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Covariance estimation for filtering in high dimension
Turčičová, Marie ; Mandel, Jan (advisor)
Estimating large covariance matrices from small samples is an important problem in many fields. Among others, this includes spatial statistics and data assimilation. In this thesis, we deal with several methods of covariance estimation with emphasis on regula- rization and covariance models useful in filtering problems. We prove several properties of estimators and propose a new filtering method. After a brief summary of basic esti- mating methods used in data assimilation, the attention is shifted to covariance models. We show a distinct type of hierarchy in nested models applied to the spectral diagonal covariance matrix: explicit estimators of parameters are computed by the maximum like- lihood method and asymptotic variance of these estimators is shown to decrease when the maximization is restricted to a subspace that contains the true parameter value. A similar result is obtained for general M-estimators. For more complex covariance mo- dels, maximum likelihood method cannot provide explicit parameter estimates. In the case of a linear model for a precision matrix, however, consistent estimator in a closed form can be computed by the score matching method. Modelling of the precision ma- trix is particularly beneficial in Gaussian Markov random fields (GMRF), which possess a sparse precision matrix. The...
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Covariance estimation for filtering in high dimension
Turčičová, Marie ; Mandel, Jan (advisor) ; van Leeuwen, Peter Jan (referee) ; Pawlas, Zbyněk (referee)
Estimating large covariance matrices from small samples is an important problem in many fields. Among others, this includes spatial statistics and data assimilation. In this thesis, we deal with several methods of covariance estimation with emphasis on regula- rization and covariance models useful in filtering problems. We prove several properties of estimators and propose a new filtering method. After a brief summary of basic esti- mating methods used in data assimilation, the attention is shifted to covariance models. We show a distinct type of hierarchy in nested models applied to the spectral diagonal covariance matrix: explicit estimators of parameters are computed by the maximum like- lihood method and asymptotic variance of these estimators is shown to decrease when the maximization is restricted to a subspace that contains the true parameter value. A similar result is obtained for general M-estimators. For more complex covariance mo- dels, maximum likelihood method cannot provide explicit parameter estimates. In the case of a linear model for a precision matrix, however, consistent estimator in a closed form can be computed by the score matching method. Modelling of the precision ma- trix is particularly beneficial in Gaussian Markov random fields (GMRF), which possess a sparse precision matrix. The...
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Some practical aspects of parallel adaptive BDDC method
Šístek, Jakub ; Mandel, J. ; Sousedík, B.
We describe a parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method enhanced by an adaptive construction of coarse problem. The method is designed for numerically difficult problems, where standard choice of continuity of arithmetic averages across faces and edges of subdomains fails to maintain the low condition number of the preconditioned system. Problems of elasticity analysis of bodies consisting of different materials with rapidly changing stiffness may represent one class of such challenging problems. The adaptive selection of constraints is shown to significantly increase the robustness of the method for this class of problems. However, since the cost of the set-up of the preconditioner with adaptive constraints is considerably larger than for the standard choices, computational feasibility of the presented implementation is obtained only for large contrasts of material coefficients.
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Online System for Fire Danger Rating in Colorado
Vejmelka, Martin ; Kochanski, A. ; Mandel, J.
A method for the data assimilation of fuel moisture surface observations has been developed for the purpose of incorporation in wildfire forecasting and fire danger rating. In this work, we describe the method itself and also an online computer system that implements the method and combines it with the Real-Time Mesoscale Analysis to track local weather conditions and estimate the fuel moisture content in the state of Colorado. We discuss the construction of the system and future development.
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