@article{ZhelavskayaAseevShprits2021, author = {Zhelavskaya, Irina and Aseev, Nikita and Shprits, Yuri}, title = {A combined neural network- and physics-based approach for modeling plasmasphere dynamics}, series = {JGR / AGU, American Geographical Union. Space Physics}, volume = {126}, journal = {JGR / AGU, American Geographical Union. Space Physics}, number = {3}, publisher = {Wiley}, address = {Hoboken, NJ}, issn = {2169-9380}, doi = {10.1029/2020JA028077}, pages = {30}, year = {2021}, abstract = {Abstract In recent years, feedforward neural networks (NNs) have been successfully applied to reconstruct global plasmasphere dynamics in the equatorial plane. These neural network-based models capture the large-scale dynamics of the plasmasphere, such as plume formation and erosion of the plasmasphere on the nightside. However, their performance depends strongly on the availability of training data. When the data coverage is limited or non-existent, as occurs during geomagnetic storms, the performance of NNs significantly decreases, as networks inherently cannot learn from the limited number of examples. This limitation can be overcome by employing physics-based modeling during strong geomagnetic storms. Physics-based models show a stable performance during periods of disturbed geomagnetic activity if they are correctly initialized and configured. In this study, we illustrate how to combine the neural network- and physics-based models of the plasmasphere in an optimal way by using data assimilation. The proposed approach utilizes advantages of both neural network- and physics-based modeling and produces global plasma density reconstructions for both quiet and disturbed geomagnetic activity, including extreme geomagnetic storms. We validate the models quantitatively by comparing their output to the in-situ density measurements from RBSP-A for an 18-month out-of-sample period from June 30, 2016 to January 01, 2018 and computing performance metrics. To validate the global density reconstructions qualitatively, we compare them to the IMAGE EUV images of the He+ particle distribution in the Earth's plasmasphere for a number of events in the past, including the Halloween storm in 2003.}, language = {en} } @article{WiljesTong2020, author = {Wiljes, Jana de and Tong, Xin T.}, title = {Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations}, series = {Nonlinearity}, volume = {33}, journal = {Nonlinearity}, number = {9}, publisher = {IOP Publ.}, address = {Bristol}, issn = {0951-7715}, doi = {10.1088/1361-6544/ab8d14}, pages = {4752 -- 4782}, year = {2020}, abstract = {Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.}, language = {en} } @article{PathirajaReichStannat2021, author = {Pathiraja, Sahani Darschika and Reich, Sebastian and Stannat, Wilhelm}, title = {McKean-Vlasov SDEs in nonlinear filtering}, series = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics}, volume = {59}, journal = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics}, number = {6}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {0363-0129}, doi = {10.1137/20M1355197}, pages = {4188 -- 4215}, year = {2021}, abstract = {Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.}, 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} } @article{deWiljesPathirajaReich2020, author = {de Wiljes, Jana and Pathiraja, Sahani Darschika and Reich, Sebastian}, title = {Ensemble transform algorithms for nonlinear smoothing problems}, series = {SIAM journal on scientific computing}, volume = {42}, journal = {SIAM journal on scientific computing}, number = {1}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1064-8275}, doi = {10.1137/19M1239544}, pages = {A87 -- A114}, year = {2020}, abstract = {Several numerical tools designed to overcome the challenges of smoothing in a non-linear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter, and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.}, language = {en} } @article{AcevedoDeWiljesReich2017, author = {Acevedo, Walter and De Wiljes, Jana and Reich, Sebastian}, title = {Second-order accurate ensemble transform particle filters}, series = {SIAM journal on scientific computing}, volume = {39}, journal = {SIAM journal on scientific computing}, number = {5}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1064-8275}, doi = {10.1137/16M1095184}, pages = {A1834 -- A1850}, year = {2017}, abstract = {Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM T. Sci. Comput., 35, (2013), pp. A2013-A2014[ replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.}, language = {en} } @article{deWiljesReichStannat2018, author = {de Wiljes, Jana and Reich, Sebastian and Stannat, Wilhelm}, title = {Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise}, series = {SIAM Journal on Applied Dynamical Systems}, volume = {17}, journal = {SIAM Journal on Applied Dynamical Systems}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1536-0040}, doi = {10.1137/17M1119056}, pages = {1152 -- 1181}, year = {2018}, abstract = {The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.}, language = {en} } @article{PathirajaMoradkhaniMarshalletal.2018, author = {Pathiraja, Sahani Darschika and Moradkhani, H. and Marshall, L. and Sharma, Ashish and Geenens, G.}, title = {Data-driven model uncertainty estimation in hydrologic data assimilation}, series = {Water resources research : WRR / American Geophysical Union}, volume = {54}, journal = {Water resources research : WRR / American Geophysical Union}, number = {2}, publisher = {American Geophysical Union}, address = {Washington}, issn = {0043-1397}, doi = {10.1002/2018WR022627}, pages = {1252 -- 1280}, year = {2018}, abstract = {The increasing availability of earth observations necessitates mathematical methods to optimally combine such data with hydrologic models. Several algorithms exist for such purposes, under the umbrella of data assimilation (DA). However, DA methods are often applied in a suboptimal fashion for complex real-world problems, due largely to several practical implementation issues. One such issue is error characterization, which is known to be critical for a successful assimilation. Mischaracterized errors lead to suboptimal forecasts, and in the worst case, to degraded estimates even compared to the no assimilation case. Model uncertainty characterization has received little attention relative to other aspects of DA science. Traditional methods rely on subjective, ad hoc tuning factors or parametric distribution assumptions that may not always be applicable. We propose a novel data-driven approach (named SDMU) to model uncertainty characterization for DA studies where (1) the system states are partially observed and (2) minimal prior knowledge of the model error processes is available, except that the errors display state dependence. It includes an approach for estimating the uncertainty in hidden model states, with the end goal of improving predictions of observed variables. The SDMU is therefore suited to DA studies where the observed variables are of primary interest. Its efficacy is demonstrated through a synthetic case study with low-dimensional chaotic dynamics and a real hydrologic experiment for one-day-ahead streamflow forecasting. In both experiments, the proposed method leads to substantial improvements in the hidden states and observed system outputs over a standard method involving perturbation with Gaussian noise.}, language = {en} } @article{Reich2012, author = {Reich, Sebastian}, title = {A Gaussian-mixture ensemble transform filter}, series = {Quarterly journal of the Royal Meteorological Society}, volume = {138}, journal = {Quarterly journal of the Royal Meteorological Society}, number = {662}, publisher = {Wiley-Blackwell}, address = {Malden}, issn = {0035-9009}, doi = {10.1002/qj.898}, pages = {222 -- 233}, year = {2012}, abstract = {We generalize the popular ensemble Kalman filter to an ensemble transform filter, in which the prior distribution can take the form of a Gaussian mixture or a Gaussian kernel density estimator. The design of the filter is based on a continuous formulation of the Bayesian filter analysis step. We call the new filter algorithm the ensemble Gaussian-mixture filter (EGMF). The EGMF is implemented for three simple test problems (Brownian dynamics in one dimension, Langevin dynamics in two dimensions and the three-dimensional Lorenz-63 model). It is demonstrated that the EGMF is capable of tracking systems with non-Gaussian uni- and multimodal ensemble distributions.}, language = {en} }