@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}
}
@phdthesis{Angwenyi2019,
  author    = {Angwenyi, David},
  title     = {Time-continuous state and parameter estimation with application to hyperbolic SPDEs},
  doi       = {10.25932/publishup-43654},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436542},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xi, 101},
  year      = {2019},
  abstract  = {Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state. The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling. The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations. The interconnection between It{\^o} integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes. We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter.},
  language  = {en}
}
@phdthesis{Stark2021,
  author    = {Stark, Markus},
  title     = {Implications of local and regional processes on the stability of metacommunities in diverse ecosystems},
  doi       = {10.25932/publishup-52639},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-526399},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 167},
  year      = {2021},
  abstract  = {Anthropogenic activities such as continuous landscape changes threaten biodiversity at both local and regional scales. Metacommunity models attempt to combine these two scales and continuously contribute to a better mechanistic understanding of how spatial processes and constraints, such as fragmentation, affect biodiversity. There is a strong consensus that such structural changes of the landscape tend to negatively effect the stability of metacommunities. However, in particular the interplay of complex trophic communities and landscape structure is not yet fully understood. In this present dissertation, a metacommunity approach is used based on a dynamic and spatially explicit model that integrates population dynamics at the local scale and dispersal dynamics at the regional scale. This approach allows the assessment of complex spatial landscape components such as habitat clustering on complex species communities, as well as the analysis of population dynamics of a single species. In addition to the impact of a fixed landscape structure, periodic environmental disturbances are also considered, where a periodical change of habitat availability, temporally alters landscape structure, such as the seasonal drying of a water body. On the local scale, the model results suggest that large-bodied animal species, such as predator species at high trophic positions, are more prone to extinction in a state of large patch isolation than smaller species at lower trophic levels. Increased metabolic losses for species with a lower body mass lead to increased energy limitation for species on higher trophic levels and serves as an explanation for a predominant loss of these species. This effect is particularly pronounced for food webs, where species are more sensitive to increased metabolic losses through dispersal and a change in landscape structure. In addition to the impact of species composition in a food web for diversity, the strength of local foraging interactions likewise affect the synchronization of population dynamics. A reduced predation pressure leads to more asynchronous population dynamics, beneficial for the stability of population dynamics as it reduces the risk of correlated extinction events among habitats. On the regional scale, two landscape aspects, which are the mean patch isolation and the formation of local clusters of two patches, promote an increase in \$\beta\$-diversity. Yet, the individual composition and robustness of the local species community equally explain a large proportion of the observed diversity patterns. A combination of periodic environmental disturbance and patch isolation has a particular impact on population dynamics of a species. While the periodic disturbance has a synchronizing effect, it can even superimpose emerging asynchronous dynamics in a state of large patch isolation and unifies trends in synchronization between different species communities. In summary, the findings underline a large local impact of species composition and interactions on local diversity patterns of a metacommunity. In comparison, landscape structures such as fragmentation have a negligible effect on local diversity patterns, but increase their impact for regional diversity patterns. In contrast, at the level of population dynamics, regional characteristics such as periodic environmental disturbance and patch isolation have a particularly strong impact and contribute substantially to the understanding of the stability of population dynamics in a metacommunity. These studies demonstrate once again the complexity of our ecosystems and the need for further analysis for a better understanding of our surrounding environment and more targeted conservation of biodiversity.},
  language  = {en}
}
@phdthesis{Perera2021,
  author    = {Perera, Upeksha},
  title     = {Solutions of direct and inverse Sturm-Liouville problems},
  doi       = {10.25932/publishup-53006},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-530064},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 109},
  year      = {2021},
  abstract  = {Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville Problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular and some singular SLPs of even orders (tested up to order eight), with a mix of boundary conditions (including non-separable and finite singular endpoints), accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of order n=2,4 is verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides methods that can be adapted successfully for solving a direct (regular/singular) or inverse SLP of an arbitrary order with arbitrary boundary conditions.},
  language  = {en}
}