@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{LeungLeutbecherReichetal.2020,
  author    = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.},
  title     = {Impact of the mesoscale range on error growth and the limits to atmospheric predictability},
  series = {Journal of the atmospheric sciences},
  volume    = {77},
  journal   = {Journal of the atmospheric sciences},
  number    = {11},
  publisher = {American Meteorological Soc.},
  address   = {Boston},
  issn      = {0022-4928},
  doi       = {10.1175/JAS-D-19-0346.1},
  pages     = {3769 -- 3779},
  year      = {2020},
  abstract  = {Global numerical weather prediction (NWP) models have begun to resolve the mesoscale k(-5/3) range of the energy spectrum, which is known to impose an inherently finite range of deterministic predictability per se as errors develop more rapidly on these scales than on the larger scales. However, the dynamics of these errors under the influence of the synoptic-scale k(-3) range is little studied. Within a perfect-model context, the present work examines the error growth behavior under such a hybrid spectrum in Lorenz's original model of 1969, and in a series of identical-twin perturbation experiments using an idealized two-dimensional barotropic turbulence model at a range of resolutions. With the typical resolution of today's global NWP ensembles, error growth remains largely uniform across scales. The theoretically expected fast error growth characteristic of a k(-5/3) spectrum is seen to be largely suppressed in the first decade of the mesoscale range by the synoptic-scale k(-3) range. However, it emerges once models become fully able to resolve features on something like a 20-km scale, which corresponds to a grid resolution on the order of a few kilometers.},
  language  = {en}
}
@article{AkhmatskayaBouRabeeReich2009,
  author    = {Akhmatskaya, Elena and Bou-Rabee, Nawaf and Reich, Sebastian},
  title     = {A comparison of generalized hybrid Monte Carlo methods with and without momentum flip},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2008.12.014},
  year      = {2009},
  abstract  = {The generalized hybrid Monte Carlo (GHMC) method combines Metropolis corrected constant energy simulations with a partial random refreshment step in the particle momenta. The standard detailed balance condition requires that momenta are negated upon rejection of a molecular dynamics proposal step. The implication is a trajectory reversal upon rejection, which is undesirable when interpreting GHMC as thermostated molecular dynamics. We show that a modified detailed balance condition can be used to implement GHMC without momentum flips. The same modification can be applied to the generalized shadow hybrid Monte Carlo (GSHMC) method. Numerical results indicate that GHMC/GSHMC implementations with momentum flip display a favorable behavior in terms of sampling efficiency, i.e., the traditional GHMC/GSHMC implementations with momentum flip got the advantage of a higher acceptance rate and faster decorrelation of Monte Carlo samples. The difference is more pronounced for GHMC. We also numerically investigate the behavior of the GHMC method as a Langevin-type thermostat. We find that the GHMC method without momentum flip interferes less with the underlying stochastic molecular dynamics in terms of autocorrelation functions and it to be preferred over the GHMC method with momentum flip. The same finding applies to GSHMC.},
  language  = {en}
}
@article{AkhmatskayaBouRabeeReich2009,
  author    = {Akhmatskaya, Elena and Bou-Rabee, Nawaf and Reich, Sebastian},
  title     = {Erratum to "A comparison of generalized hybrid Monte Carlo methods with and without momentum flip" [J. Comput. Phys. 228 (2009), S. 2256 - 2265]},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2009.06.039},
  year      = {2009},
  abstract  = {The generalized hybrid Monte Carlo (GHMC) method combines Metropolis corrected constant energy simulations with a partial random refreshment step in the particle momenta. The standard detailed balance condition requires that momenta are negated upon rejection of a molecular dynamics proposal step. The implication is a trajectory reversal upon rejection, which is undesirable when interpreting GHMC as thermostated molecular dynamics. We show that a modified detailed balance condition can be used to implement GHMC without momentum flips. The same modification can be applied to the generalized shadow hybrid Monte Carlo (GSHMC) method. Numerical results indicate that GHMC/GSHMC implementations with momentum flip display a favorable behavior in terms of sampling efficiency, i.e., the traditional GHMC/GSHMC implementations with momentum flip got the advantage of a higher acceptance rate and faster decorrelation of Monte Carlo samples. The difference is more pronounced for GHMC. We also numerically investigate the behavior of the GHMC method as a Langevin-type thermostat. We find that the GHMC method without momentum flip interferes less with the underlying stochastic molecular dynamics in terms of autocorrelation functions and it to be preferred over the GHMC method with momentum flip. The same finding applies to GSHMC.},
  language  = {en}
}
@article{LeimkuhlerReich2009,
  author    = {Leimkuhler, Benedict and Reich, Sebastian},
  title     = {A metropolis adjusted Nos{\´e}-Hoover thermostat},
  issn      = {0764-583X},
  doi       = {10.1051/M2an/2009023},
  year      = {2009},
  abstract  = {We present a Monte Carlo technique for sampling from the canonical distribution in molecular dynamics. The method is built upon the Nose-Hoover constant temperature formulation and the generalized hybrid Monte Carlo method. In contrast to standard hybrid Monte Carlo methods only the thermostat degree of freedom is stochastically resampled during a Monte Carlo step.},
  language  = {en}
}
@article{Reich2013,
  author    = {Reich, Sebastian},
  title     = {A nonparametric ensemble transform method for bayesian inference},
  series = {SIAM journal on scientific computing},
  volume    = {35},
  journal   = {SIAM journal on scientific computing},
  number    = {4},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1064-8275},
  doi       = {10.1137/130907367},
  pages     = {A2013 -- A2024},
  year      = {2013},
  abstract  = {Many applications, such as intermittent data assimilation, lead to a recursive application of Bayesian inference within a Monte Carlo context. Popular data assimilation algorithms include sequential Monte Carlo methods and ensemble Kalman filters (EnKFs). These methods differ in the way Bayesian inference is implemented. Sequential Monte Carlo methods rely on importance sampling combined with a resampling step, while EnKFs utilize a linear transformation of Monte Carlo samples based on the classic Kalman filter. While EnKFs have proven to be quite robust even for small ensemble sizes, they are not consistent since their derivation relies on a linear regression ansatz. In this paper, we propose another transform method, which does not rely on any a priori assumptions on the underlying prior and posterior distributions. The new method is based on solving an optimal transportation problem for discrete random variables.},
  language  = {en}
}
@article{BergemannGottwaldReich2009,
  author    = {Bergemann, Kay and Gottwald, Georg and Reich, Sebastian},
  title     = {Ensemble propagation and continuous matrix factorization algorithms},
  issn      = {0035-9009},
  doi       = {10.1002/qj.457},
  year      = {2009},
  abstract  = {We consider the problem of propagating an ensemble of solutions and its characterization in terms of its mean and covariance matrix. We propose differential equations that lead to a continuous matrix factorization of the ensemble into a generalized singular value decomposition (SVD). The continuous factorization is applied to ensemble propagation under periodic rescaling (ensemble breeding) and under periodic Kalman analysis steps (ensemble Kalman filter). We also use the continuous matrix factorization to perform a re-orthogonalization of the ensemble after each time-step and apply the resulting modified ensemble propagation algorithm to the ensemble Kalman filter. Results from the Lorenz-96 model indicate that the re-orthogonalization of the ensembles leads to improved filter performance.},
  language  = {en}
}
@article{CotterHamPainetal.2009,
  author    = {Cotter, Colin J. and Ham, David A. and Pain, Christopher C. and Reich, Sebastian},
  title     = {LBB stability of a mixed Galerkin finite element pair for fluid flow simulations},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2008.09.014},
  year      = {2009},
  abstract  = {We introduce a new mixed finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1(D)-P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. The advantage of this particular element choice is that the mass matrix for velocity is block diagonal so it can be trivially inverted; it also allows the order of the pressure to be increased to quadratic whilst maintaining LBB stability which has benefits in geophysical applications with Coriolis forces. We give a normal mode analysis of the semi-discrete wave equation in one dimension which shows that the element pair is stable, and demonstrate that the element is stable with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions on various meshes which shows that the element pair does not have any spurious modes. We provide convergence tests for the element pair which confirm that the element is stable since the convergence rate of the numerical solution is quadratic.},
  language  = {en}
}
@article{BergemannReich2010,
  author    = {Bergemann, Kay and Reich, Sebastian},
  title     = {A localization technique for ensemble Kalman filters},
  issn      = {0035-9009},
  doi       = {10.1002/Qj.591},
  year      = {2010},
  abstract  = {Ensemble Kalman filter techniques are widely used to assimilate observations into dynamical models. The phase- space dimension is typically much larger than the number of ensemble members, which leads to inaccurate results in the computed covariance matrices. These inaccuracies can lead, among other things, to spurious long-range correlations, which can be eliminated by Schur-product-based localization techniques. In this article, we propose a new technique for implementing such localization techniques within the class of ensemble transform/square-root Kalman filters. Our approach relies on a continuous embedding of the Kalman filter update for the ensemble members, i.e. we state an ordinary differential equation (ODE) with solutions that, over a unit time interval, are equivalent to the Kalman filter update. The ODE formulation forms a gradient system with the observations as a cost functional. Besides localization, the new ODE ensemble formulation should also find useful application in the context of nonlinear observation operators and observations that arrive continuously in time.},
  language  = {en}
}
@article{BergemannReich2010,
  author    = {Bergemann, Kay and Reich, Sebastian},
  title     = {A mollified ensemble Kalman filter},
  issn      = {0035-9009},
  doi       = {10.1002/Qj.672},
  year      = {2010},
  abstract  = {It is well recognized that discontinuous analysis increments of sequential data assimilation systems, such as ensemble Kalman filters, might lead to spurious high-frequency adjustment processes in the model dynamics. Various methods have been devised to spread out the analysis increments continuously over a fixed time interval centred about the analysis time. Among these techniques are nudging and incremental analysis updates (IAU). Here we propose another alternative, which may be viewed as a hybrid of nudging and IAU and which arises naturally from a recently proposed continuous formulation of the ensemble Kalman analysis step. A new slow-fast extension of the popular Lorenz-96 model is introduced to demonstrate the properties of the proposed mollified ensemble Kalman filter.},
  language  = {en}
}