## Hyperbolic equations on networks

Modelling and simulation of flows on networks, like traffic flow on highways, gas flow in pipeline systems or biological networks, is investigated. Detailed models based on hyperbolic partial differential equations describing the dynamics on single arcs of the network (roads, pipelines, etc.) have been developed and improved. In particular, the derivation of accurate, microscopically based, coupling conditions is a major focus of research in this field.

**Key Publications: **

- R. Borsche, A. Klar,
**Kinetic layers and coupling conditions for macroscopic equations on networks**, SIAM Scientific Computing, 40, 3, 2018

- R. Borsche, A. Klar,
**Kinetic layers and coupling conditions for nonlinear scalar equations on networks**, Nonlinearity 31 (7), 3512-3541, 2018

- R. Borsche, A. Klar, T.N.H. Pham,
**Nonlinear flux-limited models for chemotaxis on networks**, NHM 12 (3), 381-401, 2017

## Mesh-free particle methods and computational kinetic theory

Research concentrates on the development of mesh-free Lagrangian particle methods for fluid dynamic type equations and on particle methods for kinetic transport equations like the Boltzmann equation. Topics are, for example, asymptotic-induced particle methods for interacting particle systems and mean-field equations or the numerical treatment of moving boundary problems in kinetic theory. The methods are applied in the simulation of nano-technological, socioeconomic and fluid dynamical problems.

### Key publications

- Klar, S. Tiwari,
SIAM Multiscale Mod. Sim. 12 (3), 2014**A multi-scale meshfree particle method for macroscopic mean field interacting particle models,** - Klar, S. Tiwari,
**A multi-scale particle method for mean field equations: the general case** - N.K. Mahato, A. Klar, S. Tiwari,
, Appl. Math. Modeling 53, 447-461, 2018**Particle methods for multi-group pedestrian flow** - T. Baier, S. Tiwari, S. Shresta, A. Klar, S. Hardt,
, to appear in Phys. Rev. Fluids**T****hermophoresis of Janus particles at large Knudsen numbers** - S. Shresta, S. Tiwari, A. Klar, S. Hardt,
, J. Comp. Phys. 292, 239-252, 2015**Numerical simulation of a moving rigid body in rarefied gas**

## High order numerical methods

**Raul Borsche, Florian Schneider**

This research aims at finding high order numerical methods for (partial) differential equations. In general, high order methods allow a very efficient solution of smooth problems. Even though the computational effort given a fixed grid is usually increasing with the approximation order, the effort to achieve a given error tolerance with a lower order scheme on a finer grid is typically much higher.

We investigate different types of high order schemes:

Finite volume schemes using reconstruction techniques (e.g. WENO reconstruction)

Evolution methods (e.g. Discontinuous Galerkin method)

More information and Matlab scripts can be found in the corresponding subsections.

## ProMT

**René Pinnau, Axel Klar, Matthias Andres, Laura Müller**

In the proMT project, supported by the Federal Ministry of Education and Research (BMBF), researchers from ** Technische Universität Kaiserslautern, Universität Trier, Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM), Klinikum der Johann Wolfgang Goethe Universität** and

*collaborate to improve the laser-induced interstitial thermotherapy, which is a medical treatment to destroy liver tumors by thermal ablation based on laser irradiation. The goal is to develop a realistic real-time capable simulation which shall support the practitioner online in planning the therapy. The heat transfer inside the liver can be described by a PDE system consisting of the so-called bio-heat equation and a radiative transfer model. From a mathematical point of view there are several challenging questions regarding a realistic simulation:*

**Siemens Healthcare GmbH**- A suitable model for the radiative transfer has to be chosen, describing the propagation of the laser light in the tissue.
- The mathematical model for the radiative heat transfer depends on various unknown and sometimes patient specific parameters which have to be identified using inverse methods.
- In order to make the simulation applicable for online therapy planning it is necessary to investigate reduced order models to speed up the simulation.
- The simulation has to be validated based on data from a real therapy.

## Optimisation with PDE constraints

**René Pinnau, Claudia Totzeck**

We are interested in optimising structures and processes which can be described by partial differential equations (PDE). Applications are semiconductor design, cooling of glass, cake filtration or controlling a herd of sheep using dogs. The same application can often be approached from different view points, for example the particle description and the corresponding probabilistic description using measures. This arises the question if optimal controls converge in the limit of one to the other setting. In the case of particle to mean-field this would mean the limit “number of particles to infinity”.

## Asymptotic and stochastic methods for fibre dynamics

This research aims at establishing hierarchies of mathematical models for the numerical simulation of the production process of technical textiles. The models range from highly complex three-dimensional fluid-solid interactions to one-dimensional fibre dynamics with stochastic aerodynamic drag and further to efficiently dealable stochastic surrogate models for fibre lay-down. They are theoretically and numerically analyzed and coupled via asymptotic analysis, similiarity estimates and parameter identification. The model hierarchy is applicable to a wide range of industrially relevant production processes and enables the optimization, control and design of technical textiles.

**Key publications:**

• L. Kreusser, A. Klar, O. Tse, Trend to equilibrium for a delay mean field equation, SIAM Math. Anal. 49 (4), 3277-3298, 2017

• R. Borsche, A. Klar,C. Nessler, A. Roth, O. Tse, A retarded mean field approach for interacting fiber structures, SIAM Multi-scale Mod. Simul. 15 (3), 1130-1154, 2017

• S. Gramsch, A. Klar, G. Leugering, N. Marheineke, C. Nessler, C. Strohmeyer, R. Wegener. Aerodynamic web forming: Process simulation and material properties. Journal of Mathematics in Industry, 6:13, 2016.

• A. Roth, A. Klar, B. Simeon, E. Zharovsky, A semi-Lagrangian finite volume method for a 3-D Fokker-Planck equations associated to stochastic dynamical systems on the sphere, J. Scientific Comp., 61 (3), 513-532, 2014

## Consensus-Based Global Optimisation

**René Pinnau, Claudia Totzeck**

We developed a stochastic particle scheme which aims to minimise a given objective function. The scheme is based on swarm intelligence and opinion formation. In fact, particles are randomly set into the state space of the function. Then they explore the state space and discuss in order to find a consensus. The goal is that their consensus is located near the global minimum of the objective function. Currently, we develop variants of the scheme which include the handling of state constraints and other structures. Further, we want to test the performance of the scheme in non-academic problems.

### Key Publications

- R. Pinnau, C. Totzeck, O. Tse, S. Martin.
**A consensus-based model for global optimization and its mean-field limit**M3AS 27, (1), 2017. - J. A. Carrillo, Y.-P. Choi, C.Totzeck, O. Tse.
**An analytical framework for consensus-based global optimization method**M3AS 28, (06), pp. 1037-1066 (2018).

## Kinetic equations and moment methods for cell motion in fiber structures

**Gregor Corbin, Axel Klar, Florian Schneider **

This research aims at describing cell motion in tissues via kinetic and derived moment approximations. Starting from kinetic transport equations for the evolution of the cell density function under velocity reorientations, higher order macroscopic models for cell motion including equations for the mass density and momentum of the population of cells migrating through tissue are derived. The main tool to derive such equations are higher order moment closure methods.

**Key publications:**

• G. Corbin, A. Hunt, A. Klar, F. Schneider, C. Surulescu, Higher-order models for glioma invasion: from a two-scale description to effective equations for mass density and momentum, M3AS 28 (9), 1771-1800, 2018

• R. Borsche, A. Klar, F. Schneider, Kinetic and moment models for cell motion in fiber structures, to appear in Active Particles Volume 2, Theory, Models, and Applications, N. Bellomo, P. Degond and E. Tadmor Eds., MSSET series, Springer, 2018.

• R. Borsche, A. Klar, T.N.H. Pham, Kinetic and related macroscopic models for chemotaxis on networks, M3AS 26 (6), 1219-1242, 2016