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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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
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.
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 -
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 - Wormell, Caroline L.
A1 - Reich, Sebastian
T1 - Spectral convergence of diffusion maps
BT - Improved error bounds and an alternative normalization
JF - SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics
N2 - Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.
KW - diffusion maps
KW - graph Laplacian
KW - Sinkhorn problem
KW - kernel methods
Y1 - 2021
U6 - https://doi.org/10.1137/20M1344093
SN - 0036-1429
SN - 1095-7170
VL - 59
IS - 3
SP - 1687
EP - 1734
PB - Society for Industrial and Applied Mathematics
CY - Philadelphia
ER -
TY - JOUR
A1 - Molkenthin, Christian
A1 - Donner, Christian
A1 - Reich, Sebastian
A1 - Zöller, Gert
A1 - Hainzl, Sebastian
A1 - Holschneider, Matthias
A1 - Opper, Manfred
T1 - GP-ETAS: semiparametric Bayesian inference for the spatio-temporal epidemic type aftershock sequence model
JF - Statistics and Computing
N2 - The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented.
KW - Self-exciting point process
KW - Hawkes process
KW - Spatio-temporal ETAS model
KW - Bayesian inference
KW - Sampling
KW - Earthquake modeling
KW - Gaussian process
KW - Data augmentation
Y1 - 2022
U6 - https://doi.org/10.1007/s11222-022-10085-3
SN - 0960-3174
SN - 1573-1375
VL - 32
IS - 2
PB - Springer
CY - Dordrecht
ER -