaces, SIAM Journal on Scientific Computing, 39(6), 2017.

Kathrin Welker: Optimization in the space of smooth shapes, Geometric Science of Information, volume 10589 of Lecture Notes in Computer Science, Springer, 2017.


No preprints from this project yet.


Project Related News

  • May 24, 2018 : Best paper award for Kathrin Welker

    Kathrin Welker has obtained the best paper award at the 3rd conference on Geometric Science of Information (7th - 9th November 2017, Mines ParisTech, Paris, France) for her paper on "Optimization in the Space of Smooth Shapes".

  • Oct 10, 2016 : Welcome to our new project member

    Katrin Welker joins project 20.