## 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, 13 c.t. 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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Wednesday, Mar 29, 13 c.t. (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.

#### Previous Dates

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Tuesday, Feb 7, 10 c.t. (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 c.t. (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 c.t. (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 c.t. (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 c.t. (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 c.t. (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 c.t. (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 c.t. (WIAS-ESH)
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M. Hintermüller(WIAS/HU Berlin)

Bilevel optimization with applications to image processing

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Friday, Oct 21, 15:00 Uhr (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, 14:00 Uhr (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.