### „*Semi-smooth Newton methods on shape spaces*“ 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*“ (founded 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.

* Subproject „Efficient algorithms for PDE and VI constrained shape optimization problems under uncertainties“: *Shape optimization problems, as a subset of structural optimization, are particularly challenging due to the infinite-dimensional, nonlinear structure of shape spaces. One option to overcome this drawback is to choose finite dimensional shape spaces, i.e., to discretize first. Nevertheless, this limits the amount of available tools coming from function spaces, resp. the solution space itself. Therefore, shape optimization problems can be solved using one of two different approaches: discretize-then-optimize or optimize- then-discretize. In general, it is a challenging task to decide whether to use a discretize-then-optimize or optimize-then-discretize approach to solve constrained optimization problems. In this project, we investigate the benefits and drawbacks of both approaches with respect to accuracy of results and computational performance based on special shape optimization problems. In particular, we consider shape optimization problems constrained by partial differential equations (PDE) as well as variational inequalities (VI). Since classical constraint qualifications for deriving Lagrange multipliers generically fail, shape optimization problem constraints in the form of VIs are highly challenging and, thus, require new, sophisticated algorithms. Here, the choice of adjoints to compute shape sensitivities within the field of structural optimization is a topic of current research and part of this joint research project. A special focus is put on determining and updating the areas in which the inequalities are active, the so-called active sets. Moreover, uncertainties in the shape optimization problem play an important role in order to deal with realistic models. In particular, we aim at creating algorithms that are also robust to data from experimental measurements.

### „*Structural Health Monitoring*“ (founded by DTEC.Bw)

### Journals

C. Geiersbach, E. Loayza and K. Welker. Stochastic approximation for optimization in shape spaces. *SIAM Journal on Optimization, *31(1):348-376, 2021. (arXiv:2001.10786)

D. Luft, V. Schulz and K. Welker. Efficient techniques for shape optimization with variational inequalities using adjoints. *SIAM Journal on Optimization*, 30(3):1922-1953, 2020. (arXiv1904.08650)

B. Führ, V. Schulz and K. Welker. Shape optimization for interface identification with obstacle problems. *Vietnam Journal of Mathematics*, 2018. DOI: 10.1007/s10013-018-0312-0.

M. Siebenborn and K. Welker. Computational aspects of multigrid methods for optimization in shape spaces. *SIAM Journal on Scientific Computing*, 39(6):B1156-B1177, 2017. (arXiv:1611.05272)

V. 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. (arXiv:1506.02244)

V. Schulz, M. Siebenborn and K. Welker. Structured inverse modeling in parabolic diffusion problems. *SIAM Journal on Control and Optimization*, 53(6):3319-3338, 2015. (arXiv:1409.3464)

### Book chapters

V. Schulz and K. Welker. Shape optimization for variational inequalities of obstacle type: regularized and unregularized computational approaches. In: M. Hintermüller et al., SPP 1962 special issue, Birkhäuser, 2019 (accepted).

A. Panotopoulou, K. Welker, E. Ross, E. Hubert and G. Morin. Scaffolding a skeleton. In: A. Gençtav et al., editors, *Research in Shape Analysis*, Association for Woman in Mathematics, pages 17-35, Springer, 2018. DOI: 10.1007/978-3-319-77066-6. (PDF)

V. Schulz and K. Welker. On optimization transfer operators in shape spaces. In: V. Schulz and D. Seck, editors, *Shape Optimization, Homogenization and Optimal Control,* volume 169 of *International Series **of Numerical Mathematics*, pages 259–275. Springer, 2018.

V. Schulz, M. Siebenborn and K. Welker. Towards a Lagrange-Newton approach for PDE constrained shape optimization. In: A. Pratelli and G. Leugering, editors, *Trends in PDE Constrained Shape **Optimization*, volume 166 of *International Series of Numerical Mathematics*, pages 229-249, Springer, 2015. DOI: 10.1007/978-3-319-17563-8. (arXiv:1405.3266)

### Proceedings

N. Goldammer and K. Welker. Towards optimization techniques on diffeological spaces. Proceedings in Applied Mathematics and Mechanics, 2020. DOI: 10.1002/pamm.202000040

R. Bergmann, R. Herzog, E. Loayza and K. Welker. Shape optimization: what to do first, optimize or discretize? Advantages and disadvantages for PDE-constrained problems. Proceedings in Applied Mathematics and Mechanics, 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, editors, *Geometric Science of Information,* vol 11712 of* Lecture Notes in Computer Science, *pages 579-588, Springer, 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, editors, *Geometric Science of Information*, volume 10589 of *Lecture Notes in Computer Science*, pages 65-72, Springer, 2017. DOI: 10.1007/978-3-319-68445-1_8.

V. Schulz, M. Siebenborn and K. Welker. PDE constrained shape optimization as optimization on shape manifolds. In: F. Nielsen and F. Barbaresco, editors, *Geometric Science of Information*, volume 9389 of *Lecture Notes in Computer Science*, pages 499-508, Springer, 2015. DOI: 10.1007/978-3-319- 25040-3_54.

### Preprints / Submitted articles

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

C. Geiersbach, E. Loayza and K. Welker. Computational Aspects for Interface Identification Problems with Stochastic Modelling. (arXiv:1902.01160)

K. Welker. Suitable spaces for shape optimization. Submitted to Applied Mathematics and Optimization Journal, Springer, 2020. (arXiv:1702.07579)

### Thesis

K. Welker. *Efficient PDE Constrained Shape Optimization in Shape Spaces*. PhD Thesis, Universität Trier, 2016. (PDF)

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

**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

Letzte Änderung: 26. März 2021