## Joint Research Seminar on

Mathematical Optimization / Non-smooth Variational Problems and Operator Equations

The research seminar serves as a platform for the presentation of current
research results of group members at WIAS and at the HU Berlin (Mathematical
Optimization) and of invited guests.
Graduate students in Mathematical Optimization with a sound background in Optimization, Numerical Analysis,
Functional Analysis and Partial Differential Equations from HU Berlin are welcome to
take part in the seminar.
The regular date of the seminar is
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Wednesday, 1:15 p.m. at WIAS, Mohrenstraße 39, Erhard-Schmidt-Hörsaal.
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Please be aware that occasional changes in time or venue might occur, but will be announced here.

### Upcoming Dates

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Friday, January 19, 2:00 p.m. (WIAS-ESH)
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S. Bartels(Albert-Ludwigs-Universität Freiburg)

Finite element methods for nonsmooth problems and application to a problem in optimal insulationNonsmooth problems arise in the mathematical modeling of contact and obstacle problems, the description of plastic material behavior, and mathematical image processing. The unknown functions are typically characterized as minimizers of nondifferentiable functionals. Numerical schemes approximately solve these problems either via duality methods or classically by making use of appropriate regularizations. In the talk we discuss the discretization and iterative solution of a model problem defined on functions of bounded variation. The numerical analysis of finite element discretizations leads to reduced convergence rates which can be improved using adaptive mesh refinement. Suitable iterative solution procedures are ADMM schemes, for which we propose an automatic step size adjustment strategy, and gradient flows, for which we demonstrate the unconditional stability of a semi-implicit time discretization. The methods are applicable in the numerical determination of optimal insulating films for heat conducting bodies. Below a critical value of available insulation mass an unexpected break of symmetry occurs.

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Friday, February 2, 2:30 p.m. (WIAS-ESH)
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Th. Ruf(Gerolsbach)

On a variational approach to the nonlinear wave equationIn a 2012 paper, E. Serra and P. Tilli proved a conjecture by E. de Giorgi stating that global weak solutions of wave equations such as w'' - Delta w + w|w|^(p-2) = 0 on R^+ times R^d can be obtained as limits of minimizers of suitable variational functionals. We generalize this proof to equations of the form w'' - Delta w + f_w (t; x; w) = 0 with p-growth conditions on f, also replacing R^d by an arbitrary open set O subset R^d and suitable boundary conditions.

### Previous Dates

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Thursday, January 11, 10:00 a.m. (WIAS-ESH)
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T. Kluth(Universität Bremen)

Model-based magnetic particle imagingMagnetic particle imaging (MPI) is a tracer-based imaging modality developed to detect the concentration of superparamagnetic iron oxide nanoparticles. It is highly sensitive to the nanoparticle's nonlinear response to a dynamic applied magnetic field. Model-based reconstruction techniques are still not able to reach the quality of data-based approaches in which the linear system function is determined by a time-consuming measurement process. Possible reasons include the relaxation behavior of nanoparticles in fast changing magnetic fields. However, the equilibrium model described by the Langevin function is still used to predict the system behavior. In this context we discuss the ill-posedness of the imaging problem. We further focus on the model-based MPI reconstruction problem incorporating deviations in the forward operator. This is illustrated by initial results from real data using a regularized total least squares approach.

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Thursday, December 21, 2:15 p.m. (WIAS, HVP11A, 4.13)
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Guozhi Dong(HU Berlin)

Regularization methods and nonlinear PDEs for solving inverse and imaging problemsIn this talk, I will present some recent developments on regularization methods in image sciences. I shall show a tiny background and also mention some of the state of the art in this area. The focus will be then to discuss some non-convex regularization models for specific problems which suffer from displacement errors. The non-convex energy functionals reveal to have tight connections to some nonlinear (geometric) partial differential equations, e.g. mean curvature flows. Finally, I will show some numerical results with some discussions on the algorithms.

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Monday, November 27, 9:30 a.m. (WIAS, HVP11A, 4.13)
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Andrea Ceretani(HU Berlin)

Anomalous diffusion with free boundariesRecently, time fractional Stefan-like problems have been used to model anomalous diffusion with free boundaries and long memory retention. Nevertheless, the physical foundations of this usage is still unclear. We present a model for acid water neutralization with anomalous and fast diffusion. Though this problem presents short memory retention, it is a first step in deriving mathematical models for anomalous diffusion under memory effects based on commonly accepted physical laws. The problem consists in the neutralization of an acid solution in which the hydrogen ions are transported according to Cattaneo's diffusion, and we consider the specific case of a marble slab reacting with a sulphuric acid solution in a one-dimensional geometry. The mathematical problem is reduced to a hyperbolic free boundary problem where the consumption of the slab is described by a nonlinear differential equation. We prove global well-posedness and present some numerical simulations.

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Thursday, October 12, 10:30 a.m. (HU Adlershof, room 2.417)
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Stephan Schmidt(University of Wuerzburg)

SQP Methods for shape optimization based on weak shape HessiansMany PDE constrained optimization problems fall into the category of shape optimization, meaning the geometry of the domain is the unknown to be found. Most natural applications are drag minimization in fluid dynamics, but many tomography and image reconstruction problems also fall into this category. The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems. The main focus here will be on generating Newton-like methods for large scale applications. The key for this endeavor is the derivation of the shape Hessian, that is the second directional derivative of a cost functional with respect to geometry changes in a weak form based on material derivatives instead of classical local shape derivatives. To avoid human errors, a computer aided derivation system is also introduced. The methodologies are tested on problem from fluid dynamics and geometric inverse problems.

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Wednesday, July 26, 1:15 p.m. (WIAS-ESH)
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Hasnaa Zidani(ENSTA ParisTech)

Multi-objective control problems under state constraintsIn this talk, we shall present a new approach, based on Hamilton-Jacobi theory, for characterizing the Pareto front for multi-objective optimal control problems in presence of state constraints. We define an auxiliary control problem for an augmented dynamical system and show that the pareto front is a subset of the zero level set of the auxiliary value function. This characterization allows to deduce an efficient numerical procedure for computing the entire Pareto front and the corresponding optimal trajectories. Moreover, the approach allows to consider objective functions of different structures (minimum time cost, Bolza cost and/or infinite horizon objective). A numerical example will be considered to show the relevance of this approach.

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Thursday, June 22, 9:30 a.m. (HU Berlin, Adlershof, RUD25, 2.417)
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Caroline Löbhard(WIAS)

An adaptive space-time discretization for parabolic optimal control problem with state constraintsWe present a space-time discretization which is based on a reformulation of the stationarity conditions of a Moreau-Yosida regularized parabolic optimal control problem, which involves only the state variable. The resulting nonlinear partial differential equation is of fourth order in space, and of second order in time. In order to cope with the disbalance of regularity of the respective solutions, we develop a taylored discontinuous Galerkin scheme and derive convergence rates in the mesh size as well as an integrated update strategy for the regularization parameter related to the state constraints. We also propose an adaptive mesh refinement strategy and illustrate the performance of our method in numerical test cases.

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Wednesday, June 21, 3:30 p.m. (WIAS, HVP11A, 4.13)
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Mattia Bongini(Barcelona Graduate School of Economics)

Optimal Control Problems in Transport DynamicsIn this talk we shall discuss several results concerning the "indirect control of populations", i.e., how to influence a group of individuals by means of external agents with a directly controlled dynamics. By using the general theory of functionals defined on spaces of measures, we give sufficient conditions for the existence of optimal control strategies and then we present a Pontryagin Maximum Principle for such controls in the form of an Hamiltonian flow in the Wasserstein space of probability measure. Finally, we present an application of the above framework to the evacuation problem of a crowd from an unknown environment with the help of undercover stewards.

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Wednesday, Mai 31, 1:15 p.m. (WIAS, HVP11A, 4.13)
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Jo Andrea Brüggemann(HU Berlin / WIAS)

Solution methods for a contact model motivated by the human heart's pericardiumIntroducing a function space description of the contact within the human heart's pericardium, in this talk, we will study a relaxation of the latter. The quasi-variational inequality (QVI) in the focus of interest is attacked with a fixed-point approach and the hereby arising sequence of variational problems can be efficiently solved with a path-following semi-smooth Newton method.

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Wednesday, Mai 24, 1:15 p.m. (WIAS-ESH)
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Hongpeng Sun(Renmin University of China)

Weak convergence of proximal ADMM and its relaxations in Hilbert spacesADMM (Alternating Direction Method of Multipliers) is a popular first order method for mathematical imaging and inverse problems. However, the weak convergence of ADMM in infinite dimensional spaces is not clear yet, which is different from the classical Augmented Lagrangian Method. We will discuss the weak convergence of ADMM and its proximal variants with relaxations in Hilbert spaces.

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Tuesday, Mai 9, 3:30 p.m. (WIAS, HVP11A, 4.13)
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Kazufumi Ito(North Carolina State University, USA)

Value function calculus and applicationsIn this talk the sensitivity analysis is discussed for the parameter-dependent optimization. The sensitivity of the optimal value function with respect to the change in parameters plays a significant role in the optimization theory, including economics, finance, the Hamilton-Jacobi theory, the inf-sup duality and the structural design and the bi-level optimization. We develop the calculus for the value function and present its applications in the variational calculus, the bi-level optimization and the optimal control and optimal design and inverse problems.

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Friday, Mai 5, 2:15 p.m. (WIAS-ESH)
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Behzad Azmi(Univ. Graz, Austria)

On the stabilizability of infinite dimensional systems via receding horizon controlOne efficient strategy for dealing with optimal control problems on an infinite time horizon is the receding horizon framework. In this approach, an infinite horizon optimal control problem is approximated by a sequence of finite horizon problems in a receding horizon fashion. Stability is not generally ensured due to the use of a finite prediction horizon. Thus, in order to ensure the asymptotic stability of the controlled system, additional terminal cost functions and/or terminal constraints are often needed to add to the finite horizon problems. In this presentation, we are concerned with the stabilization of several classes of infinite-dimensional controlled systems by means of a Receding Horizon Control (RHC) scheme. In this scheme, no terminal costs or terminal constraints are used to ensured the stability. The key assumption is the stabilizability of the underlying system. Based on this condition the suboptimality and stability of RHC are investigated. To justify the applicability of this framework, we consider controlled systems governed by different types of partial differential equations. Numerical examples are presented as well.

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Thursday, Mai 4, 9:30 a.m. (HU Berlin, Adlershof, RUD25, 2.417)
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Axel Kröner(HU Berlin / CMAP, Ecole Polytechnique, Paris-Saclay)

Optimal control of infinite dimensional systemsIn this talk we analyze second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. The general framework allows the application to heat, wave, and Schr\"odinger equation. This is joint work with S. Aronna und F. Bonnans.

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Wednesday, Mar 29, 1:15 p.m. (WIAS-ESH)
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Christian Clason(Universität Duisburg-Essen)

Convex relaxation of hybrid discrete-continuous control problemsWe consider control problems for partial differential equations where the distributed control should take on values only from a given discrete and hence non-convex set. Such problems occur for example in parameter identification or topology optimization. Similar to their use in sparse optimization, L1-type norms can be used to formulate a convex relaxation which can be solved by semi-smooth Newton methods. We illustrate this approach using linear model problems and discuss the extension to vector-valued and nonlinear control problems.

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Tuesday, Feb 7, 10:15 a.m. (WIAS-ESH)
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Sara Merino-Aceituno(Imperial College London)

Kinetic theory to study emergent phenomena in biology: an example on swarmingClassical methods in kinetic theory are challenged when studying emergent phenomena in the biological and social sciences. New methodologies are needed to study the problems at hand which typically involve many agents that interact locally. The aim of this talk is to introduce the general framework of kinetic theory and some of the new challenges of applying it to biological systems. We illustrate it in the case of the so-called collective motion of self-propelled particles, like swarming of birds. Particularly, based on the Vicsek model, we study systems of agents (birds) that move at a constant speed while trying to align their body orientation with those of their neighbors. Starting from a particle description, we find the macroscopic dynamics.

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Tuesday, Jan 17, 10:15 a.m. (WIAS-ESH)
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Tim Sullivan(FU Berlin)

Well-Posedness of Bayesian Inverse Problems - Stable Priors on Quasi-Banach SpacesThe Bayesian perspective on inverse problems has attracted much mathematical attention in recent years, and particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite-dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ordinary or partial differential equation. Numerical solution of such infinite-dimensional BIPs must necessarily be performed in an approximate manner on a finite-dimensional subspace, but it is profitable to delay discretisation to the last possible moment and consider the original infinite-dimensional problem as the primary object of study, since infinite-dimensional well-posedness results and algorithms descend to any finite-dimensional subspace in a discretisation-independent way, whereas careless early discretisation may lead to a sequence of well-posed finite-dimensional BIPs or algorithms whose stability properties degenerate as the discretisation dimension increases.

This presentation will give an introduction to the framework of well-posed BIPs in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451-559, 2010) and others. Recently, this framework has been extended to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen-Loeve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data.

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Tuesday, Jan 10, 10:15 a.m. (WIAS-ESH)
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Andrew Lam(Uni Regensburg)

Diffuse interface models of tumor growth and optimizing cancer treatment timesThere has been a recent focus in modeling tumor growth with diffuse interface models, due to their ability to capture topological transitions and the nature of the equations allows for further mathematical treatment. In the first part of this talk I will introduce a class of Cahn-Hilliard systems that are used to capture the basic dynamics of tumor growth. Then, we will discuss an optimal control problem for chemotherapy, which is a cancer treatment using drugs to eliminate tumor cells. Treatments are usually conducted in cycles, and long treatment times may cause harm to the patient. Thus, it is important to optimize both the treatment time and drug dosage to minimize patient suffering. In the second part of this talk, we will analyze an optimal control problem with an objective functional depending on a free time variable, which represents the unknown treatment time to be optimized.

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Tuesday, Dec 13, 10:15 a.m. (WIAS-ESH)
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Amal Alphonse(WIAS)

Existence for a fractional porous medium equation on an evolving surfaceIn this talk, which is based on joint work with Prof. Charlie Elliott, I will present an existence theory for a porous medium equation with a fractional diffusion on an evolving surface. The nonlocal nature of the fractional diffusion (which in our case is the square root of the Laplacian) in combination with the nonlinearity and the moving domain makes the problem interesting. After defining the fractional Laplacian and giving a Dirichlet-to-Neumann characterisation of it in a general setting of closed Riemannian manifolds, I will define what we mean by a weak solution and then proceed with the proof of existence. This will involve harmonic extensions on semi-infinite and truncated cylinders, convergence/decay estimates and some technical results in order to deal with the time-evolving surface. I will finish by discussing some ideas for further work.

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Tuesday, Dec 6, 10:15 a.m. (WIAS-ESH)
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Luca Calatroni(Ecole Polytechnique, Paris)

Infimal convolution of data discrepancies for mixed noise removal in imagesIn several real-world imaging applications such as microscopy, astronomy and medical imaging, transmission and/or acquisition faults result in a combination of multiple noise statistics in the observed image. Variational data discrepancy models designed to deal with such mixtures linearly combine standard data fidelities used for single-noise removal or make use of either approximated and cheap or exact but computationally expensive log-likelihood functionals. Via a joint MAP estimation, we derive a statistically consistent variational model combining data fidelities associated to single noise distributions in a handy infimal convolution fashion by which individual noise components corrupting the data are modeled appropriately and decomposed from each other. After showing the well-posedness of the model in suitable function spaces, we propose a semi-smooth Newton-type scheme to compute its numerical solution efficiently.

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Tuesday, Nov 29, 10:15 a.m. (WIAS, HVP11A 4.13)
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Martin Kanitsar(University of Graz)

Numerical shape optimization for an industrial applicationFor industrial applications in fluid dynamics, finding an optimal shape with respect to some cost functional is important. Moreover, constraints due to inflow and outflow boundaries as well as a respecified construction space need to be taken care of. For the implementation of a gradient descent scheme, the shape sensitivity calculus is used to derive the shape gradient of the cost functional with respect to changes in the shape. The underlying physics are described by the stationary Navier-Stokes equation, the primal equation, and an adjoint equation is used for calculating the shape derivative. Highlighting some details on the numerical realization for a 3D application is the main part of this talk, but results on existence of an optimal shape will be pointed out too. In addition to illustrating interests of the industry, hints are given for appropriate usage of the software OpenFOAM and Star-CD CCM+.

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Tuesday, Nov 22, 10:15 a.m. (WIAS-ESH)
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Soheil Hajian(WIAS/HU Berlin)

Total variation diminishing RK methods for the optimal control of conservation lawsOptimal control problems subject to conservation laws are among challenging optimization problems due to difficulties that arise when shocks are present in the solution. Such optimization problems have application, for instance, in gas networks. Beside theoretical difficulties at the continuous level, discretization of such optimal control problems should be done with care in order to guarantee convergence. In this talk, we present stability results of the total variation diminishing (TVD) Runge-Kutta (RK) methods for such optimal control problems. In particular we show that enforcing strong stability preserving (SSP) for both forward and adjoint problem results to a first order time-discretization. However, requiring SSP only for forward problem is sufficient to obtain a stable discrete adjoint. We also present order-conditions for the TVD-RK methods in the optimal control context.

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Tuesday, Nov 15, 10:15 a.m. (WIAS-ESH)
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5M. Hintermüller(WIAS/HU Berlin)

Bilevel optimization with applications to image processing

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Friday, Oct 21, 3:00 p.m (WIAS-ESH)
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M. Holler(Karl-Franzens-Universität Graz, Austria)

Higher order regularization and applications to medical image processing and data decompressionVariational methods are a powerful tool for tackling ill-posed problems in image processing. As such, they rely heavily on appropriate regularization terms which render a stable recovery possible and strongly influence qualitative solution properties. In this talk, we consider regularization concepts for both static and dynamic data that are based on higher order differentiation. Beginning with the static setting, we first discuss analytical properties of Total Generalized Variation (TGV) regularization which allow for well-posedness results for standard inverse problems. We then consider the application of TGV in the context of a variational model for image decompression, being in particular applicable to JPEG or JPEG 2000 compressed images. As second application, we introduce a nuclear-norm-based vectorial TGV functional for joint MR-PET reconstruction that exploits structural similarities between the two modalities. Moving to the dynamic setting, we motivate and introduce a suitable extension of derivative based regularization for spatio-temporal data. After establishing essential analytical properties, we deal with applications to the reconstruction of highly subsampled dynamic MR data and the decompression of MPEG compressed movies.

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Wednesday, Oct 12, 2:00 p.m. (WIAS-HVP11A 4.01)
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C. Geldhauser(WIAS)

Scaling limits of interacting diffusionsAbstract: In this talk we consider a system of N coupled stochastic differential equations, which we interpret as a system of N particles evolving according to the dynamics given by the SDEs. Due to the properties of the driving force and the noise, the limit as N goes to infinity does not lead in general to a well-posed equation. We develop conditions on the interaction strength between the particles to ensure existence of solutions to the limiting stochastic PDE. Moreover, we investigate the long-time behaviour of the solution. This is joint work with Anton Bovier.

Carina Geldhauser studied Mathematics, Protestant Theology and Philosophy in Tübingen, Pisa and Paris. She received her Ph.D. from the Institute for Applied Mathematics Universität Bonn. Since September 2016, she is a Postdoc at the Weierstrass Institute.