AMH5 Multi-scale Methods with Applications

This page relates to the Applied Mathematics Honours course "Multi-scale Methods with Applications".

The lecturer for this course is Georg Gottwald.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

Most physical and biological systems are too complex to allow for an analytical treatment and very often even for a numerical treatment. This is due to the high dimensionality of the underlying systems and the occurrence of wide ranges of spatial and temporal scales. However, often one is not interested in the description of the full system but rather in some "interesting" subspace. For example, in numerical weather forecasting one is interested in large scale phenomena such as high and low pressure fields but not in all the small-scale effects such as turbulent wind gusts in your back garden. In the case of time-scale separation and weak coupling, one can try to find the dynamics on a reduced subspace which describes only the dynamics of the "interesting" dynamics. But for this we need to know how to model the accumulative effect of all these "uninteresting" scales.

This unit looks at generic universal methods to distill the "interesting" dynamics from a highly complex system. The unit will present deterministic methods of reduction as well as stochastic ones.

• You will learn how and why scale separation and weak coupling allow for dimension reduction and an effective reduced dynamics.

• You will learn several perturbation techniques which you can apply to physical, chemical, biological and engineering type applications.

• You will learn how certain high-dimensional deterministic and stochastic systems can be effectively reduced to low-dimensional stochastic dynamical systems.

• You will be exposed to objects from stochastic dynamics and will understand through a reformulation of deterministic dynamics, called Mori-Zwanzig formalism, how stochastic terms can be used to parametrize unresolved scales.