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 simulation

  Abstract:
  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 self-preservative 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 law

  Abstract:
  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, Tel-Aviv 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 k-positive 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 Poincare-Bendixon property for certain trajectories.

  This is joint work with Eyal Weiss (Tel Aviv).

Abstracts for the first part (Sep 24-25, 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 optimization

  Abstract:
  For (directional) derivatives of set-valued 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 (set-valued) 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, Tel-Aviv University)
  Set operations, regularity of set-valued maps and stability of dynamical systems

  Abstract:
  Following a general framework, we introduce various notions of continuity (or other regularity) of set-valued 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 set-valued 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)
  High-order spatial approximation for stationary Hamilton-Jacobi equations

  Abstract:
  Stationary Hamilton-Jacobi 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 high-order spatial discretization schemes, these iterations do not converge in general. The reason is the lack of monotonicity induced by high-order 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 high-dimensional functions

  Abstract:
  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 well-known kernel multi-level 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 high-dimensional interpolation schemes. In this talk, I will give an introduction to the topics of kernel multi-level methods and anisotropic Smolyak algorithms before providing a convergence result for this new Kernel Tensor-Product Multi-Level 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, Tel-Aviv University)
  Visiting Whitney

  Abstract (see also the pdf file):
  Let x₀ < x₁ < ⋯ < x_m-1, and set I := [x₀,x_m-1] and |I| := x_m-1 - x₀. Assume that for some 0 < λ ≤ 1, we have x_j+1 - x_j ≥ λ |I|, for all 0 ≤ j ≤ m-2. The Lagrange-Hermite polynomial of a function f ∈ C^r(I), L_m-1(x; f; x₀, …, x_m-1) is the unique polynomial of degree m-1, interpolating f at the points x₀, …, x_m-1. 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_m-1(x; f; x₀, …, x_m-1) | ≤ C(m; λ) |I|^r ω_m-r(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 x₀ ≤ x₁ ≤ ⋯ ≤ x_m-1 such that we only have x_j+r+1 - x_j ≥ λ |I|, for all 0 ≤ j ≤ m-r-2. In other words, we assume that the Lagrange-Hermite 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 local-type estimate.

• Alona Mokhov (Mathematics Unit, Afeka Tel Aviv Academic College of Engineering)
  Representations of multifunctions by metric selections and the metric set-valued integral

  Abstract:
  In this work we consider representations of set-valued functions (SVFs) with compact images in IRⁿ by special selections through each point of the graph. Using representations by selections, the approximation of the set-valued function is reduced to the approximation of the corresponding collection of representing single-valued functions. Thus well-known tools from constructive function theory can be applied directly. The main effort here is the design of an appropriate representation consisting of single-valued functions with regularity properties "inherited" from those of the approximated SVF. The case of convex-valued multifunction with the support functions is well-developed. 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 set-valued function.

  We also discuss the recently introduced notion of the metric integral of set-valued functions. The metric integral is different from the classical one - the Aumann integral, and may be non-convex. 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 (Tel-Aviv University).

• Jörg Rambau (Chair of Mathematics in Economy, University of Bayreuth)
  Camera surveillance and the visibility function

  Abstract:
  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 supervised-volume function must be analytically well-behaved. In this talk, oriented matroids are used to show that this function is continuous and piecewise smooth.