@article{LeungLeutbecherReichetal.2019,
  author    = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.},
  title     = {Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier},
  series = {Journal of the atmospheric sciences},
  volume    = {76},
  journal   = {Journal of the atmospheric sciences},
  number    = {12},
  publisher = {American Meteorological Soc.},
  address   = {Boston},
  issn      = {0022-4928},
  doi       = {10.1175/JAS-D-19-0057.1},
  pages     = {3883 -- 3892},
  year      = {2019},
  abstract  = {The accepted idea that there exists an inherent finite-time barrier in deterministically predicting atmospheric flows originates from Edward N. Lorenz's 1969 work based on two-dimensional (2D) turbulence. Yet, known analytic results on the 2D Navier-Stokes (N-S) equations suggest that one can skillfully predict the 2D N-S system indefinitely far ahead should the initial-condition error become sufficiently small, thereby presenting a potential conflict with Lorenz's theory. Aided by numerical simulations, the present work reexamines Lorenz's model and reviews both sides of the argument, paying particular attention to the roles played by the slope of the kinetic energy spectrum. It is found that when this slope is shallower than -3, the Lipschitz continuity of analytic solutions (with respect to initial conditions) breaks down as the model resolution increases, unless the viscous range of the real system is resolved—which remains practically impossible. This breakdown leads to the inherent finite-time limit. If, on the other hand, the spectral slope is steeper than -3, then the breakdown does not occur. In this way, the apparent contradiction between the analytic results and Lorenz's theory is reconciled.},
  language  = {en}
}
@article{StaniforthWoodReich2006,
  author    = {Staniforth, Andrew and Wood, Nigel and Reich, Sebastian},
  title     = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {132},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {621C},
  publisher = {Wiley},
  address   = {Weinheim},
  issn      = {0035-9009},
  doi       = {10.1256/qj.06.30},
  pages     = {3107 -- 3116},
  year      = {2006},
  abstract  = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes.},
  language  = {en}
}
@article{Reich2006,
  author    = {Reich, Sebastian},
  title     = {Linearly implicit time stepping methods for numerical weather prediction},
  series = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  volume    = {46},
  journal   = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0006-3835},
  doi       = {10.1007/s10543-006-0065-0},
  pages     = {607 -- 616},
  year      = {2006},
  abstract  = {The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.},
  language  = {en}
}
@article{SomogyvariReich2020,
  author    = {Somogyv{\´a}ri, M{\´a}rk and Reich, Sebastian},
  title     = {Convergence tests for transdimensional Markov chains in geoscience imaging},
  series = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences},
  volume    = {52},
  journal   = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences},
  number    = {5},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1874-8961},
  doi       = {10.1007/s11004-019-09811-x},
  pages     = {651 -- 668},
  year      = {2020},
  abstract  = {Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.},
  language  = {en}
}
@article{TaghvaeideWiljesMehtaetal.2017,
  author    = {Taghvaei, Amirhossein and de Wiljes, Jana and Mehta, Prashant G. and Reich, Sebastian},
  title     = {Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem},
  series = {Journal of dynamic systems measurement and control},
  volume    = {140},
  journal   = {Journal of dynamic systems measurement and control},
  number    = {3},
  publisher = {ASME},
  address   = {New York},
  issn      = {0022-0434},
  doi       = {10.1115/1.4037780},
  pages     = {11},
  year      = {2017},
  abstract  = {This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman-Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.},
  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{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}
}
@misc{AscherChinReich1994,
  author    = {Ascher, Uri M. and Chin, Hongsheng and Reich, Sebastian},
  title     = {Stabilization of DAEs and invariant manifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15625},
  year      = {1994},
  abstract  = {Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.},
  language  = {en}
}
@misc{Reich1995,
  author    = {Reich, Sebastian},
  title     = {Smoothed dynamics of highly oscillatory Hamiltonian systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15639},
  year      = {1995},
  abstract  = {We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.},
  language  = {en}
}
@misc{LeimkuhlerReich1994,
  author    = {Leimkuhler, Benedict and Reich, Sebastian},
  title     = {Symplectic integration of constrained Hamiltonian systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15653},
  year      = {1994},
  abstract  = {A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.},
  language  = {en}
}
@misc{AscherChinPetzoldetal.1994,
  author    = {Ascher, Uri M. and Chin, Hongsheng and Petzold, Linda R. and Reich, Sebastian},
  title     = {Stabilization of constrained mechanical systems with DAEs and invariant manifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15698},
  year      = {1994},
  abstract  = {Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious of these have mild instabilities and drift problems. Consequently, stabilization techniques have been proposed A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice. Some of the simulation methods that have been proposed and used in computations are reviewed here, from a stability point of view. This involves concepts of differential-algebraic equation (DAE) and ordinary differential equation (ODE) invariants. An explanation of the difficulties that may be encountered using Baumgarte's method is given, and a discussion of why a further quest for better parameter values for this method will always remain frustrating is presented. It is then shown how Baumgarte's method can be improved. An efficient stabilization technique is proposed, which may employ explicit ODE solvers in case of nonstiff or highly oscillatory problems and which relates to coordinate projection methods. Examples of a two-link planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.},
  language  = {en}
}
@misc{Reich1980,
  author    = {Reich, Sebastian},
  title     = {Algebrodifferentialgleichungen und Vektorfelder auf Mannigfaltigkeiten},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-47290},
  year      = {1980},
  abstract  = {In diesem Beitrag wird der Zusammenhang zwischen Algebrodifferentialgleichungen (ADGL) und Vektorfeldern auf Mannigfaltigkeiten untersucht. Dazu wird zun{\"a}chst der Begriff der regul{\"a}ren ADGL eingef{\"u}hrt, wobei unter eirter regul{\"a}ren ADGL eine ADGL verstanden wird, deren L{\"o}sungsmenge identisch mit der L{\"o}sungsmenge eines Vektorfeldes ist. Ausgehend von bekannten Aussagen {\"u}ber die L{\"o}sungsmenge eines Vektorfeldes werden analoge Aussagen f{\"u}r die L{\"o}sungsmenge einer regul{\"a}ren ADGL abgeleitet. Es wird eine Reduktionsmethode angegeben, die auf ein Kriterium f{\"u}r die Begularit{\"a}t einer ADGL und auf die Definition des Index einer nichtlinearen ADGL f{\"u}hrt. Außerdem wird gezeigt, daß beliebige Vektorfelder durch regul{\"a}re ADGL so realisiert werden k{\"o}nnen, daß die L{\"o}sungsmenge des Vektorfeldes mit der der realisierenden ADGL identisch ist. Abschließend werden die f{\"u}r autonome ADGL gewonnenen Aussagen auf den Fall der nichtautonomen ADGL {\"u}bertragen.},
  language  = {de}
}
@misc{Reich1992,
  author    = {Reich, Sebastian},
  title     = {Differential-algebraic equations and applications in circuit theory},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46646},
  year      = {1992},
  abstract  = {Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples.},
  language  = {en}
}
@misc{NueskenReichRozdeba2019,
  author    = {N{\"u}sken, Nikolas and Reich, Sebastian and Rozdeba, Paul J.},
  title     = {State and parameter estimation from observed signal increments},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {916},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-44260},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-442609},
  pages     = {25},
  year      = {2019},
  abstract  = {The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.},
  language  = {en}
}
@book{VanLeeuwenChengReich2015,
  author    = {Van Leeuwen, Peter Jan and Cheng, Yuan and Reich, Sebastian},
  title     = {Nonlinear data assimilation},
  series = {Frontiers in applied dynamical systems: reviews and tutorials ; 2},
  journal   = {Frontiers in applied dynamical systems: reviews and tutorials ; 2},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-319-18346-6},
  doi       = {10.1007/978-3-319-18347-3},
  pages     = {xii, 118},
  year      = {2015},
  abstract  = {This book contains two review articles on nonlinear data assimilation that deal with closely related topics but were written and can be read independently. Both contributions focus on so-called particle filters. The first contribution by Jan van Leeuwen focuses on the potential of proposal densities. It discusses the issues with present-day particle filters and explorers new ideas for proposal densities to solve them, converging to particle filters that work well in systems of any dimension, closing the contribution with a high-dimensional example. The second contribution by Cheng and Reich discusses a unified framework for ensemble-transform particle filters. This allows one to bridge successful ensemble Kalman filters with fully nonlinear particle filters, and allows a proper introduction of localization in particle filters, which has been lacking up to now.},
  language  = {en}
}
@article{Reich2019,
  author    = {Reich, Sebastian},
  title     = {Data assimilation},
  series = {Acta numerica},
  volume    = {28},
  journal   = {Acta numerica},
  publisher = {Cambridge Univ. Press},
  address   = {New York},
  issn      = {0962-4929},
  doi       = {10.1017/S0962492919000011},
  pages     = {635 -- 711},
  year      = {2019},
  abstract  = {Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete- and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schr{\"o}dinger's boundary value problem for stochastic processes in particular.},
  language  = {en}
}
@article{NueskenReichRozdeba2019,
  author    = {N{\"u}sken, Nikolas and Reich, Sebastian and Rozdeba, Paul J.},
  title     = {State and parameter estimation from observed signal increments},
  series = {Entropy : an international and interdisciplinary journal of entropy and information studies},
  volume    = {21},
  journal   = {Entropy : an international and interdisciplinary journal of entropy and information studies},
  number    = {5},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e21050505},
  pages     = {23},
  year      = {2019},
  abstract  = {The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.},
  language  = {en}
}
@article{GarbunoInigoNueskenReich2020,
  author    = {Garbuno-Inigo, Alfredo and N{\"u}sken, Nikolas and Reich, Sebastian},
  title     = {Affine invariant interacting Langevin dynamics for Bayesian inference},
  series = {SIAM journal on applied dynamical systems},
  volume    = {19},
  journal   = {SIAM journal on applied dynamical systems},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1536-0040},
  doi       = {10.1137/19M1304891},
  pages     = {1633 -- 1658},
  year      = {2020},
  abstract  = {We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.},
  language  = {en}
}
@article{MaoutsaReichOpper2020,
  author    = {Maoutsa, Dimitra and Reich, Sebastian and Opper, Manfred},
  title     = {Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation},
  series = {Entropy},
  volume    = {22},
  journal   = {Entropy},
  number    = {8},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e22080802},
  pages     = {35},
  year      = {2020},
  abstract  = {Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.},
  language  = {en}
}
@misc{Reich1990,
  author    = {Reich, Sebastian},
  title     = {On a geometrical interpretation of differential-algebraic equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46683},
  year      = {1990},
  abstract  = {The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.},
  language  = {en}
}
@misc{Reich1994,
  author    = {Reich, Sebastian},
  title     = {Momentum conserving symplectic integrators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-16824},
  year      = {1994},
  abstract  = {In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).},
  language  = {en}
}
@misc{Reich1995,
  author    = {Reich, Sebastian},
  title     = {On the local qualitative behavior of differential-algebraic equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46739},
  year      = {1995},
  abstract  = {A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.},
  language  = {en}
}
@misc{Reich1991,
  author    = {Reich, Sebastian},
  title     = {On an existence and uniqueness theory for nonlinear differential-algebraic equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46706},
  year      = {1991},
  abstract  = {An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.},
  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}
}
@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{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{SeeligRabeMalemShinitskietal.2020,
  author    = {Seelig, Stefan A. and Rabe, Maximilian Michael and Malem-Shinitski, Noa and Risse, Sarah and Reich, Sebastian and Engbert, Ralf},
  title     = {Bayesian parameter estimation for the SWIFT model of eye-movement control during reading},
  series = {Journal of mathematical psychology},
  volume    = {95},
  journal   = {Journal of mathematical psychology},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0022-2496},
  doi       = {10.1016/j.jmp.2019.102313},
  pages     = {32},
  year      = {2020},
  abstract  = {Process-oriented theories of cognition must be evaluated against time-ordered observations. Here we present a representative example for data assimilation of the SWIFT model, a dynamical model of the control of fixation positions and fixation durations during natural reading of single sentences. First, we develop and test an approximate likelihood function of the model, which is a combination of a spatial, pseudo-marginal likelihood and a temporal likelihood obtained by probability density approximation Second, we implement a Bayesian approach to parameter inference using an adaptive Markov chain Monte Carlo procedure. Our results indicate that model parameters can be estimated reliably for individual subjects. We conclude that approximative Bayesian inference represents a considerable step forward for computational models of eye-movement control, where modeling of individual data on the basis of process-based dynamic models has not been possible so far.},
  language  = {en}
}
@article{MalemShinitskiOpperReichetal.2020,
  author    = {Malem-Shinitski, Noa and Opper, Manfred and Reich, Sebastian and Schwetlick, Lisa and Seelig, Stefan A. and Engbert, Ralf},
  title     = {A mathematical model of local and global attention in natural scene viewing},
  series = {PLoS Computational Biology : a new community journal},
  volume    = {16},
  journal   = {PLoS Computational Biology : a new community journal},
  number    = {12},
  publisher = {PLoS},
  address   = {San Fransisco},
  issn      = {1553-734X},
  doi       = {10.1371/journal.pcbi.1007880},
  pages     = {21},
  year      = {2020},
  abstract  = {Author summary <br /> Switching between local and global attention is a general strategy in human information processing. We investigate whether this strategy is a viable approach to model sequences of fixations generated by a human observer in a free viewing task with natural scenes. Variants of the basic model are used to predict the experimental data based on Bayesian inference. Results indicate a high predictive power for both aggregated data and individual differences across observers. The combination of a novel model with state-of-the-art Bayesian methods lends support to our two-state model using local and global internal attention states for controlling eye movements. <br /> Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.},
  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{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{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{WormellReich2021,
  author    = {Wormell, Caroline L. and Reich, Sebastian},
  title     = {Spectral convergence of diffusion maps},
  series = {SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics},
  volume    = {59},
  journal   = {SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {0036-1429},
  doi       = {10.1137/20M1344093},
  pages     = {1687 -- 1734},
  year      = {2021},
  abstract  = {Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.},
  language  = {en}
}
@article{MolkenthinDonnerReichetal.2022,
  author    = {Molkenthin, Christian and Donner, Christian and Reich, Sebastian and Z{\"o}ller, Gert and Hainzl, Sebastian and Holschneider, Matthias and Opper, Manfred},
  title     = {GP-ETAS: semiparametric Bayesian inference for the spatio-temporal epidemic type aftershock sequence model},
  series = {Statistics and Computing},
  volume    = {32},
  journal   = {Statistics and Computing},
  number    = {2},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0960-3174},
  doi       = {10.1007/s11222-022-10085-3},
  pages     = {25},
  year      = {2022},
  abstract  = {The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented.},
  language  = {en}
}