In many applications, initial data or model parameters of partial differential equations
are not known precisely. This issue can be addressed in several ways:
If the probability distribution of the unknown data is available one can investigate how this probability distribution is propagated via the solution operator of the PDE. This is possible by intrusive methods (e.g. stochastic Galerkin) as well as by non-intrusive methods (e.g. Monte-Carlo or stochastic collocation). We construct reliable and efficient numerical schemes for this task and derive error estimates for them.
Another scenario is that measurement data of the system state (e.g. values of the solution of the PDE at certain points) are available. Data assimilation aims at combining measurement data and models in such a way that reliable estimates of the system state can be obtained. We approach this task based on so-called observers, i.e. modified versions of the PDE that include measurement data. Our goal is to show that for long times the observer state is close to the state of the original system.
