Second Bilateral Workshop on Applied Mathematics – Abstracts
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Abstracts for the second part (Oct 17, 2019)
This list contains a first list of abstracts, more are likely to come.

Kurt Chudej
(Chair of Scientific Computing
and
Research Center for Modeling and Simulation (MODUS),
University of Bayreuth)
Dengue fever: mathematical modelling, analysis, and numerical simulationAbstract:
Dengue fever is a disease with about 390 Mio. cases per year (WHO, 2016). It is a vector borne disease which is transmitted by Asian tiger mosquitoes. Though dengue fever is originally a tropical and subtropical disease the Asian tiger mosquitoes spread across (south) Europe. In the Mediterranean area the tiger mosquito is already established and unfortunately selfpreservative populations are already observed in Germany in Freiburg im Breisgau, Sinsheim, Heidelberg (in 2018) and Jena (2019). Adult asian tiger mosquitos were found recently in Fürth (Sept. 2019) and previously in Regensburg, Erding and between Rosenheim and Kufstein.With the spread of the asian tiger mosquito is coupled an increasing danger of an outbreak of Dengue fever. Due to the fact that a second infection can cause Dengue shock Syndrom (DSS), a severe form of the disease, we consider a new dengue fever model with a vaccination of seropositive humans. We present numerical simulations and optimal control strategies to combat the disease in order to predict possible outbreak scenarios.

Thomas Kriecherbauer
(Chair of Nonlinear Analysis and Mathematical Physics,
University of Bayreuth)
Proving Wigner's semicircle lawAbstract:
For a large class of random matrices the appropriately rescaled eigenvalues converge to a deterministic distribution – with the semicircle as the graph of its density – if the dimension of the matrices tends to infinity. In this talk we discuss two ways to prove such a result using either, like Eugene Wigner in his original proof from 1955, the moments of the spectral distribution or its Stieltjes transform. 
Michael Margaliot
(Department of Systems,
School of Electrical Engineering,
TelAviv University)
A generalization of linear positive systems with applications to nonlinear systems: invariant sets and the Poincare–Bendixon property
The dynamics of linear positive systems maps the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. This raises the following question: what linear systems map the set of vectors with k sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called kpositive linear systems, that reduces to positive systems for k = 1. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case k = 2 establish the PoincareBendixon property for certain trajectories.
This is joint work with Eyal Weiss (Tel Aviv).
Abstracts for the first part (Sep 2425, 2019)
This list contains only the abstracts of the confirmed speakers.

Robert Baier
(Chair of Applied Mathematics,
University of Bayreuth)
The numerical calculation of a directional derivative in set optimizationAbstract:
For (directional) derivatives of setvalued maps with convex, compact images a good definition of a set difference is essential. We suggest a vector space approach with generalized Steiner sets which is based on families of Steiner points of supporting faces. It differs from known approaches with equivalence classes of pairs of sets or with continuous functions.Generalized Steiner sets are parametrized by unit directions and their arithmetic operations generalize the Minkowski sum and the scalar multiplication of convex sets. Convex sets can be "interpreted" as generalized Steiner sets, their visualization – consisting of a positive, a negative and a nonconvex part – allows a geometric interpretation as a (usually) nonconvex set and has close links to known convex set differences.
The concept is applied to set optimization problems which are briefly introduced and optimality conditions are stated via the new (setvalued) directional derivative based on the difference of generalized Steiner sets.
The talk is based on a joint work with Gabriele Eichfelder and Tobias Gerlach (both: Ilmenau).

Elza Farkhi
(Department of Applied Mathematics,
TelAviv University)
Set operations, regularity of setvalued maps and stability of dynamical systemsAbstract:
Following a general framework, we introduce various notions of continuity (or other regularity) of setvalued maps using various differences of two sets. For instance, any notion of a difference of two sets, A ⊖ B, induces Lipschitz continuity of the map F as ‖ F(x) ⊖ F(y) ‖ ≤ L ‖x − y‖.In some cases, continuity (regularity) in such a sense is equivalent to the existence of a family of uniformly continuous (regular) selections passing through each point of the graph of the given setvalued map. This may lead to stability theorems and regularity properties for solutions of differential inclusions.
The talk is based on joint works with Robert Baier (Bayreuth), Tzanko Donchev (Sofia) and Simon Reich (Haifa).

Lars Grüne
(Chair of Applied Mathematics,
University of Bayreuth)
Highorder spatial approximation for stationary HamiltonJacobi equationsAbstract:
Stationary HamiltonJacobi equations are first or second order PDEs that appear in many applications, such as infinite horizon optimal control or level set methods. For the numerical solution of the fully discretized equation, several iterative methods are known, the simplest one being the so called value iteration. It is known that when using highorder spatial discretization schemes, these iterations do not converge in general. The reason is the lack of monotonicity induced by highorder spatial reconstruction schemes.In this talk we show how this problem can be circumvented using a relaxed notion of approximate monotonicity. We present a corresponding "relaxed" convergence result for the value iteration, discuss numerical schemes that satisfy our assumptions and illustrate them by numerical experiments.
The talk is based on joint work with Olivier Bokanowski (Paris), Maurizio Falcone (Rome), Roberto Ferretti (Rome), Dante Kalise (Nottingham, UK), and Hasnaa Zidani (Palaiseau Cedex, France).

Rüdiger Kempf
(Chair of Applied and Numerical Analysis,
University of Bayreuth)
A new method to reconstruct highdimensional functionsAbstract:
In applications such as machine learning and uncertainty quantification, one of the main tasks is to reconstruct an unknown function from given data with data sites lying in a high dimensional domain. This task is usually even for relatively small domain dimensions numerically difficult. We propose a new reconstruction scheme by combining the wellknown kernel multilevel technique in low dimensional domains with the anisotropic Smolyak algorithm, which allows us to derive a high dimensional interpolation scheme. This new method has significantly lower complexity than traditional highdimensional interpolation schemes. In this talk, I will give an introduction to the topics of kernel multilevel methods and anisotropic Smolyak algorithms before providing a convergence result for this new Kernel TensorProduct MultiLevel method. If time permits, I will also give numerical examples.The talk is based on joint works with Holger Wendland (Bayreuth).

Dany Leviatan
(Department of Pure Mathematics,
TelAviv University)
Visiting WhitneyAbstract (see also the pdf file):
Let x_{0} < x_{1} < ⋯ < x_{m1}, and set I := [x_{0},x_{m1}] and I := x_{m1}  x_{0}. Assume that for some 0 < λ ≤ 1, we have x_{j+1}  x_{j} ≥ λ I, for all 0 ≤ j ≤ m2. The LagrangeHermite polynomial of a function f ∈ C^{r}(I), L_{m1}(x; f; x_{0}, …, x_{m1}) is the unique polynomial of degree m1, interpolating f at the points x_{0}, …, x_{m1}. For m ≥ max{r + 1, 2}, the classical Whitney estimate of how well this polynomial approximates f in I is given by f(x)  L_{m1}(x; f; x_{0}, …, x_{m1})  ≤ C(m; λ) I^{r} ω_{mr}(f^{(r)}, I, I), x ∈ I,
where ω_{k} is that kth modulus of smoothness.
We allow some of the points to coalesce, specifically, we assume x_{0} ≤ x_{1} ≤ ⋯ ≤ x_{m1} such that we only have x_{j+r+1}  x_{j} ≥ λ I, for all 0 ≤ j ≤ mr2. In other words, we assume that the LagrangeHermite polynomial interpolates f and its derivatives at x_{j} according to the multiplicity of the appearance of x_{j} in the the collection of points (but note that no x_{j} may appear more than r+1 times). We prove the above Whitney inequality for this situation.
Further, we will discuss an extension of Whitney's inequality to a localtype estimate.

Alona Mokhov
(Mathematics Unit,
Afeka Tel Aviv Academic College of Engineering)
Representations of multifunctions by metric selections and the metric setvalued integralAbstract:
In this work we consider representations of setvalued functions (SVFs) with compact images in IR^{n} by special selections through each point of the graph. Using representations by selections, the approximation of the setvalued function is reduced to the approximation of the corresponding collection of representing singlevalued functions. Thus wellknown tools from constructive function theory can be applied directly. The main effort here is the design of an appropriate representation consisting of singlevalued functions with regularity properties "inherited" from those of the approximated SVF. The case of convexvalued multifunction with the support functions is welldeveloped. The case of SVFs with general images is less studied. We consider a representation of a multifunction by specific metric selections with low variation and with regularity properties as those of the setvalued function.We also discuss the recently introduced notion of the metric integral of setvalued functions. The metric integral is different from the classical one  the Aumann integral, and may be nonconvex. For SVFs of bounded variation the metric integral consists of integrals of the metric selections. We study some properties of the metric integral and give examples. We also introduce an extension of the metric integral to the weighted metric integral. We plan to use it in our future work.
The talk is based on joint works with Nira Dyn and Elza Farkhi (TelAviv University).

Jörg Rambau
(Chair of Mathematics in Economy,
University of Bayreuth)
Camera surveillance and the visibility functionAbstract:
Given are a room with a set of movable obstacles and configurable cameras. How should the camera parameters (positions, directions, zoom) and enviroment parameters (positions) be set so as to maximize the supervised volume in the room? In order to apply randomized local search methods to this hard optimization problem, the supervisedvolume function must be analytically wellbehaved. In this talk, oriented matroids are used to show that this function is continuous and piecewise smooth.