This course will introduce students to numerical methods for ordinary differential equations (ODEs) and their analysis. In particular, we will study in detail initial value problems (IVP) that play a big role in the time evolution of physical, biological or economical systems. Students will first be reminded and introduced to some basic properties of ODEs that will be important for their stable and efficient numerical approximation and will then learn about explicit and implicit one-step methods as well as linear multistep methods for IVPs. In the later parts of the course, we will also start discussing two-point boundary value problems (BVPs) and their discretisation via finite difference methods.
The aim of this semester is to find synergies and common interests in Heidelberg in the area of data-driven numerical simulation and inverse problems and to present to each other the core ideas of our respective research fields in this area. To make the title somewhat more precise, our key objective is to find efficient and robust ways to merge numerical and statistical methods, as well as methods from machine learning to allow for a more effective and mathematically sound estimation of problem parameters, predictions with quantified uncertainties, optimisation, modelling, etc, especially in areas where the data is sparse, highly uncertain, heterogeneous or based on indirect measurements, but where the underlying physical, biological, chemical or engineering processes are well understood and described by mathematical models.