@article{LeungLeutbecherReichetal.2021, author = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.}, title = {Forecast verification}, series = {Quarterly journal of the Royal Meteorological Society}, volume = {147}, journal = {Quarterly journal of the Royal Meteorological Society}, number = {739}, publisher = {Wiley}, address = {Hoboken}, issn = {0035-9009}, doi = {10.1002/qj.4120}, pages = {3124 -- 3134}, year = {2021}, abstract = {The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS-RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met.}, language = {en} } @article{ReichWeissmann2021, author = {Reich, Sebastian and Weissmann, Simon}, title = {Fokker-Planck particle systems for Bayesian inference: computational approaches}, series = {SIAM ASA journal on uncertainty quantification}, volume = {9}, journal = {SIAM ASA journal on uncertainty quantification}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {2166-2525}, doi = {10.1137/19M1303162}, pages = {446 -- 482}, year = {2021}, abstract = {Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.}, 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{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Supervised learning from noisy observations}, series = {Physica : D, Nonlinear phenomena}, volume = {423}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2021.132911}, pages = {15}, year = {2021}, abstract = {Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data -driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems. (C) 2021 Elsevier B.V. All rights reserved.}, language = {en} } @article{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {10}, publisher = {AIP}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0066080}, pages = {8}, year = {2021}, abstract = {We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework, a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens's embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data.}, language = {en} } @article{EngbertRabeKliegletal.2021, author = {Engbert, Ralf and Rabe, Maximilian Michael and Kliegl, Reinhold and Reich, Sebastian}, title = {Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics}, series = {Bulletin of mathematical biology : official journal of the Society for Mathematical Biology}, volume = {83}, journal = {Bulletin of mathematical biology : official journal of the Society for Mathematical Biology}, number = {1}, publisher = {Springer}, address = {New York}, issn = {0092-8240}, doi = {10.1007/s11538-020-00834-8}, pages = {16}, year = {2021}, abstract = {Newly emerging pandemics like COVID-19 call for predictive models to implement precisely tuned responses to limit their deep impact on society. Standard epidemic models provide a theoretically well-founded dynamical description of disease incidence. For COVID-19 with infectiousness peaking before and at symptom onset, the SEIR model explains the hidden build-up of exposed individuals which creates challenges for containment strategies. However, spatial heterogeneity raises questions about the adequacy of modeling epidemic outbreaks on the level of a whole country. Here, we show that by applying sequential data assimilation to the stochastic SEIR epidemic model, we can capture the dynamic behavior of outbreaks on a regional level. Regional modeling, with relatively low numbers of infected and demographic noise, accounts for both spatial heterogeneity and stochasticity. Based on adapted models, short-term predictions can be achieved. Thus, with the help of these sequential data assimilation methods, more realistic epidemic models are within reach.}, language = {en} }