Um Spielkarten zu mischen gibt es unterschiedliche Techniken, die sich sowohl in ihrem Zeitaufwand, als auch in der Güte der Durchmischung unterscheiden. Der folgende Artikel vermittelt, wie man die Frage nach einer besonders guten Mischtechnik nutzen kann, um mathematische Modellierung anhand einer alltagsnahen Fragestellung in den Unterricht einzubinden. Dabei können verschiedene Aspekte der Stochastik angesprochen werden, und es bietet sich ein breites Potential, auf unterschiedlichen Niveaus Computer zum Generieren von Zufallsexperimenten zu verwenden.
Die Planung von Bushaltestellen in Innenstädten ist ein authentisches Thema, welches sich für den Einsatz in einem realitätsbezogenen Unterricht in unterschiedlichen Klassenstufen eignet. Verschiedene Interessen und Gegebenheiten müssen in einem Modell und in einer Lösungsstrategie vereint werden. Durch eine sehr offen gewählte Fragestellung sind verschiedene Ansätze und Modelle möglich. Somit wird mathematisches Modellieren trainiert und das Durchlaufen eines Modellierungsprozesses in einem interessanten Projekt ermöglicht. Die mathematischen Hintergründe sowie das vielseitige Lösungsspektrum von Schülerinnen und Schülern unterschiedlicher Jahrgangsstufen zu derselben Fragestellung werden im Folgenden vorgestellt.
Die Autoren befassen sich mit der Ableitung und Bearbeitung eines Modellierungsprojektes aus der populären Sportart Fußball: Ein Freistoß wird unter Beachtung der gegebenen physikalischen Effekte mathematisch modelliert und simuliert. Der Fokus liegt auf der möglichen Durchführung dieses Modellierungsprojekts mit Schülerinnen und Schülern der Sekundarstufe II.
We investigate the long-term behaviour of diffusions on the non-negative real numbers under killing at some random time. Killing can occur at zero as well as in the interior of the state space. The diffusion follows a stochastic differential equation driven by a Brownian motion. The diffusions we are working with will almost surely be killed. In large parts of this thesis we only assume the drift coefficient to be continuous. Further, we suppose that zero is regular and that infinity is natural. We condition the diffusion on survival up to time t and let t tend to infinity looking for a limiting behaviour.
For the prediction of digging forces from a granular material simulation, the
Nonsmooth Contact Dynamics Method is examined. First, the equations of motion
for nonsmooth mechanical systems are laid out. They are a differential
variational inequality that has the same structure as classical discrete algebraic equations. Using a Galerkin projection in time, it becomes possible to derive
nonsmooth versions of the classical SHAK and RATTLE integrators.
A matrix-free Interior Point Method is used for the complementarity
problems that need to be solved in every time step. It is shown that this method
outperforms the Projected Gauss-Jacobi method by several orders of magnitude
and produces the same digging force result as the Discrete Element Method in comparable computing time.
We consider the problem to evacuate several regions due to river flooding, where sufficient time is given to plan ahead. To ensure a smooth evacuation procedure, our model includes the decision which regions to assign to which shelter, and when evacuation orders should be issued, such that roads do not become congested.
Due to uncertainty in weather forecast, several possible scenarios are simultaneously considered in a robust optimization framework. To solve the resulting integer program, we apply a Tabu search algorithm based on decomposing the problem into better tractable subproblems. Computational experiments on random instances and an instance based on Kulmbach, Germany, data show considerable improvement compared to an MIP solver provided with a strong starting solution.
We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker.
We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding.
We investigate a PDE-ODE system describing cancer cell invasion in a tissue network. The model is an extension of the multiscale setting in [28,40], by considering two subpopulations of tumor cells interacting mutually and with the surrounding tissue. According to the go-or-grow hypothesis, these subpopulations consist of moving and proliferating cells, respectively. The mathematical setting also accommodates the effects of some therapy approaches. We prove the global existence of weak solutions to this model and perform numerical simulations to illustrate its behavior for different therapy strategies.
In a financial market we consider three types of investors trading with a finite
time horizon with access to a bank account as well as multliple stocks: the
fully informed investor, the partially informed investor whose only source of
information are the stock prices and an investor who does not use this infor-
mation. The drift is modeled either as following linear Gaussian dynamics
or as being a continuous time Markov chain with finite state space. The
optimization problem is to maximize expected utility of terminal wealth.
The case of partial information is based on the use of filtering techniques.
Conditions to ensure boundedness of the expected value of the filters are
developed, in the Markov case also for positivity. For the Markov modulated
drift, boundedness of the expected value of the filter relates strongly to port-
folio optimization: effects are studied and quantified. The derivation of an
equivalent, less dimensional market is presented next. It is a type of Mutual
Fund Theorem that is shown here.
Gains and losses eminating from the use of filtering are then discussed in
detail for different market parameters: For infrequent trading we find that
both filters need to comply with the boundedness conditions to be an advan-
tage for the investor. Losses are minimal in case the filters are advantageous.
At an increasing number of stocks, again boundedness conditions need to be
met. Losses in this case depend strongly on the added stocks. The relation
of boundedness and portfolio optimization in the Markov model leads here to
increasing losses for the investor if the boundedness condition is to hold for
all numbers of stocks. In the Markov case, the losses for different numbers
of states are negligible in case more states are assumed then were originally
present. Assuming less states leads to high losses. Again for the Markov
model, a simplification of the complex optimal trading strategy for power
utility in the partial information setting is shown to cause only minor losses.
If the market parameters are such that shortselling and borrowing constraints
are in effect, these constraints may lead to big losses depending on how much
effect the constraints have. They can though also be an advantage for the
investor in case the expected value of the filters does not meet the conditions
All results are implemented and illustrated with the corresponding numerical
In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first–discretize then setting. For the discretization we use
isogeometric analysis (iga) to solve the state equation, and search optimal designs in a space of admissible b-spline or nurbs combinations. Thus a quite
general class of functions for representing optimal shapes is available. For the
gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between iga and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.