Research

Optimization

• Shape optimization problems and their numerical treatment / optimization methods in shape spaces
• Analytical and numerical treatment of constrained optimization problems (in particular, constraints in form of partial differential equations and variational inequalities)
• Stochastic approximation / optimization under uncertainties
• Modelling of optimization problems

Shape Spaces and their Structures

• Riemannian manifolds
• Shape spaces as diffeological spaces

Optimization Applications in Civil Engineering

• Optimal experimental design (e.g., optimal sensor placement)
• Fracture modelling by optimization techniques

Structural Health Monitoring (funded by dtec.bw)

The overarching goal of the SHM project is to develop new and innovative methods to monitor infrastructure buildings and to continuously evaluate their structural conditions. In this project, an interdisciplinary team consisting of engineers and mathematicians works together with industrial companies. This projects aims to develop methods which are eligible for the detection of any kind of damage in various building structures. The developed methods should allow the reliability-based evaluation of existing infrastructure buildings using sensor data.

Semi-Smooth Newton Methods on Shape Spaces (funded by the German Research Fundation within DFG Priority Programme SPP 1962/2)

The main aim of this project is to set up an approach for investigating analytically and solving computationally shape optimization problems constrained by variational inequalities (VI) in shape spaces. Shape optimization problem constraints in the form of VIs are challenging, since classical constraint qualifications for deriving Lagrange multipliers generically fail.

In this project, we consider Newton-shape derivatives instead of classical shape derivatives in order to formulate first-order necessary optimality conditions. Setting up a Newton-shape derivative scheme is the guiding principle for the analytical and numerical investigations within this project. More precisely, the resulting scheme enables the analytical and computational treatment of shape optimization problems constrained by VIs which are non-shape differentiable in the classical sense such that these can handled and solved without any regularization techniques leading often only to approximated shape solutions. Further goals of this project are investigations in the area of shape optimization for VIs regarding appropriate shape space formulations, existence and well-posedness of solutions including stationary concepts in shape spaces, semi-smooth Newton methods in shape spaces, mesh independent algorithmic approaches, robust treatment of uncertainties and solution approaches to application problems like, e.g., from the field of (thermo-)mechanics.

Simulation-Based Design Optimization of Dynamic Systems under Uncertainties (funded by Landesforschungsförderung Hamburg)

The main aim of the project is to develop new innovative simulation methods for the robust optimization of complex components. By combining methods from applied mathematics and theoretical mechanical engineering, mathematical models, which involve dynamic operating conditions and uncertain manufacturing processes, will be developed. In particular, a robust design is important for maintenance-intensive and maintenance-free products from the Hamburg aviation and medical technology environment.

Journals

C. Geiersbach, T. Suchan, and K. Welker, Stochastic Augmented Lagrangian Method in Riemannian Shape Manifolds. Journal of Optimization Theory and Applications (2024). DOI: 10.1007/s10957-024-02488-1.

T. Suchan, C. Kandekar, W.E. Weber, and K. Welker. Crack propagation in anisotropic brittle materials: From a phase-field model to a shape optimization approach. Engineering Fracture Mechanics, 303:110065 (2024). DOI: 10.1016/j.engfracmech.2024.110065.

L. Radtke, G. Bletsos, N. Kühl, T. Suchan, T. Rung, A. Düster, and K. Welker. Parameter-Free Shape Optimization: Various Shape Updates for Engineering Applications. Aerospace, 10(9):751. 2023. DOI: 10.3390/aerospace10090751.

K. Welker. Suitable Spaces for Shape Optimization. Applied Mathematics and Optimization, 84(S1):869–902, Springer, 2021. DOI: 10.1007/s00245-021-09788-2.

C. Geiersbach, E. Loayza-Romero and K. Welker. Stochastic Approximation for Optimization in Shape Spaces. SIAM Journal on Optimization, 31(1):348-376, 2021. DOI: 10.1137/20m1316111.

D. Luft, V.H. Schulz, and K. Welker. Efficient Techniques for Shape Optimization with Variational Inequalities Using Adjoints. SIAM Journal on Optimization, 30(3):1922-1953, 2020. DOI: 10.1137/19m1257226.

B. Führ, V.H. Schulz and K. Welker. Shape Optimization for Interface Identification with Obstacle Problems. Vietnam Journal of Mathematics, 46(4):967–985, 2018. DOI: 10.1007/s10013-018-0312-0.

M. Siebenborn and K. Welker. Algorithmic Aspects of Multigrid Methods for Optimization in Shape Spaces. SIAM Journal on Scientific Computing, 39(6):B1156-B1177, 2017. DOI: 10.1137/16m1104561.

V.H. Schulz, M. Siebenborn, and K. Welker. Efficient PDE Constrained Shape Optimization Based on Steklov–Poincaré-Type Metrics. SIAM Journal on Optimization, 26(4):2800-2819, 2016. DOI: 10.1137/15m1029369.

V.H. Schulz, M. Siebenborn, and K. Welker. Structured Inverse Modeling in Parabolic Diffusion Problems. SIAM Journal on Control and Optimization, 53(6):3319-3338, 2015. DOI: 10.1137/140985883.

Books and Special Issues

P. Gangl and K. Welker (eds.) Proceedings in Applied Mathematics and Mechanics, Special Issue: 8th GAMM Juniors’ Summer School 21(S1). Wiley: e202100901 (2021). DOI: 10.1002/pamm.202100901.

I. Demir, Y. Lou, X. Wang, and K. Welker (eds.) Advances in Data Science. Springer International Publishing, Cham (2021). DOI: 10.1007/978-3-030-79891-8.

Book chapters

C. Geiersbach, E. Loayza-Romero, and K. Welker, PDE-Constrained Shape Optimization: Toward Product Shape Spaces and Stochastic Models. In: K. Chen, C.-B. Schönlieb, X.-C. Tai, and L. Younes (eds.) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer International Publishing, Cham: 1585-1630 (2023). DOI: 10.1007/978-3-030-98661-2_120.

W. Weber, N. Rauter, R. Lammering, and K. Welker, Räumliche Auflösung des Schadenszustandes aus mechanischer und mathematischer Sicht. In: D. Schulz, A. Fay, W. Matiaske, and M. Schulz (eds.) dtec.bw-Beiträge der Helmut-Schmidt-Universität / Universität der Bundeswehr Hamburg: Forschungsaktivitäten im Zentrum für Digitalisierungs- und Technologieforschung der Bundeswehr dtec.bw 1. Helmut-Schmidt-Universität, Hamburg: 281-286 (2022). DOI: 10.24405/14565.

V.H. Schulz and K. Welker, Shape Optimization for Variational Inequalities of Obstacle Type: Regularized and Unregularized Computational Approaches. In: M. Hintermüller et al. (eds.) Non-Smooth and Complementarity-Based Distributed Parameter Systems, International Series of Numerical Mathematics 172. Birkhäuser, Cham: 397-420 (2021). DOI: 10.1007/978-3-030-79393-7_16.

V.H. Schulz and K. Welker, On Optimization Transfer Operators in Shape Spaces. In: V.H. Schulz and D. Seck (eds.) Shape Optimization, Homogenization and Optimal Control, International Series of Numerical Mathematics 169. Birkhäuser, Cham: 259-275 (2018). DOI: 10.1007/978-3-319-90469-6_13.

A. Panotopoulou, E. Ross, K. Welker, E. Hubert, and G. Morin, Scaffolding a Skeleton. In: A. Gençtav et al. (eds.) Research in Shape Analysis, Association for Woman in Mathematics 12. Springer International Publishing, Cham: 17-35 (2018). DOI: 10.1007/978-3-319-77066-6_2.

V.H. Schulz, M. Siebenborn, and K. Welker, Towards a Lagrange-Newton Approach for PDE Constrained Shape Optimization. In: A. Pratelli and G. Leugering (eds.) Trends in PDE Constrained Shape Optimization, International Series of Numerical Mathematics 166. Birkhäuser, Cham: 229-249 (2015). DOI: 10.1007/978-3-319-17563-8_10.

Proceedings

L. Pryymak, T. Suchan, and K. Welker, A Product Shape Manifold Approach for Optimizing Piecewise-Smooth Shapes, In: F. Nielsen and F. Barbaresco (eds.) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science 14071. Springer International Publishing, Cham: 21-30 (2023). DOI: 10.1007/978-3-031-38271-0_3.

T. Suchan, K. Welker, and W. Wollner, A new shape optimization approach for fracture propagation. In: Proceedings in Applied Mathematics and Mechanics 22: e202200124 (2023). DOI: 10.1002/pamm.202200124.

T. Suchan, R. Najafi Koopas, N. Rauter, and K. Welker, Tracking of fracture‐state displacement data generated by cohesive zone modeling using shape optimization. In: Proceedings in Applied Mathematics and Mechanics 22: e202200284 (2023). DOI: 10.1002/pamm.202200284.

T. Suchan and K. Welker, Viscous energy dissipation reduction by optimization of multiple shapes. In: Proceedings in Applied Mathematics and Mechanics 21: e202100261 (2021). DOI: 10.1002/pamm.202100261.

N. Goldammer and K. Welker, Optimization on diffeological spaces. In: Proceedings in Applied Mathematics and Mechanics 21: e202100260 (2021). DOI: 10.1002/pamm.202100260.

N. Goldammer and K. Welker, Towards optimization techniques on diffeological spaces. In: Proceedings in Applied Mathematics and Mechanics 20: e202000040 (2021). DOI: 10.1002/pamm.202000040.

R. Bergmann, R. Herzog, E. Loayza-Romero, and K. Welker, Shape optimization: what to do first, optimize or discretize? In: Proceedings in Applied Mathematics and Mechanics 19: e201900067 (2019). DOI: 10.1002/pamm.201900067.

D. Luft and K. Welker, Computational Investigations of an Obstacle-Type Shape Optimization Problem in the Space of Smooth Shapes. In: F. Nielsen and F. Barbaresco (eds.) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science 11712. Springer International Publishing, Cham: 579-588 (2019). DOI: 10.1007/978-3-030-26980-7_60.

K. Welker, Optimization in the Space of Smooth Shapes. In: F. Nielsen and F. Barbaresco (eds.) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science 10589. Springer International Publishing, Cham: 65-72 (2017). DOI: 10.1007/978-3-319-68445-1_8.

V.H. Schulz, M. Siebenborn, and K. Welker, PDE Constrained Shape Optimization as Optimization on Shape Manifolds. In: F. Nielsen and F. Barbaresco (eds.) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science 9389. Springer Nature Switzerland, Cham: 499-508 (2015). DOI: 10.1007/978-3-319-25040-3_54.

Preprints / Submitted articles

T. Suchan, V. Schulz, and K. Welker, Shape optimization in the space of piecewise-smooth shapes for the Bingham flow variational inequality, arXiv, 2024. DOI: 10.48550/arXiv.2403.02106.

C. Geiersbach, T. Suchan, and K. Welker, Optimization of piecewise smooth shapes under uncertainty using the example of Navier-Stokes flow, arXiv, 2023. DOI: 10.48550/arXiv.2308.07742.

N. Goldammer, V.H. Schulz, and K. Welker, Gâteaux semiderivative approach applied to shape optimization for contact problems, arXiv, 2022. DOI: 10.48550/arXiv.2208.03687.

N. Goldammer and K. Welker, Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts, arXiv, 2020. DOI: 10.48550/arXiv.2009.04262.

C. Geiersbach, E. Loayza, and K. Welker, Computational Aspects for Interface Identification Problems with Stochastic Modelling, arXiv, 2019. DOI: 10.48550/arXiv.1902.01160.

Thesis

K. Welker. Efficient PDE Constrained Shape Optimization in Shape Spaces. PhD Thesis, Universität Trier, 2016. DOI: 10.25353/ubtr-xxxx-6575-788c/.

K. Welker. Riemannsche Metriken auf dem Raum der Formen. Diploma Thesis, Universität Trier, 2013.