@phdthesis{Reinhardt2020,
  author    = {Reinhardt, Maria},
  title     = {Hybrid filters and multi-scale models},
  doi       = {10.25932/publishup-47435},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-474356},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xiii, 102},
  year      = {2020},
  abstract  = {This thesis is concerned with Data Assimilation, the process of combining model predictions with observations. So called filters are of special interest. One is inter- ested in computing the probability distribution of the state of a physical process in the future, given (possibly) imperfect measurements. This is done using Bayes' rule. The first part focuses on hybrid filters, that bridge between the two main groups of filters: ensemble Kalman filters (EnKF) and particle filters. The first are a group of very stable and computationally cheap algorithms, but they request certain strong assumptions. Particle filters on the other hand are more generally applicable, but computationally expensive and as such not always suitable for high dimensional systems. Therefore it exists a need to combine both groups to benefit from the advantages of each. This can be achieved by splitting the likelihood function, when assimilating a new observation and treating one part of it with an EnKF and the other part with a particle filter. The second part of this thesis deals with the application of Data Assimilation to multi-scale models and the problems that arise from that. One of the main areas of application for Data Assimilation techniques is predicting the development of oceans and the atmosphere. These processes involve several scales and often balance rela- tions between the state variables. The use of Data Assimilation procedures most often violates relations of that kind, which leads to unrealistic and non-physical pre- dictions of the future development of the process eventually. This work discusses the inclusion of a post-processing step after each assimilation step, in which a minimi- sation problem is solved, which penalises the imbalance. This method is tested on four different models, two Hamiltonian systems and two spatially extended models, which adds even more difficulties.},
  language  = {en}
}
@article{HastermannReinhardtKleinetal.2021,
  author    = {Hastermann, Gottfried and Reinhardt, Maria and Klein, Rupert and Reich, Sebastian},
  title     = {Balanced data assimilation for highly oscillatory mechanical systems},
  series = {Communications in applied mathematics and computational science : CAMCoS},
  volume    = {16},
  journal   = {Communications in applied mathematics and computational science : CAMCoS},
  number    = {1},
  publisher = {Mathematical Sciences Publishers},
  address   = {Berkeley},
  issn      = {1559-3940},
  doi       = {10.2140/camcos.2021.16.119},
  pages     = {119 -- 154},
  year      = {2021},
  abstract  = {Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, this filter also has limitations due to its inherent assumptions of Gaussianity and linearity, which can manifest themselves in the form of dynamically inconsistent state estimates. This issue is investigated here for balanced, slowly evolving solutions to highly oscillatory Hamiltonian systems which are prototypical for applications in numerical weather prediction. It is demonstrated that the standard ensemble Kalman filter can lead to state estimates that do not satisfy the pertinent balance relations and ultimately lead to filter divergence. Two remedies are proposed, one in terms of blended asymptotically consistent time-stepping schemes, and one in terms of minimization-based postprocessing methods. The effects of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for balanced motions of two prototypical Hamiltonian reference systems.},
  language  = {en}
}