TY  - JOUR
A1  - Asgarimehr, Milad
A1  - Wickert, Jens
A1  - Reich, Sebastian
T1  - TDS-1 GNSS Reflectometry
BT  - Development and Validation of Forward Scattering Winds
JF  - IEEE journal of selected topics in applied earth observations and remote sensing
N2  - This study presents the development and a systematic evaluation study of GNSS reflectometry wind speeds. After establishing a wind speed retrieval algorithm, UK TechDemoSat-1 (TDS-1) derived winds, from May 2015 to July 2017, are compared to the Advanced Scatterometer (ASCAT). ERA-Interim wind fields of the European Centre for Medium-range Weather Forecasts (ECMWF) and in situ observation from Tropical Atmosphere Ocean buoy array in the Pacific are taken as reference. One-year averaged TDS-1 global winds demonstrate small differences with ECMWF in a majority of areas as well as discuss under- and overestimations. The pioneering TDS-1 winds demonstrate a root-mean-squared error (RMSE) and bias of 2.77 and -0.33 m/s, which are comparable to the RMSE and bias derived by ASCAT winds, as large as 2.31 and 0.25 m/s, respectively. Using buoys measurements as reference, RMSE and bias of 2.23 and -0.03 m/s for TDS-1 as well as 1.40 and -0.68 m/s for ASCAT are obtained. Utilizing rain microwave-infrared estimates of the Tropical Rainfall Measuring Mission, rain-affected observation of both ASCAT and TDS-1 are collected and evaluated. Although ASCAT winds show a significant performance degradation resulting in an RMSE and bias of 3.16 and 1.03 m/s, respectively, during rain condition, TDS-1 shows a more reliable performance with an RMSE and bias of 2.94 and -0.21 m/s, respectively, which indicates the promising capability of GNSS forward scattering for wind retrievals during rain. A decrease in TDS-1-derived bistatic radar cross sections during rain events, at weak winds, is also demonstrated.
KW  - Advanced scatterometer (ASCAT)
KW  - European Centre for Medium-Range Weather Forecasts (ECMWF)
KW  - GNSS forward scatterometry
KW  - GNSS reflectometry
KW  - TechDemoSat-1 (TDS-1)
KW  - wind speed
Y1  - 2018
U6  - https://doi.org/10.1109/JSTARS.2018.2873241
SN  - 1939-1404
SN  - 2151-1535
VL  - 11
IS  - 11
SP  - 4534
EP  - 4541
PB  - Inst. of Electr. and Electronics Engineers
CY  - Piscataway
ER  - 
TY  - JOUR
A1  - Asgarimehr, Milad
A1  - Zavorotny, Valery
A1  - Wickert, Jens
A1  - Reich, Sebastian
T1  - Can GNSS Reflectometry Detect Precipitation Over Oceans?
JF  - Geophysical research letters
N2  - For the first time, a rain signature in Global Navigation Satellite System Reflectometry (GNSS-R) observations is demonstrated. Based on the argument that the forward quasi-specular scattering relies upon surface gravity waves with lengths larger than several wavelengths of the reflected signal, a commonly made conclusion is that the scatterometric GNSS-R measurements are not sensitive to the surface small-scale roughness generated by raindrops impinging on the ocean surface. On the contrary, this study presents an evidence that the bistatic radar cross section sigma(0) derived from TechDemoSat-1 data is reduced due to rain at weak winds, lower than approximate to 6 m/s. The decrease is as large as approximate to 0.7 dB at the wind speed of 3 m/s due to a precipitation of 0-2 mm/hr. The simulations based on the recently published scattering theory provide a plausible explanation for this phenomenon which potentially enables the GNSS-R technique to detect precipitation over oceans at low winds.
KW  - GNSS Reflectometry
KW  - rain detection
KW  - rain splash
KW  - TDS-1
KW  - ocean surface
KW  - electromagnetic scattering
Y1  - 2018
U6  - https://doi.org/10.1029/2018GL079708
SN  - 0094-8276
SN  - 1944-8007
VL  - 45
IS  - 22
SP  - 12585
EP  - 12592
PB  - American Geophysical Union
CY  - Washington
ER  - 
TY  - JOUR
A1  - Leung, Tsz Yan
A1  - Leutbecher, Martin
A1  - Reich, Sebastian
A1  - Shepherd, Theodore G.
T1  - Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier
JF  - Journal of the atmospheric sciences
N2  - 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.
KW  - Atmosphere
KW  - Turbulence
KW  - Error analysis
KW  - Spectral analysis
KW  - models
KW  - distribution
KW  - Numerical weather prediction
KW  - forecasting
Y1  - 2019
U6  - https://doi.org/10.1175/JAS-D-19-0057.1
SN  - 0022-4928
SN  - 1520-0469
VL  - 76
IS  - 12
SP  - 3883
EP  - 3892
PB  - American Meteorological Soc.
CY  - Boston
ER  - 
TY  - JOUR
A1  - Staniforth, Andrew
A1  - Wood, Nigel
A1  - Reich, Sebastian
T1  - A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations
JF  - Quarterly journal of the Royal Meteorological Society
N2  - 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.
KW  - regularization
KW  - temporal discretization
Y1  - 2006
U6  - https://doi.org/10.1256/qj.06.30
SN  - 0035-9009
VL  - 132
IS  - 621C
SP  - 3107
EP  - 3116
PB  - Wiley
CY  - Weinheim
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
T1  - Linearly implicit time stepping methods for numerical weather prediction
JF  - BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians
N2  - 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.
KW  - numerical weather prediction
KW  - linearly implicit time stepping methods
KW  - semi-Lagrangian method
KW  - Stormer-Verlet method
KW  - shallow-water equations
Y1  - 2006
U6  - https://doi.org/10.1007/s10543-006-0065-0
SN  - 0006-3835
VL  - 46
SP  - 607
EP  - 616
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Somogyvári, Márk
A1  - Reich, Sebastian
T1  - Convergence tests for transdimensional Markov chains in geoscience imaging
JF  - Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences
N2  - 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.
KW  - transdimensional inversion
KW  - MCMC modelling
KW  - convergence assessment
Y1  - 2019
U6  - https://doi.org/10.1007/s11004-019-09811-x
SN  - 1874-8961
SN  - 1874-8953
VL  - 52
IS  - 5
SP  - 651
EP  - 668
PB  - Springer
CY  - Heidelberg
ER  - 
TY  - JOUR
A1  - Taghvaei, Amirhossein
A1  - de Wiljes, Jana
A1  - Mehta, Prashant G.
A1  - Reich, Sebastian
T1  - Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem
JF  - Journal of dynamic systems measurement and control
N2  - 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.
Y1  - 2017
U6  - https://doi.org/10.1115/1.4037780
SN  - 0022-0434
SN  - 1528-9028
VL  - 140
IS  - 3
PB  - ASME
CY  - New York
ER  - 
TY  - JOUR
A1  - de Wiljes, Jana
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise
JF  - SIAM Journal on Applied Dynamical Systems
N2  - 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.
KW  - data assimilation
KW  - Kalman Bucy filter
KW  - ensemble Kalman filter
KW  - stability
KW  - accuracy
KW  - asymptotic behavior
Y1  - 2018
U6  - https://doi.org/10.1137/17M1119056
SN  - 1536-0040
VL  - 17
IS  - 2
SP  - 1152
EP  - 1181
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Acevedo, Walter
A1  - De Wiljes, Jana
A1  - Reich, Sebastian
T1  - Second-order accurate ensemble transform particle filters
JF  - SIAM journal on scientific computing
N2  - 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.
KW  - Bayesian inference
KW  - data assimilation
KW  - particle filter
KW  - ensemble Kalman filter
KW  - Sinkhorn approximation
Y1  - 2017
U6  - https://doi.org/10.1137/16M1095184
SN  - 1064-8275
SN  - 1095-7197
SN  - 2168-3417
VL  - 39
IS  - 5
SP  - A1834
EP  - A1850
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Frank, Jason
A1  - Moore, Brian E.
A1  - Reich, Sebastian
T1  - Linear PDEs and numerical methods that preserve a multisymplectic conservation law
N2  - Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]
Y1  - 2006
UR  - http://epubs.siam.org/sisc/
U6  - https://doi.org/10.1137/050628271
SN  - 1064-8275
ER  - 
TY  - JOUR
A1  - Shin, Seoleun
A1  - Sommer, Matthias
A1  - Reich, Sebastian
A1  - Névir, Peter
T1  - Evaluation of three spatial discretization schemes with the Galewsky et al. test
N2  - We evaluate the Hamiltonian particle methods (HPM) and the Nambu discretization applied to shallow-water equations on the sphere using the test suggested by Galewsky et al. (2004). Both simulations show excellent conservation of energy and are stable in long-term simulation. We repeat the test also using the ICOSWP scheme to compare with the two conservative spatial discretization schemes. The HPM simulation captures the main features of the reference solution, but wave 5 pattern is dominant in the simulations applied on the ICON grid with relatively low spatial resolutions. Nevertheless, agreement in statistics between the three schemes indicates their qualitatively similar behaviors in the long-term integration.
Y1  - 2010
UR  - http://www3.interscience.wiley.com/cgi-bin/jhome/106562719
U6  - https://doi.org/10.1002/Asl.279
SN  - 1530-261X
ER  - 
TY  - GEN
A1  - Ascher, Uri M.
A1  - Chin, Hongsheng
A1  - Reich, Sebastian
T1  - Stabilization of DAEs and invariant manifolds
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 030 
Y1  - 1994
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15625
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - Smoothed dynamics of highly oscillatory Hamiltonian systems
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 031 
Y1  - 1995
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15639
ER  - 
TY  - GEN
A1  - Leimkuhler, Benedict
A1  - Reich, Sebastian
T1  - Symplectic integration of constrained Hamiltonian systems
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 032 
KW  - differential-algebraic equations
KW  - constrained Hamiltonian systems
KW  - canonical discretization schemes
KW  - symplectic methods
Y1  - 1994
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15653
ER  - 
TY  - GEN
A1  - Ascher, Uri M.
A1  - Chin, Hongsheng
A1  - Petzold, Linda R.
A1  - Reich, Sebastian
T1  - Stabilization of constrained mechanical systems with DAEs and invariant manifolds
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 033 
Y1  - 1994
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15698
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - Algebrodifferentialgleichungen und Vektorfelder auf Mannigfaltigkeiten
N2  - In diesem Beitrag wird der Zusammenhang zwischen Algebrodifferentialgleichungen (ADGL) und Vektorfeldern auf Mannigfaltigkeiten untersucht. Dazu wird zunächst der Begriff der regulären ADGL eingeführt, wobei unter eirter regulären ADGL eine ADGL verstanden wird, deren Lösungsmenge identisch mit der Lösungsmenge eines Vektorfeldes ist. Ausgehend von bekannten Aussagen über die Lösungsmenge eines Vektorfeldes werden analoge Aussagen für die Lösungsmenge einer regulären ADGL abgeleitet. Es wird eine Reduktionsmethode angegeben, die auf ein Kriterium für die Begularität einer ADGL und auf die Definition des Index einer nichtlinearen ADGL führt. Außerdem wird gezeigt, daß beliebige Vektorfelder durch reguläre ADGL so realisiert werden können, daß die Lösungsmenge des Vektorfeldes mit der der realisierenden ADGL identisch ist. Abschließend werden die für autonome ADGL gewonnenen Aussagen auf den Fall der nichtautonomen ADGL übertragen.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 160 
Y1  - 1980
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-47290
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - Differential-algebraic equations and applications in circuit theory
N2  - 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.
N2  - Die mathematische Modellierung technisch physikalischer Systeme wie elektrische Netzwerke, führt häufig auf ein System von Differentialgleichungen und nichtlinearen impliziten Gleichungen sogenannten Algebrodifferentialgleichungen (ADGL). Es zeigt sich, daß die numerischen und analytischen Eigenschaften von ADGL durch den Index des Problems charakterisiert werden können. Insbesondere können die bekannten Integrationsformeln von Gear im allgemeinen nur auf ADGL mit dem Index eins angewendet werden. In diesem Beitrag wird eine geometrische Interpretation von ADGL mit einem höheren Index gegeben sowie auf Probleme im Zusammenhang mit derartigen ADGL an Hand verschiedener Beispiele hingewiesen.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 156 
Y1  - 1992
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46646
ER  - 
TY  - GEN
A1  - Asgarimehr, Milad
A1  - Wickert, Jens
A1  - Reich, Sebastian
T1  - Evaluating impact of rain attenuation on space-borne GNSS Reflectometry wind speeds
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - The novel space-borne Global Navigation Satellite System Reflectometry (GNSS-R) technique has recently shown promise in monitoring the ocean state and surface wind speed with high spatial coverage and unprecedented sampling rate. The L-band signals of GNSS are structurally able to provide a higher quality of observations from areas covered by dense clouds and under intense precipitation, compared to those signals at higher frequencies from conventional ocean scatterometers. As a result, studying the inner core of cyclones and improvement of severe weather forecasting and cyclone tracking have turned into the main objectives of GNSS-R satellite missions such as Cyclone Global Navigation Satellite System (CYGNSS). Nevertheless, the rain attenuation impact on GNSS-R wind speed products is not yet well documented. Evaluating the rain attenuation effects on this technique is significant since a small change in the GNSS-R can potentially cause a considerable bias in the resultant wind products at intense wind speeds. Based on both empirical evidence and theory, wind speed is inversely proportional to derived bistatic radar cross section with a natural logarithmic relation, which introduces high condition numbers (similar to ill-posed conditions) at the inversions to high wind speeds. This paper presents an evaluation of the rain signal attenuation impact on the bistatic radar cross section and the derived wind speed. This study is conducted simulating GNSS-R delay-Doppler maps at different rain rates and reflection geometries, considering that an empirical data analysis at extreme wind intensities and rain rates is impossible due to the insufficient number of observations from these severe conditions. Finally, the study demonstrates that at a wind speed of 30 m/s and incidence angle of 30 degrees, rain at rates of 10, 15, and 20 mm/h might cause overestimation as large as approximate to 0.65 m/s (2%), 1.00 m/s (3%), and 1.3 m/s (4%), respectively, which are still smaller than the CYGNSS required uncertainty threshold. The simulations are conducted in a pessimistic condition (severe continuous rainfall below the freezing height and over the entire glistening zone) and the bias is expected to be smaller in size in real environments.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1132 
KW  - GNSS Reflectometry
KW  - wind speed
KW  - rain effect
KW  - rain attenuation
KW  - DDM simulation
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-473441
SN  - 1866-8372
IS  - 1132
ER  - 
TY  - GEN
A1  - Nüsken, Nikolas
A1  - Reich, Sebastian
A1  - Rozdeba, Paul J.
T1  - State and parameter estimation from observed signal increments
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 916 
KW  - parameter estimation
KW  - continuous-time data assimilation
KW  - ensemble Kalman filter
KW  - correlated noise
KW  - multi-scale diffusion processes
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-442609
SN  - 1866-8372
IS  - 916
ER  - 
TY  - JOUR
A1  - van Leeuwen, Peter Jan
A1  - Kunsch, Hans R.
A1  - Nerger, Lars
A1  - Potthast, Roland
A1  - Reich, Sebastian
T1  - Particle filters for high-dimensional geoscience applications: A review
JF  - Quarterly journal of the Royal Meteorological Society
N2  - Particle filters contain the promise of fully nonlinear data assimilation. They have been applied in numerous science areas, including the geosciences, but their application to high-dimensional geoscience systems has been limited due to their inefficiency in high-dimensional systems in standard settings. However, huge progress has been made, and this limitation is disappearing fast due to recent developments in proposal densities, the use of ideas from (optimal) transportation, the use of localization and intelligent adaptive resampling strategies. Furthermore, powerful hybrids between particle filters and ensemble Kalman filters and variational methods have been developed. We present a state-of-the-art discussion of present efforts of developing particle filters for high-dimensional nonlinear geoscience state-estimation problems, with an emphasis on atmospheric and oceanic applications, including many new ideas, derivations and unifications, highlighting hidden connections, including pseudo-code, and generating a valuable tool and guide for the community. Initial experiments show that particle filters can be competitive with present-day methods for numerical weather prediction, suggesting that they will become mainstream soon.
KW  - hybrids
KW  - localization
KW  - nonlinear data assimilation
KW  - particle filters
KW  - proposal densities
Y1  - 2019
U6  - https://doi.org/10.1002/qj.3551
SN  - 0035-9009
SN  - 1477-870X
VL  - 145
IS  - 723
SP  - 2335
EP  - 2365
PB  - Wiley
CY  - Hoboken
ER  - 
TY  - BOOK
A1  - Van Leeuwen, Peter Jan
A1  - Cheng, Yuan
A1  - Reich, Sebastian
T1  - Nonlinear data assimilation
T3  - Frontiers in applied dynamical systems: reviews and tutorials ; 2
N2  - 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.
Y1  - 2015
SN  - 978-3-319-18346-6
SN  - 978-3-319-18347-3
U6  - https://doi.org/10.1007/978-3-319-18347-3
PB  - Springer
CY  - Cham
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
T1  - Data assimilation
BT  - the Schrödinger perspective
JF  - Acta numerica
N2  - 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 Schrödinger’s boundary value problem for stochastic processes in particular.
Y1  - 2019
U6  - https://doi.org/10.1017/S0962492919000011
SN  - 0962-4929
SN  - 1474-0508
VL  - 28
SP  - 635
EP  - 711
PB  - Cambridge Univ. Press
CY  - New York
ER  - 
TY  - JOUR
A1  - Nüsken, Nikolas
A1  - Reich, Sebastian
A1  - Rozdeba, Paul J.
T1  - State and parameter estimation from observed signal increments
JF  - Entropy : an international and interdisciplinary journal of entropy and information studies
N2  - 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.
KW  - parameter estimation
KW  - continuous-time data assimilation
KW  - ensemble Kalman filter
KW  - correlated noise
KW  - multi-scale diffusion processes
Y1  - 2019
U6  - https://doi.org/10.3390/e21050505
SN  - 1099-4300
VL  - 21
IS  - 5
PB  - MDPI
CY  - Basel
ER  - 
TY  - JOUR
A1  - Asgarimehr, Milad
A1  - Wickert, Jens
A1  - Reich, Sebastian
T1  - Evaluating impact of rain attenuation on space-borne GNSS reflectometry wind speeds
JF  - Remote Sensing
N2  - The novel space-borne Global Navigation Satellite System Reflectometry (GNSS-R) technique has recently shown promise in monitoring the ocean state and surface wind speed with high spatial coverage and unprecedented sampling rate. The L-band signals of GNSS are structurally able to provide a higher quality of observations from areas covered by dense clouds and under intense precipitation, compared to those signals at higher frequencies from conventional ocean scatterometers. As a result, studying the inner core of cyclones and improvement of severe weather forecasting and cyclone tracking have turned into the main objectives of GNSS-R satellite missions such as Cyclone Global Navigation Satellite System (CYGNSS). Nevertheless, the rain attenuation impact on GNSS-R wind speed products is not yet well documented. Evaluating the rain attenuation effects on this technique is significant since a small change in the GNSS-R can potentially cause a considerable bias in the resultant wind products at intense wind speeds. Based on both empirical evidence and theory, wind speed is inversely proportional to derived bistatic radar cross section with a natural logarithmic relation, which introduces high condition numbers (similar to ill-posed conditions) at the inversions to high wind speeds. This paper presents an evaluation of the rain signal attenuation impact on the bistatic radar cross section and the derived wind speed. This study is conducted simulating GNSS-R delay-Doppler maps at different rain rates and reflection geometries, considering that an empirical data analysis at extreme wind intensities and rain rates is impossible due to the insufficient number of observations from these severe conditions. Finally, the study demonstrates that at a wind speed of 30 m/s and incidence angle of 30 degrees, rain at rates of 10, 15, and 20 mm/h might cause overestimation as large as approximate to 0.65 m/s (2%), 1.00 m/s (3%), and 1.3 m/s (4%), respectively, which are still smaller than the CYGNSS required uncertainty threshold. The simulations are conducted in a pessimistic condition (severe continuous rainfall below the freezing height and over the entire glistening zone) and the bias is expected to be smaller in size in real environments.
KW  - GNSS Reflectometry
KW  - wind speed
KW  - rain effect
KW  - rain attenuation
KW  - DDM simulation
Y1  - 2019
U6  - https://doi.org/10.3390/rs11091048
SN  - 2072-4292
VL  - 11
IS  - 9
PB  - MDPI
CY  - Basel
ER  - 
TY  - JOUR
A1  - Garbuno-Inigo, Alfredo
A1  - Nüsken, Nikolas
A1  - Reich, Sebastian
T1  - Affine invariant interacting Langevin dynamics for Bayesian inference
JF  - SIAM journal on applied dynamical systems
N2  - 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.
KW  - Langevin dynamics
KW  - interacting particle systems
KW  - Bayesian inference
KW  - gradient flow
KW  - multiplicative noise
KW  - affine invariance
KW  - gradient-free
Y1  - 2020
U6  - https://doi.org/10.1137/19M1304891
SN  - 1536-0040
VL  - 19
IS  - 3
SP  - 1633
EP  - 1658
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - GEN
A1  - Acevedo, Walter
A1  - Reich, Sebastian
A1  - Cubasch, Ulrich
T1  - Towards the assimilation of tree-ring-width records using ensemble Kalman filtering techniques
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
N2  - This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for tree-ring-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 892 
KW  - proxy forward modeling
KW  - data assimilation
KW  - fuzzy logic
KW  - ensemble Kalman filter
KW  - paleoclimate reconstruction
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436363
SN  - 1866-8372
VL  - 46
IS  - 892
SP  - 1909
EP  - 1920
ER  - 
TY  - JOUR
A1  - Maoutsa, Dimitra
A1  - Reich, Sebastian
A1  - Opper, Manfred
T1  - Interacting particle solutions of Fokker–Planck equations through gradient–log–density estimation
JF  - Entropy
N2  - 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.
KW  - stochastic systems
KW  - Fokker-Planck equation
KW  - interacting particles
KW  - multiplicative noise
KW  - gradient flow
KW  - stochastic differential equations
Y1  - 2020
U6  - https://doi.org/10.3390/e22080802
SN  - 1099-4300
VL  - 22
IS  - 8
PB  - MDPI
CY  - Basel
ER  - 
TY  - JOUR
A1  - Akhmatskaya, Elena
A1  - Bou-Rabee, Nawaf
A1  - Reich, Sebastian
T1  - A comparison of generalized hybrid Monte Carlo methods with and without momentum flip
N2  - 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.
Y1  - 2009
UR  - http://www.sciencedirect.com/science/journal/00219991
U6  - https://doi.org/10.1016/j.jcp.2008.12.014
SN  - 0021-9991
ER  - 
TY  - JOUR
A1  - Akhmatskaya, Elena
A1  - Bou-Rabee, Nawaf
A1  - Reich, Sebastian
T1  - Erratum to "A comparison of generalized hybrid Monte Carlo methods with and without momentum flip" [J. Comput. Phys. 228 (2009), S. 2256 - 2265]
N2  - 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.
Y1  - 2009
UR  - http://www.sciencedirect.com/science/journal/00219991
U6  - https://doi.org/10.1016/j.jcp.2009.06.039
SN  - 0021-9991
ER  - 
TY  - JOUR
A1  - Leimkuhler, Benedict
A1  - Reich, Sebastian
T1  - A metropolis adjusted Nosé-Hoover thermostat
N2  - 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.
Y1  - 2009
UR  - http://www.edpsciences.org/journal/index.cfm?edpsname=m2an
U6  - https://doi.org/10.1051/M2an/2009023
SN  - 0764-583X
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
T1  - A nonparametric ensemble transform method for bayesian inference
JF  - SIAM journal on scientific computing
N2  - 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.
KW  - Bayesian inference
KW  - Monte Carlo method
KW  - sequential data assimilation
KW  - linear programming
KW  - resampling
Y1  - 2013
U6  - https://doi.org/10.1137/130907367
SN  - 1064-8275
VL  - 35
IS  - 4
SP  - A2013
EP  - A2024
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Gottwald, Georg
A1  - Reich, Sebastian
T1  - Ensemble propagation and continuous matrix factorization algorithms
N2  - 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.
Y1  - 2009
UR  - http://onlinelibrary.wiley.com/journal/10.1002/%28ISSN%291477-870X
U6  - https://doi.org/10.1002/qj.457
SN  - 0035-9009
ER  - 
TY  - JOUR
A1  - Cotter, Colin J.
A1  - Ham, David A.
A1  - Pain, Christopher C.
A1  - Reich, Sebastian
T1  - LBB stability of a mixed Galerkin finite element pair for fluid flow simulations
N2  - 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.
Y1  - 2009
UR  - http://www.sciencedirect.com/science/journal/00219991
U6  - https://doi.org/10.1016/j.jcp.2008.09.014
SN  - 0021-9991
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Reich, Sebastian
T1  - A localization technique for ensemble Kalman filters
N2  - 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.
Y1  - 2010
UR  - http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1477-870X
U6  - https://doi.org/10.1002/Qj.591
SN  - 0035-9009
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Reich, Sebastian
T1  - A mollified ensemble Kalman filter
N2  - 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.
Y1  - 2010
UR  - http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1477-870X
U6  - https://doi.org/10.1002/Qj.672
SN  - 0035-9009
ER  - 
TY  - JOUR
A1  - Bridges, Thomas J.
A1  - Reich, Sebastian
T1  - Numerical methods for Hamiltonian PDEs
N2  - The paper provides an introduction and survey of conservative discretization methods for Hamiltonian partial differential equations. The emphasis is on variational, symplectic and multi-symplectic methods. The derivation of methods as well as some of their fundamental geometric properties are discussed. Basic principles are illustrated by means of examples from wave and fluid dynamics
Y1  - 2006
UR  - http://iopscience.iop.org/1751-8121/
U6  - https://doi.org/10.1088/0305-4470/39/19/S02
SN  - 0305-4470
ER  - 
TY  - JOUR
A1  - Shin, Seoleun
A1  - Zöller, Gert
A1  - Holschneider, Matthias
A1  - Reich, Sebastian
T1  - A multigrid solver for modeling complex interseismic stress fields
JF  - Computers & geosciences : an international journal devoted to the publication of papers on all aspects of geocomputation and to the distribution of computer programs and test data sets ; an official journal of the International Association for Mathematical Geology
N2  - We develop a multigrid, multiple time stepping scheme to reduce computational efforts for calculating complex stress interactions in a strike-slip 2D planar fault for the simulation of seismicity. The key elements of the multilevel solver are separation of length scale, grid-coarsening, and hierarchy. In this study the complex stress interactions are split into two parts: the first with a small contribution is computed on a coarse level, and the rest for strong interactions is on a fine level. This partition leads to a significant reduction of the number of computations. The reduction of complexity is even enhanced by combining the multigrid with multiple time stepping. Computational efficiency is enhanced by a factor of 10 while retaining a reasonable accuracy, compared to the original full matrix-vortex multiplication. The accuracy of solution and computational efficiency depend on a given cut-off radius that splits multiplications into the two parts. The multigrid scheme is constructed in such a way that it conserves stress in the entire half-space.
KW  - Multigrid
KW  - Multiple time stepping
KW  - Strike-slip fault model
Y1  - 2011
U6  - https://doi.org/10.1016/j.cageo.2010.11.011
SN  - 0098-3004
VL  - 37
IS  - 8
SP  - 1075
EP  - 1082
PB  - Elsevier
CY  - Oxford
ER  - 
TY  - JOUR
A1  - Grewe, Volker
A1  - Brinkop, Sabine
A1  - Joeckel, Patrick
A1  - Shin, Seoleun
A1  - Reich, Sebastian
A1  - Yserentant, Harry
T1  - On the theory of mass conserving transformations for Lagrangian methods in 3D atmosphere-chemistry models
JF  - Meteorologische Zeitschrift
KW  - Lagrangian modelling
KW  - chemistry
KW  - transformations
Y1  - 2014
U6  - https://doi.org/10.1127/0941-2948/2014/0552
SN  - 0941-2948
SN  - 1610-1227
VL  - 23
IS  - 4
SP  - 441
EP  - 447
PB  - Schweizerbart
CY  - Stuttgart
ER  - 
TY  - JOUR
A1  - Shin, Seoleun
A1  - Reich, Sebastian
A1  - Frank, Jason
T1  - Hydrostatic Hamiltonian particle-mesh (HPM) methods for atmospheric modelling
JF  - Quarterly journal of the Royal Meteorological Society
N2  - We develop a hydrostatic Hamiltonian particle-mesh (HPM) method for efficient long-term numerical integration of the atmosphere. In the HPM method, the hydrostatic approximation is interpreted as a holonomic constraint for the vertical position of particles. This can be viewed as defining a set of vertically buoyant horizontal meshes, with the altitude of each mesh point determined so as to satisfy the hydrostatic balance condition and with particles modelling horizontal advection between the moving meshes. We implement the method in a vertical-slice model and evaluate its performance for the simulation of idealized linear and nonlinear orographic flow in both dry and moist environments. The HPM method is able to capture the basic features of the gravity wave to a degree of accuracy comparable with that reported in the literature. The numerical solution in the moist experiment indicates that the influence of moisture on wave characteristics is represented reasonably well and the reduction of momentum flux is in good agreement with theoretical analysis.
KW  - conservative discretization
KW  - Lagrangian modeling
KW  - holonomic constraints
KW  - fluid mechanics
Y1  - 2012
U6  - https://doi.org/10.1002/qj.982
SN  - 0035-9009
VL  - 138
IS  - 666
SP  - 1388
EP  - 1399
PB  - Wiley-Blackwell
CY  - Hoboken
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
T1  - A dynamical systems framework for intermittent data assimilation
JF  - BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians
N2  - We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.
KW  - Data assimilation
KW  - Ensemble Kalman filter
KW  - Dynamical systems
KW  - Nonlinear filters
KW  - Optimal transportation
Y1  - 2011
U6  - https://doi.org/10.1007/s10543-010-0302-4
SN  - 0006-3835
VL  - 51
IS  - 1
SP  - 235
EP  - 249
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Skeel, R. D.
A1  - Reich, Sebastian
T1  - Corrected potential energy functions for constrained molecular dynamics
JF  - European physical journal special topics
N2  - Atomic oscillations present in classical molecular dynamics restrict the step size that can be used. Multiple time stepping schemes offer only modest improvements, and implicit integrators are costly and inaccurate. The best approach may be to actually remove the highest frequency oscillations by constraining bond lengths and bond angles, thus permitting perhaps a 4-fold increase in the step size. However, omitting degrees of freedom produces errors in statistical averages, and rigid angles do not bend for strong excluded volume forces. These difficulties can be addressed by an enhanced treatment of holonomic constrained dynamics using ideas from papers of Fixman (1974) and Reich (1995, 1999). In particular, the 1995 paper proposes the use of "flexible" constraints, and the 1999 paper uses a modified potential energy function with rigid constraints to emulate flexible constraints. Presented here is a more direct and rigorous derivation of the latter approach, together with justification for the use of constraints in molecular modeling. With rigor comes limitations, so practical compromises are proposed: simplifications of the equations and their judicious application when assumptions are violated. Included are suggestions for new approaches.
Y1  - 2011
U6  - https://doi.org/10.1140/epjst/e2011-01518-8
SN  - 1951-6355
VL  - 200
IS  - 1
SP  - 55
EP  - 72
PB  - Springer
CY  - Heidelberg
ER  - 
TY  - JOUR
A1  - Amezcua, Javier
A1  - Ide, Kayo
A1  - Kalnay, Eugenia
A1  - Reich, Sebastian
T1  - Ensemble transform Kalman-Bucy filters
JF  - Quarterly journal of the Royal Meteorological Society
N2  - Two recent works have adapted the Kalman-Bucy filter into an ensemble setting. In the first formulation, the ensemble of perturbations is updated by the solution of an ordinary differential equation (ODE) in pseudo-time, while the mean is updated as in the standard Kalman filter. In the second formulation, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-time. Neither requires matrix inversions except for the frequently diagonal observation error covariance.
 We analyse the behaviour of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background error to observational error variance, and that using the integration scheme proposed in both formulations can lead to failure. A numerical integration scheme that is both stable and is not computationally expensive is proposed. We develop transform-based alternatives for these Bucy-type approaches so that the integrations are computed in ensemble space where the variables are weights (of dimension equal to the ensemble size) rather than model variables.
 Finally, the performance of our ensemble transform Kalman-Bucy implementations is evaluated using three models: the 3-variable Lorenz 1963 model, the 40-variable Lorenz 1996 model, and a medium complexity atmospheric general circulation model known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques.
KW  - Kalman-Bucy Filter
KW  - Ensemble Kalman Filter
KW  - stiff ODE
KW  - weight-based formulations
Y1  - 2014
U6  - https://doi.org/10.1002/qj.2186
SN  - 0035-9009
SN  - 1477-870X
VL  - 140
IS  - 680
SP  - 995
EP  - 1004
PB  - Wiley-Blackwell
CY  - Hoboken
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Reich, Sebastian
T1  - An ensemble Kalman-Bucy filter for continuous data assimilation
JF  - Meteorologische Zeitschrift
N2  - The ensemble Kalman filter has emerged as a promising filter algorithm for nonlinear differential equations subject to intermittent observations. In this paper, we extend the well-known Kalman-Bucy filter for linear differential equations subject to continous observations to the ensemble setting and nonlinear differential equations. The proposed filter is called the ensemble Kalman-Bucy filter and its performance is demonstrated for a simple mechanical model (Langevin dynamics) subject to incremental observations of its velocity.
Y1  - 2012
U6  - https://doi.org/10.1127/0941-2948/2012/0307
SN  - 0941-2948
VL  - 21
IS  - 3
SP  - 213
EP  - 219
PB  - Schweizerbart
CY  - Stuttgart
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
T1  - A Gaussian-mixture ensemble transform filter
JF  - Quarterly journal of the Royal Meteorological Society
N2  - 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.
KW  - data assimilation
KW  - ensemble Kalman filter
KW  - nonlinear filtering
KW  - Gaussian mixtures
KW  - Gaussian kernel estimators
Y1  - 2012
U6  - https://doi.org/10.1002/qj.898
SN  - 0035-9009
VL  - 138
IS  - 662
SP  - 222
EP  - 233
PB  - Wiley-Blackwell
CY  - Malden
ER  - 
TY  - JOUR
A1  - Acevedo, Walter
A1  - Reich, Sebastian
A1  - Cubasch, Ulrich
T1  - Towards the assimilation of tree-ring-width records using ensemble Kalman filtering techniques
JF  - Climate dynamics : observational, theoretical and computational research on the climate system
N2  - This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for tree-ring-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes.
KW  - Proxy forward modeling
KW  - Data assimilation
KW  - Fuzzy logic
KW  - Ensemble Kalman filter
KW  - Paleoclimate reconstruction
Y1  - 2016
U6  - https://doi.org/10.1007/s00382-015-2683-1
SN  - 0930-7575
SN  - 1432-0894
VL  - 46
SP  - 1909
EP  - 1920
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Gregory, A.
A1  - Cotter, C. J.
A1  - Reich, Sebastian
T1  - MULTILEVEL ENSEMBLE TRANSFORM PARTICLE FILTERING
JF  - SIAM journal on scientific computing
N2  - This paper extends the multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles.
KW  - multilevel Monte Carlo
KW  - sequential data assimilation
KW  - optimal transport
Y1  - 2016
U6  - https://doi.org/10.1137/15M1038232
SN  - 1064-8275
SN  - 1095-7197
VL  - 38
SP  - A1317
EP  - A1338
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Escribano, Bruno
A1  - Akhmatskaya, Elena
A1  - Reich, Sebastian
A1  - Azpiroz, Jon M.
T1  - Multiple-time-stepping generalized hybrid Monte Carlo methods
JF  - Journal of computational physics
N2  - Performance of the generalized shadow hybrid Monte Carlo (GSHMC) method [1], which proved to be superior in sampling efficiency over its predecessors [2-4], molecular dynamics and hybrid Monte Carlo, can be further improved by combining it with multi-time-stepping (MTS) and mollification of slow forces. We demonstrate that the comparatively simple modifications of the method not only lead to better performance of GSHMC itself but also allow for beating the best performed methods, which use the similar force splitting schemes. In addition we show that the same ideas can be successfully applied to the conventional generalized hybrid Monte Carlo method (GHMC). The resulting methods, MTS-GHMC and MTS-GSHMC, provide accurate reproduction of thermodynamic and dynamical properties, exact temperature control during simulation and computational robustness and efficiency. MTS-GHMC uses a generalized momentum update to achieve weak stochastic stabilization to the molecular dynamics (MD) integrator. MTS-GSHMC adds the use of a shadow (modified) Hamiltonian to filter the MD trajectories in the HMC scheme. We introduce a new shadow Hamiltonian formulation adapted to force-splitting methods. The use of such Hamiltonians improves the acceptance rate of trajectories and has a strong impact on the sampling efficiency of the method. Both methods were implemented in the open-source MD package ProtoMol and were tested on a water and a protein systems. Results were compared to those obtained using a Langevin Molly (LM) method [5] on the same systems. The test results demonstrate the superiority of the new methods over LM in terms of stability, accuracy and sampling efficiency. This suggests that putting the MTS approach in the framework of hybrid Monte Carlo and using the natural stochasticity offered by the generalized hybrid Monte Carlo lead to improving stability of MTS and allow for achieving larger step sizes in the simulation of complex systems.
KW  - Force splitting
KW  - Mollification
KW  - Generalized hybrid Monte Carlo
KW  - Molecular dynamics
KW  - Modified Hamiltonians
Y1  - 2015
U6  - https://doi.org/10.1016/j.jcp.2014.08.052
SN  - 0021-9991
SN  - 1090-2716
VL  - 280
SP  - 1
EP  - 20
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Aizinger, Vadym
A1  - Korn, Peter
A1  - Giorgetta, Marco
A1  - Reich, Sebastian
T1  - Large-scale turbulence modelling via alpha-regularisation for atmospheric simulations
JF  - Journal of turbulence
N2  - We study the possibility of obtaining a computational turbulence model by means of non-dissipative regularisation of the compressible atmospheric equations for climate-type applications. We use an -regularisation (Lagrangian averaging) of the atmospheric equations. For the hydrostatic and compressible atmospheric equations discretised using a finite volume method on unstructured grids, deterministic and non-deterministic numerical experiments are conducted to compare the individual solutions and the statistics of the regularised equations to those of the original model. The impact of the regularisation parameter is investigated. Our results confirm the principal compatibility of -regularisation with atmospheric dynamics and encourage further investigations within atmospheric model including complex physical parametrisations.
KW  - hydrostatic atmosphere
KW  - non-dissipative regularisations
KW  - Lagrangian-averaged equations
Y1  - 2015
U6  - https://doi.org/10.1080/14685248.2014.991443
SN  - 1468-5248
VL  - 16
IS  - 4
SP  - 367
EP  - 391
PB  - Routledge, Taylor & Francis Group
CY  - Abingdon
ER  - 
TY  - GEN
A1  - Acevedo, Walter
A1  - Fallah, Bijan
A1  - Reich, Sebastian
A1  - Cubasch, Ulrich
T1  - Assimilation of pseudo-tree-ring-width observations into an atmospheric general circulation model
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
N2  - Paleoclimate data assimilation (DA) is a promising technique to systematically combine the information from climate model simulations and proxy records. Here, we investigate the assimilation of tree-ring-width (TRW) chronologies into an atmospheric global climate model using ensemble Kalman filter (EnKF) techniques and a process-based tree-growth forward model as an observation operator. Our results, within a perfect-model experiment setting, indicate that the "online DA" approach did not outperform the "off-line" one, despite its considerable additional implementation complexity. On the other hand, it was observed that the nonlinear response of tree growth to surface temperature and soil moisture does deteriorate the operation of the time-averaged EnKF methodology. Moreover, for the first time we show that this skill loss appears significantly sensitive to the structure of the growth rate function, used to represent the principle of limiting factors (PLF) within the forward model. In general, our experiments showed that the error reduction achieved by assimilating pseudo-TRW chronologies is modulated by the magnitude of the yearly internal variability in themodel. This result might help the dendrochronology community to optimize their sampling efforts.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 627 
KW  - high resolution paleoclimatology
KW  - sparse proxy data
KW  - climate reconstructions
KW  - limiting factors
KW  - Kalman filter
KW  - co-limitation
KW  - ensemble
KW  - variability
KW  - reanalysis
KW  - framework
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-418743
SN  - 1866-8372
IS  - 627
SP  - 545
EP  - 557
ER  - 
TY  - JOUR
A1  - Acevedo, Walter
A1  - Fallah, Bijan
A1  - Reich, Sebastian
A1  - Cubasch, Ulrich
T1  - Assimilation of pseudo-tree-ring-width observations into an atmospheric general circulation model
JF  - Climate of the past : an interactive open access journal of the European Geosciences Union
N2  - Paleoclimate data assimilation (DA) is a promising technique to systematically combine the information from climate model simulations and proxy records. Here, we investigate the assimilation of tree-ring-width (TRW) chronologies into an atmospheric global climate model using ensemble Kalman filter (EnKF) techniques and a process-based tree-growth forward model as an observation operator. Our results, within a perfect-model experiment setting, indicate that the "online DA" approach did not outperform the "off-line" one, despite its considerable additional implementation complexity. On the other hand, it was observed that the nonlinear response of tree growth to surface temperature and soil moisture does deteriorate the operation of the time-averaged EnKF methodology. Moreover, for the first time we show that this skill loss appears significantly sensitive to the structure of the growth rate function, used to represent the principle of limiting factors (PLF) within the forward model. In general, our experiments showed that the error reduction achieved by assimilating pseudo-TRW chronologies is modulated by the magnitude of the yearly internal variability in themodel. This result might help the dendrochronology community to optimize their sampling efforts.
Y1  - 2017
U6  - https://doi.org/10.5194/cp-13-545-2017
SN  - 1814-9324
SN  - 1814-9332
VL  - 13
SP  - 545
EP  - 557
PB  - Copernicus
CY  - Göttingen
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - On a geometrical interpretation of differential-algebraic equations
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 157 
Y1  - 1990
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46683
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - Momentum conserving symplectic integrators
N2  - 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).
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 044 
Y1  - 1994
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-16824
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - On the local qualitative behavior of differential-algebraic equations
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 159 
Y1  - 1995
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46739
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - On an existence and uniqueness theory for nonlinear differential-algebraic equations
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 158 
Y1  - 1991
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46706
ER  - 
TY  - JOUR
A1  - Gottwald, Georg A.
A1  - Reich, Sebastian
T1  - Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations
JF  - Chaos : an interdisciplinary journal of nonlinear science
N2  - 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.
Y1  - 2021
U6  - https://doi.org/10.1063/5.0066080
SN  - 1054-1500
SN  - 1089-7682
VL  - 31
IS  - 10
PB  - AIP
CY  - Melville
ER  - 
TY  - JOUR
A1  - Gottwald, Georg A.
A1  - Mitchell, Lewis
A1  - Reich, Sebastian
T1  - Controlling overestimation of error covariance in ensemble kalman filters with sparse observations a variance-limiting kalman filter
JF  - Monthly weather review
N2  - The problem of an ensemble Kalman filter when only partial observations are available is considered. In particular, the situation is investigated where the observational space consists of variables that are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. To limit the variance of the latter poorly resolved variables a variance-limiting Kalman filter (VLKF) is derived in a variational setting. The VLKF for a simple linear toy model is analyzed and its range of optimal performance is determined. The VLKF is explored in an ensemble transform setting for the Lorenz-96 system, and it is shown that incorporating the information of the variance of some unobservable variables can improve the skill and also increase the stability of the data assimilation procedure.
Y1  - 2011
U6  - https://doi.org/10.1175/2011MWR3557.1
SN  - 0027-0644
VL  - 139
IS  - 8
SP  - 2650
EP  - 2667
PB  - American Meteorological Soc.
CY  - Boston
ER  - 
TY  - JOUR
A1  - Gottwald, Georg A.
A1  - Reich, Sebastian
T1  - Supervised learning from noisy observations
BT  - Combining machine-learning techniques with data assimilation
JF  - Physica : D, Nonlinear phenomena
N2  - 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.
KW  - Data-driven modelling
KW  - Random feature maps
KW  - Data assimilation
Y1  - 2021
U6  - https://doi.org/10.1016/j.physd.2021.132911
SN  - 0167-2789
SN  - 1872-8022
VL  - 423
PB  - Elsevier
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Engbert, Ralf
A1  - Rabe, Maximilian Michael
A1  - Kliegl, Reinhold
A1  - Reich, Sebastian
T1  - Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics
JF  - Bulletin of mathematical biology : official journal of the Society for Mathematical Biology
N2  - 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.
KW  - Stochastic epidemic model
KW  - Sequential data assimilation
KW  - Ensemble Kalman
KW  - filter
KW  - COVID-19
Y1  - 2020
U6  - https://doi.org/10.1007/s11538-020-00834-8
SN  - 0092-8240
SN  - 1522-9602
VL  - 83
IS  - 1
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Schütt, Heiko Herbert
A1  - Rothkegel, Lars Oliver Martin
A1  - Trukenbrod, Hans Arne
A1  - Reich, Sebastian
A1  - Wichmann, Felix A.
A1  - Engbert, Ralf
T1  - Likelihood-based parameter estimation and comparison of dynamical cognitive models
JF  - Psychological Review
N2  - Dynamical models of cognition play an increasingly important role in driving theoretical and experimental research in psychology. Therefore, parameter estimation, model analysis and comparison of dynamical models are of essential importance. In this article, we propose a maximum likelihood approach for model analysis in a fully dynamical framework that includes time-ordered experimental data. Our methods can be applied to dynamical models for the prediction of discrete behavior (e.g., movement onsets); in particular, we use a dynamical model of saccade generation in scene viewing as a case study for our approach. For this model, the likelihood function can be computed directly by numerical simulation, which enables more efficient parameter estimation including Bayesian inference to obtain reliable estimates and corresponding credible intervals. Using hierarchical models inference is even possible for individual observers. Furthermore, our likelihood approach can be used to compare different models. In our example, the dynamical framework is shown to outperform nondynamical statistical models. Additionally, the likelihood based evaluation differentiates model variants, which produced indistinguishable predictions on hitherto used statistics. Our results indicate that the likelihood approach is a promising framework for dynamical cognitive models.
KW  - likelihood
KW  - model fitting
KW  - dynamical model
KW  - eye movements
KW  - model comparison
Y1  - 2017
U6  - https://doi.org/10.1037/rev0000068
SN  - 0033-295X
SN  - 1939-1471
VL  - 124
IS  - 4
SP  - 505
EP  - 524
PB  - American Psychological Association
CY  - Washington
ER  - 
TY  - JOUR
A1  - Engbert, Ralf
A1  - Rabe, Maximilian Michael
A1  - Schwetlick, Lisa
A1  - Seelig, Stefan A.
A1  - Reich, Sebastian
A1  - Vasishth, Shravan
T1  - Data assimilation in dynamical cognitive science
JF  - Trends in cognitive sciences
N2  - Dynamical models make specific assumptions about cognitive processes that generate human behavior. In data assimilation, these models are tested against timeordered data. Recent progress on Bayesian data assimilation demonstrates that this approach combines the strengths of statistical modeling of individual differences with the those of dynamical cognitive models.
Y1  - 2022
U6  - https://doi.org/10.1016/j.tics.2021.11.006
SN  - 1364-6613
SN  - 1879-307X
VL  - 26
IS  - 2
SP  - 99
EP  - 102
PB  - Elsevier
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
A1  - Weissmann, Simon
T1  - Fokker-Planck particle systems for Bayesian inference: computational approaches
JF  - SIAM ASA journal on uncertainty quantification
N2  - 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.
KW  - Bayesian inverse problems
KW  - Fokker-Planck equation
KW  - gradient flow
KW  - affine
KW  - invariance
KW  - gradient-free sampling methods
KW  - localization
Y1  - 2021
U6  - https://doi.org/10.1137/19M1303162
SN  - 2166-2525
VL  - 9
IS  - 2
SP  - 446
EP  - 482
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Hastermann, Gottfried
A1  - Reinhardt, Maria
A1  - Klein, Rupert
A1  - Reich, Sebastian
T1  - Balanced data assimilation for highly oscillatory mechanical systems
JF  - Communications in applied mathematics and computational science : CAMCoS
N2  - 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.
KW  - data assimilation
KW  - ensemble Kalman filter
KW  - balanced dynamics
KW  - highly
KW  - oscillatory systems
KW  - Hamiltonian dynamics
KW  - geophysics
Y1  - 2021
U6  - https://doi.org/10.2140/camcos.2021.16.119
SN  - 1559-3940
SN  - 2157-5452
VL  - 16
IS  - 1
SP  - 119
EP  - 154
PB  - Mathematical Sciences Publishers
CY  - Berkeley
ER  - 
TY  - JOUR
A1  - Leung, Tsz Yan
A1  - Leutbecher, Martin
A1  - Reich, Sebastian
A1  - Shepherd, Theodore G.
T1  - Impact of the mesoscale range on error growth and the limits to atmospheric predictability
JF  - Journal of the atmospheric sciences
N2  - 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.
KW  - mesoscale forecasting
KW  - numerical weather prediction/forecasting
KW  - short-range prediction
KW  - numerical analysis/modeling
Y1  - 2020
U6  - https://doi.org/10.1175/JAS-D-19-0346.1
SN  - 0022-4928
SN  - 1520-0469
VL  - 77
IS  - 11
SP  - 3769
EP  - 3779
PB  - American Meteorological Soc.
CY  - Boston
ER  - 
TY  - JOUR
A1  - Malic, E.
A1  - Weber, C.
A1  - Richter, M.
A1  - Atalla, V.
A1  - Klamroth, Tillmann
A1  - Saalfrank, Peter
A1  - Reich, Sebastian
A1  - Knorr, A.
T1  - Microscopic model of the optical absorption of carbon nanotubes functionalized with molecular spiropyran photoswitches
JF  - Physical review letters
N2  - The adsorption of molecules to the surface of carbon nanostructures opens a new field of hybrid systems with distinct and controllable properties. We present a microscopic study of the optical absorption in carbon nanotubes functionalized with molecular spiropyran photoswitches. The switching process induces a change in the dipole moment leading to a significant coupling to the charge carriers in the nanotube. As a result, the absorption spectra of functionalized tubes reveal a considerable redshift of transition energies depending on the switching state of the spiropyran molecule. Our results suggest that carbon nanotubes are excellent substrates for the optical readout of spiropyran-based molecular switches. The gained insights can be applied to other noncovalently functionalized one-dimensional nanostructures in an externally induced dipole field.
Y1  - 2011
U6  - https://doi.org/10.1103/PhysRevLett.106.097401
SN  - 0031-9007
VL  - 106
IS  - 9
PB  - American Physical Society
CY  - College Park
ER  - 
TY  - JOUR
A1  - Leung, Tsz Yan
A1  - Leutbecher, Martin
A1  - Reich, Sebastian
A1  - Shepherd, Theodore G.
T1  - Forecast verification
BT  - relating deterministic and probabilistic metrics
JF  - Quarterly journal of the Royal Meteorological Society
N2  - 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.
KW  - CRPS
KW  - ensembles
KW  - idealised turbulence
KW  - NWP
KW  - RMSE
KW  - verification
Y1  - 2021
U6  - https://doi.org/10.1002/qj.4120
SN  - 0035-9009
SN  - 1477-870X
VL  - 147
IS  - 739
SP  - 3124
EP  - 3134
PB  - Wiley
CY  - Hoboken
ER  - 
TY  - JOUR
A1  - Seelig, Stefan A.
A1  - Rabe, Maximilian Michael
A1  - Malem-Shinitski, Noa
A1  - Risse, Sarah
A1  - Reich, Sebastian
A1  - Engbert, Ralf
T1  - Bayesian parameter estimation for the SWIFT model of eye-movement control during reading
JF  - Journal of mathematical psychology
N2  - 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.
KW  - dynamical models
KW  - reading
KW  - eye movements
KW  - saccades
KW  - likelihood function
KW  - Bayesian inference
KW  - MCMC
KW  - interindividual differences
Y1  - 2020
U6  - https://doi.org/10.1016/j.jmp.2019.102313
SN  - 0022-2496
SN  - 1096-0880
VL  - 95
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Malem-Shinitski, Noa
A1  - Opper, Manfred
A1  - Reich, Sebastian
A1  - Schwetlick, Lisa
A1  - Seelig, Stefan A.
A1  - Engbert, Ralf
T1  - A mathematical model of local and global attention in natural scene viewing
JF  - PLoS Computational Biology : a new community journal
N2  - 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.
Y1  - 2020
U6  - https://doi.org/10.1371/journal.pcbi.1007880
SN  - 1553-734X
SN  - 1553-7358
VL  - 16
IS  - 12
PB  - PLoS
CY  - San Fransisco
ER  - 
TY  - JOUR
A1  - de Wiljes, Jana
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
T1  - Ensemble transform algorithms for nonlinear smoothing problems
JF  - SIAM journal on scientific computing
N2  - 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.
KW  - data assimilation
KW  - smoother
KW  - localization
KW  - optimal transport
KW  - adaptive
KW  - spread correction
Y1  - 2019
U6  - https://doi.org/10.1137/19M1239544
SN  - 1064-8275
SN  - 1095-7197
VL  - 42
IS  - 1
SP  - A87
EP  - A114
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - McKean-Vlasov SDEs in nonlinear filtering
JF  - SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics
N2  - 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.
KW  - data assimilation
KW  - feedback particle filter
KW  - Poincare inequality
KW  - well-posedness
KW  - nonlinear filtering
KW  - McKean-Vlasov
KW  - mean-field equations
Y1  - 2022
U6  - https://doi.org/10.1137/20M1355197
SN  - 0363-0129
SN  - 1095-7138
VL  - 59
IS  - 6
SP  - 4188
EP  - 4215
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -