Introduction to Stochastic Differential Equations and Data Assimilation

Randomness and uncertainty are inherent aspects of many processes in the environmental and life sciences. Examples are plentiful and exist at all scales, including the spreading of viral infections, water pollution in rivers, optimal foraging strategies of animals, brain activity, crop growth and plant breeding, weather prediction, and climate fluctuations like Dansgaard-Oescher events. To effectively capture the stochastic nature of these processes in simulations, mathematical models should be combined with data and statistics. In this course two approaches to do so are introduced, Stochastic Differential Equations (SDEs) and Data Assimilation (DA).

Although SDEs and DA both deal with uncertainty in dynamic systems, they address different aspects of the modeling process. SDE-based models combine deterministic mathematical process descriptions with explicit descriptions of stochastic components. DA approaches, like (ensemble) Kalman filtering and variational methods, integrate observational data with model-based predictions to obtain the best possible estimates of uncertain model states, parameters and/or controls.

This course starts with a short recap of essential concepts of ordinary differential equations (ODE) and probability theory, after which a hands-on and example-driven introduction to the implementation of SDEs and DA for modelling in the life sciences domains is provided, with a focus on real-world examples. Topics that will be discussed include bifurcation theory, linear statistical modelling, Brownian motion, stochastic integrals (Itô, Stratonovich), the Fokker-Planck equation, statistical modelling of ODE systems, parameter estimation, ensemble methods, and Kalman filtering.

After successful completion of this course students are expected to be able to:

• Explain and interpret commonly used concepts, methods, and techniques from ordinary and stochastic differential equations and data assimilation
• Recognize research questions and situations that can profit from an approach using stochastic differential equations or data assimilation
• Apply modern computational approaches for implementing and solving stochastic differential equations and data assimilation
• Effectively set up, perform, and communicate about stochastic differential equations and data assimilation

Assumed Knowledge:
Mathematics 2 (MAT14903), Mathematics 3 (MAT15003) and Statistics 2 (MAT15403) or equivalent