Seminar Scientific Computing

The application of sampling techniques is widespread in the field of uncertainty quantification. In some cases, e.g. when not much is known about the relation between input and output, it makes sense to generate samples which fill the sample-space in a homogeneous manner. A feature of technical interest is that the generated samples are not collapsing when they are projected onto lower dimensions. In this seminar topic techniques shall be presented which generate space-filling and non-collapsing sample-points.

References:

• Markov chain importance sampling with applications to rare event probability estimation, Zdravko I. Botev, Pierre L'Ecuyer, Bruno Tuffin.
• Markov chain Monte Carlo for rare-event simulation in heavy-tailed settings, Th. Gudmundsson.
• Monte Carlo Methods, Adam M. Johansen and Ludger Evers, 2007.

Data mining and uncertainty quantification both generate high-dimensional data sets that are usually too large to store in a classical manner. For this task tensor based representations have been developed that try to make this manageable. In addition to the rather classic formats, i.e. the canonical and the Tucker format, new schemes like e.g. the tensor train and the hierarchical Tucker format are under active development. In this topic either an overview over recent developments or a an in-depth discussion of a selected tensor format shall be given.

The solution of ODEs, DAEs and time-dependent PDEs needs a good discretisation method to obtain a stable numerical method. But it is well-known that onestep methods have order reduction if they are applied on stiff ODEs like the example of Prothero--Robinson. Possible topics might be:

• Time adaptivity for Hamiltonian systems
• General linear methods
• Implicit-explicit (IMEX) methods
• Rosenbrock--Wanner W-methods for semilinear parabolic equations
• Parallel time discretisation methods
• Adaptive timestep control

Literature:

• Ascher, Ruuth and Spiteri: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, pp. 151-167, 1997.
• Butcher: General linear methods. Acta Numerica 15, pp. 157-256, 2006
• Calvo and Hairer: Accurate long-term integration of dynamical systems. Appl. Numer. Math. 18, pp. 95-105, 1995
• Krogh: Stepsize selection for ordinary differential equations. ACM Trans. Math. Softw. 37, pp. 15:1--15:11, 2010.
• Schwitzer: W-methods for semilinear parabolic equations. Appl. Numer. Math. 18, pp. 351-366, 1995.
• Voss: Fourth-order parallel Rosenbrock formulae for stiff systems. Math. Comput. Modelling 40, pp. 1193-1198, 2004.

Often multifield problem like fluid-structure interaction (FSI) are solved with a partitioned approach, i.e. the subproblems are discretised by different methods and solved with different codes. Possible topics might be:

• Multirate methods
• Adaptive partitioning techniques for ODEs

Literature:

• Skelboe: Stability properties of backward differentiation multirate formulas. Appl. Numer. Math. 5, pp. 151-160, 1989.
• Skelboe: Adaptive partitioning techniques for ordinary differential equations. BIT 46, pp. 617-629, 2006.