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.

The design of new efficient structures and materials requires to use special approaches and techniques. This work is focused on topology optimisation of materials that can lead to the so-called metamaterials matter that exhibit some extraordinary properties: negative Poisson ratio, suppression of electromagnetic or acoustic waves, etc. The optimisation is based on finding proper balance between a construction price and material properties such as weight or stiffness.

• Sigmund, O., & Maute, K. (2013). Topology optimization approaches: A comparative review. Structural and Multidisciplinary Optimization, 48(6), 1031-1055. doi.org/10.1007/s00158-013-0978-6

Material models undergo softening behaviour such as fracture or damage as a result of extreme events. At the initial loading stage, typical material respond with the ability to carry the external loads. At the later stage, the material looses the integrity accompanied by occurrence of e.g. microcracks or microvoids. This behaviour that is characterised by lower ability to carry loads is called softening behaviour. A mathematical modelling and numerical approximation is very interesting from the engineering point of view because it can describe a character and behaviour of structures during failure.

• Farrell, P. E., & Maurini, C. (2015). Linear and nonlinear solvers for variational phase-field models of brittle fracture. Numerical Analysis. Retrieved from arxiv.org/abs/1511.08463
• Marigo, J. J., Maurini, C., & Pham, K. (2016). An overview of the modelling of fracture by gradient damage models. Meccanica, 51(12), 3107-3128. doi.org/10.1007/s11012-016-0538-4

This topic focuses on a classification of (uncertain) parameters. There are many possible criteria and needs for parameter classifications. One can filter out the non-physical parameters that lead to a breakdown of a mathematical and numerical model. Another option is the classification and estimation of uncertain parameters leading to structural collapse can describe the probability of failure, which is a delicate task especially for small failure probabilities.

• Pedregosa, F., Varoquaux, G., Gramfort, A., et. al. (2011). Scikit-learn: Machine Learning in Python. Journal of Machine Learning Research, 12(Oct), 2825-2830. Retrieved from www.jmlr.org/papers/v12/pedregosa11a.html
• Gorodetsky, A., & Marzouk, Y. (2014). Efficient Localization of Discontinuities in Complex Computational Simulations. SIAM Journal on Scientific Computing, 36(6), A2584-A2610. doi.org/10.1137/140953137
• Smola, A. J., & Schölkopf, B. (2004). A tutorial on support vector regression. Statistics and Computing, 14(3), 199-222. doi.org/10.1023/B:STCO.0000035301.49549.88

The eigenvalue problems naturally occur in engineering problems when one study propagation of electromagnetic and elastic waves, or particularly stability of structures. The undergoing equations can also describe mysterious features of wave propagation such as invisibility or sound filters. Here, we will focus on simple eigenvalue problems with parameters that can be of random character or that can be a object of optimisation to deliver e.g. more stable structure.

• Sousedík, B., & Elman, H. C. (2016). Inverse subspace iteration for spectral stochastic finite element methods. SIAM/ASA Journal on Uncertainty Quantification, 4(1), 163-189. Numerical Analysis; Probability. doi.org/10.1137/140999359
• Lewis, A. (2003). The mathematics of eigenvalue optimization. Mathematical Programming, 176, 155-176. Retrieved from link.springer.com/article/10.1007/s10107-003-0441-3
• Wang, P., & Dai, H. (2016). Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse. Journal of Engineering Mathematics, 96(1), 201-210. doi.org/10.1007/s10665-015-9790-1