Uncertainty Quantification and Data Assimilation (UQDA)
University of Amsterdam, fall semester 2020. Lecturer: Daan Crommelin.
Mathematical models in the form of PDEs or ODEs arise in many fields of science and engineering, for instance physics, chemistry, biology and climate science. The development of efficient and accurate algorithms to simulate these systems is a classical topic in numerical analysis. A much more recent topic is to investigate the impact of errors and uncertainties in e.g. model parameters, initial conditions and boundary conditions. Even with very accurate numerical algorithms, such uncertainties can have major impact and lead to large uncertainties in numerical results.
In this course we will look at modern mathematical techniques that have been developed to deal with uncertainties in numerical simulation. In Uncertainty Quantification (UQ), a prototypical problem is to characterize the probability distribution of a model output variable given the distribution of an input parameter, and to do so in an efficient way.
For Data Assimilation (DA), the main question is how to incorporate data (e.g. from physical measurements) in models in a suitable way, in order to improve model predictions and quantify prediction uncertainty. Here, the focus is on the prediction of nonlinear dynamical systems (the classical application example being weather forecasting).
UQ and DA are both very active areas of research and have relevance for a wide range applications. Both involve a modern combination of elements from numerical analysis, dynamical systems, probability and statistics.
The focus of this course will be on the mathematical techniques and methodologies developed for UQ and DA. We will briefly review some topics from numerical analysis (orthogonal polynomials, approximation and interpolation, quadrature). These provide a basis for several main UQ methods that we will cover (stochastic Galerkin, polynomial chaos expansion, stochastic collocation). In the last part of the course we will discuss basic DA approaches (such as Kalman filter and variational data assimilation).
More information:
Time and place: The course starts on 31 August 2020. Classes are Monday 11:00-13:00. Because of the Corona measures, they are currenly planned as online class meetings for weeks 36-42 followed by classes in room SP G3.05 of the Amsterdam Science Park campus of UvA during weeks 44-50. However, the schedule is still subject to change, please check the Datanose page for the latest updates regarding the course schedule (time/location, online/offline).
Literature:
Main text:
D. Xiu, Numerical Methods for Stochastic Computations. A Spectral Method Approach. Princeton University Press, 2010.
Additional literature:
T.J. Sullivan, Introduction to Uncertainty Quantification. Springer, 2015.
K. Law, A. Stuart, K. Zygalakis, Data Assimilation: A Mathematical Introduction. Springer, 2015.