## Student Projects

Several projects for students are available. These range from assistant (HiWi) jobs (graphics, animations, data structures and interface programming) to undergraduate research through UROP, internships, and projects to theses.

Possible fields are any of the research projects, or in a subject of your own choice provided that it has sufficient mathematical content.

### Open Projects

Development of a multipeak material analysis method (HiWi job)
Please check the detailed job description.

Implementation of a cloud application (Bachelor thesis) → Prof. Martin Frank
Solar towers use many mirrors to concentrate sun light on a central, tower-mounted receiver. The receiver then transfers the resulting heat to a fluid (i.e. molten salt or air) that, in turn, exchanges the heat to steam which powers a turbine, generating electricity. The placement of the mirrors may lead to individual mirrors being blocked and shaded; this affects the efficiency (and therefo- re costs) of the power plant. The model is later used for an optimisation process which finds the most efficient arrangement of mirrors. In this thesis a web application has to be implemented. Detailed information on this project can be found here.
Contact: Pascal Richter

Optimisation of solar tower power plant (Master thesis)
The exploitation of solar power for energy production is of increasing importance. Solar towers use many flat mirrors to concentrate sun light on the absorber, which is mounted on a tower. This absorber warms a medium (i.e. water) that powers a turbine which in turn generates electricity. The mirrors' placement may lead to individual mirrors being blocked and shaded; this affects the efficiency (and therefore costs) of the power plant. The goal is to find the most efficient arrangement of mirrors that balances power production against construction costs. Detailed information on this project can be found here.
Contact: Pascal Richter

Direct steam simulation of solar thermal power plants (Bachelor or Master thesis)
In this project we consider the flow of water inside an absorber tube. This two-phase flow is governed by the 1D Euler equations which are coupled with a heat equation modelling the solar irradiance. Possible topics for theses include implementation of various numerical schemes.
Contact: Pascal Richter

Simplifying kinetic simulations by optimization methods (Project Work)
Kinetic equations have a vast number of applications—including gas dynamics, neutron transport, supernovas, semiconductors—but are hard to simulate numerically. One way to simplify their simulation, a family of methods known as entropy-based moment closures, involves repeatedly solving low-dimensional optimization problems. Even though these problems are low-dimensional, convex, and unconstrained, some can cause standard optimization algorithms (like Newton's method) to fail in numerical implementations.
We would like to apply a new state-of-the-art optimization algorithm and see if it gives improved performance on our most challenging moment-closure problems. This new algorithm, the cubic overestimation method, has shown promising performance in many cases, and it would be helpful to see if it can help tackle the unique challenges presented by our toughest problems.
Contact: Graham Alldredge

Simulation of Tumor Oxygenation (Project work)
Knowledge of the oxygen content of tumor tissue will in the future become essential for cancer treatment by radiation. The oxygen enters the tumor tissue through the blood vessels, which are very unevenly distributed. The aim of this project is to efficiently solve a nonlinear Poisson-like equation for the oxygen content on a very complex domain (tumor without the blood vessels). In previous projects, a finite difference code has been developed which can be extended in several ways: parallelization, increased order, online computation on a visualization cluster, inverse problem, etc.
Contact: Martin Frank

Accelerating Lattice Boltzmann Methods using nVidia Kepler GPUs or Intel Xeon Phi Coprocessors (Bachelor Thesis)
The Lattice Boltzmann Method (LBM) is becoming more and more popular. Up to this day most parallel LBM solvers are implemented using distributed memory parallelism only. Due to the rise of multi-core CPUs and accelerators, hybrid parallelism is of increasing importance. In a first step, the code shall be modified such that it uses hybrid parallelism (MPI and OpenMP). Later, the code shall be extended to support accelerators. MathCCES hosts two accelerator-based system: a 16 core, dual nVidia Kepler GPU system; and a 16 core, dual Intel Xeon Phi coprocessor system.
Detailed information on this project can be found here.
Contact: Philipp Otte

Mean-Field Models of Financial Markets (Thesis, Seminar or Project)
The application of kinetic theory to financial markets is rather new and started in the last decade. Many financial market models in economy are designed as systems of multi-agent interactions. The goal of these market models is to give explanations of the appearance of so called stylized facts. Computer simulations of econophysical models have shown that irrational behavior of financial agents might be the reason for stylized facts. Starting from microscopic agent-based models one can derive mesoscopic/kinetic models in which not the actions of the agents themselves are studied but the statistical probabilities of agents behaving in certain ways. These kinetic models can be studied by analyzing PDEs.
Detailed information on this project can be found here.
Contact: Torsten Trimborn

Asynchronous finite-difference schemes for PDEs (Bachelor/Master thesis)
One of the main challenges for grid-based discretization schemes on massively parallel computer architectures is that processors have to wait in order to synchronize, for example when they exchange information over neighboring grid points. This leads to a loss of efficiency. Current research investigates the behavior of numerical schemes when no synchronization is enforced, i.e. the algorithm keeps computing even if necessary information is not yet available. The message arrivals are then modeled as random variables, which brings stochasticity into the analysis.
In this thesis, based on the paper (Donzis & Aditya, 2014), standard finite-difference schemes for the heat and advection equation are analyzed, and numerical experiments are performed. The hypothesis is that average errors drop to first-order accuracy, independent of the order of the scheme. The project is an ideal continuation of Mathematics IV (CES) or Numerical Analysis IV (Mathematics).
Detailed information on this project can be found here.
Contact: Martin Frank

Unified Gas-Kinetic Schemes
The so-called unified gas-kinetic schemes (UGKS) have been a successful discretization technique for the Boltzmann equation of gas dynamics, and its fluid-dynamic limits, especially the Navier-Stokes equations. Recently, the methodology has been applied to radiation transport by Mieussens. The paper also contains a careful analysis. Especially, the scheme was shown to be asymptotic-preserving, i.e. it becomes a consistent and stable approximation in the diffusive limit. The description and the numerical results so far exist only in one dimension.
The project is an ideal continuation of the course Introduction to Transport Theory. Detailed information on this project can be found here.
Contact: Martin Frank

Entropy Viscosity Schemes
Solving systems of nonlinear hyperbolic equations is very challenging, because the solution can develop discontinuities (shocks) in finite time. This requires a stabilization of the numerical method, which is usually achieved by adding a sufficient amount of numerical viscosity. This, however, becomes difficult for higher-order schemes, which rely on an engineering fix (limiters). Recently, an alternative methodology has been proposed. Starting from any scheme (even a naive unstable scheme), one adds sufficient viscosity so as to satisfy the entropy dissipation inequality at shocks. The task in this project is to implement, test, and possibly to extend this methodology. Detailed information on this project can be found here.
Contact: Martin Frank

PDE-Constrained Optimization under Uncertainties
In most applications, one does not only ask for a simulation, but for some kind of decision or design based on a simulation, e.g. optimal shapes of airplane wings or the temperature control of a melting process. This can be mathematically formulated as a constrained optimization problem. Such constrai- ned optimization problems (especially if the model is described by a partial differential equation) are a subject of current research. Many times, however, some parameters or input data of the model are not exactly known (e.g. uncertain flow profile or uncertain heat conductivity). It is therefore of interest to study the influence of these uncertain parameters on the optimization. The toolset to study the pro- pagation of uncertainties through models is provided by the field of Uncertainty Quantification. The uncertain parameters are viewed as random variables, and are parametrized. The parametrized system is solved with methods that are similar to PDE numerics (stochastic Galerkin, stochastic collocation). Combining these techniques provides a way to answer the question as to how sensitive the optimization is to uncertainties, by constraining the optimization functional by a stochastic Galerkin or collocation system. This approach has been followed by Tiesler et al.
The project is an ideal continuation of the course Uncertainty Quantification. A background in PDE-constrained optimization is required.
Detailed information on this project can be found here. Contact: Martin Frank

Zero-Variance Monte Carlo Schemes
Monte-Carlo (MC) schemes are used for many applications, mostly to compute high-dimensional integrals. Applications include Uncertainty Quantification, and particle transport (where the integral equation formulation of the Boltzmann transport equation is solved). The convergence rate of MC schemes comes from the central limit theorem, and is thus fixed at 1. However, MC schemes can 2 be accelerated in practice by variance reduction. This is essentially done by biasing the random experiment toward the desired quantity of interest. In particle transport, this is achieved by so-called weight windows, and the subsequent generation and destruction of MC particles. The choice of these weight windows is often done by hand. However, it can be shown that the weight windows can be extracted from the exact adjoint transport solution. Knowing the adjoint solution, one can construct an exact MC scheme for the forward problem that uses just one particle/random experiment. This is a so-called zero-variance scheme. However, computing the exact adjoint solution is as difficult as solving the exact forward problem. There may be an optimal combination, though, of an inexact (grid-based) adjoint solution, and the corresponding weighted MC particle game.
The project is an ideal continuation of the courses Uncertainty Quantification or Introduction to Transport Theory. Detailed information on this project can be found here. Contact: Martin Frank

### Completed Projects

• Modeling and simulation of offshore wind farms, Gregor Heiming, CES Bachelor, 2015
• Didaktisch-methodische Ausarbeitung eines Lernmoduls zum Thema GPS, Markus Wiener, Math Education, 2015
• Didaktisch-methodische Ausarbeitung eines Lernmoduls zum Thema Google, Sarah Schönbrodt, Math Education Bachelor, 2015
• Theory and application of numerical methods for fractional diffusion equations, Thomas Camminady, CES Master, 2015
• Mathematical Modeling and numerical methods for non-classical transport in correlated media, Kai Krycki, Mathematics PhD
• Analyse der Luftkurzschlussgefahr bei Luftbehandlungsgeräten, Atilla Sezen, CES Master, 2015
• Agent-based economic market models, Allison Betley, UROP International, 2015
• New methods for radiotherapy dose calculation, Louis Graup, UROP International, 2015
• Fast iterative methods for non-classical particle transport, Andreas Schmitt, CES Master, 2015
• Numerical investigations of melting phenomena of AdBlue in vehicle tank, Akash Pakanati, Simulation Sciences Master, 2015
• Entwicklung eines kompetenzorientierten Anforderungsprofils für die Teilnahme an der Modellierungswoche CAMMP week, Maren Hattebuhr, Math Education, 2014
• Implementation of a Finite Difference Solver for the Fokker-Planck equation, Hamza Mesbahi, Internship, 2014
• Performance Analysis of the Lattice Boltzmann Solver Musubi on the Intel Xeon Phi Accelerator, Bilal Himafi, Internship, 2014
• Application of the Pn method on staggered grids to the Boltzmann continuous slowing-down equation, CES Bachelor, 2014
• A multi-dimensional PN-staggered-grid radiative transfer solver for photon transport in atmospheric clouds, Martin Beeger, Gregor Corbin, Philipp Menke, Computational Engineering Science Project, 2013
• Kinetic modeling of a heterogeneous financial market, Torsten Trimborn, Mathematics Master, 2013
• Image processing techniques applied to reconstruction algorithms for neutron imaging, Nikhil Bhandari, Internship, 2013
• Simulation of Tumor Oxygenation, Sinan Gezgin, Jonas Lange, Kai Meschede, Computational Engineering Science Project, 2012
• Moment realizability and Kershaw closures in Radiative Transfer, Philipp Monreal, Mathematics PhD, 2012
• Modeling and Optimization of Solar Tower Power Plants , Michelle Hauer (University of Alberta), UROP International, 2012
• Computing Reachable Sets for Higher Dimensional Control Systems, Allison Betley (Washington University in St. Louis), UROP International, 2012
• Modelling and simulation of a solar tower power plant, Matthias Ewert, Omnieldis Navarro Fuentes, Computer Science Project, 2012
• Models and Numerical Methods for Time- and Energy-Dependent Particle Transport, Edgar Olbrant, Mathematics PhD, 2012
• Solving a non-linear partial differential equation for the simulation of tumour oxygenation: second order accuracy, Fouad Benhaida (ENSEIRB MATMECA Bordeaux), Internship, 2011
• Optimal Police Patrols, Andre Sutanto (Carnegie Mellon University), UROP International, 2011
• Solving a Nonlinear Partial Differential Equation for the Simulation of Tumour Oxygenation, Julian Köllermeier, Lisa Kusch, Thorsten Lajewski, Computational Engineering Science Project, 2011. The results were also published as a paper in SIAM Undergraduate Research Online 5 (2012).
• A Radiative Transfer Code for Global Climate Models, Loïc Viennois (ENSEIRB MATMECA Bordeaux), Internship, 2010