Particle filtering with Lagrangian data in a point vortex model

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 131-138).Particle filtering is a technique used for state estimation from noisy measurements. In fluid dynamics, a popular problem called Lagrangian data assimilation (LaDA) uses Lagrangian measurements in the form of tracer positions to learn about the changing flow field. Particle filtering can be applied to LaDA to track the flow field over a period of time. As opposed to techniques like Extended Kalman Filter (EKF) and Ensemble Kalman Filter (EnKF), particle filtering does not rely on linearization of the forward model and can provide very accurate estimates of the state, as it represents the true Bayesian posterior distribution using a large number of weighted particles. In this work, we study the performance of various particle filters for LaDA using a two-dimensional point vortex model; this is a simplified fluid dynamics model wherein the positions of vortex singularities (point vortices) define the state. We consider various parameters associated with algorithm and examine their effect on filtering performance under several vortex configurations. Further, we study the effect of different tracer release positions on filtering performance. Finally, we relate the problem of optimal tracer deployment to the Lagrangian coherent structures (LCS) of point vortex system.by Subhadeep Mitra.S.M

A hybrid method for computing Lyapunov exponents

Beyn W-J, Lust A. A hybrid method for computing Lyapunov exponents. Numerische Mathematik. 2009;113(3):357-375.In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223-237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec' multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation

A hybrid method for computing Lyapunov exponents

Lust A. Eine hybride Methode zur Berechnung von Liapunow-Exponenten. Bielefeld (Germany): Bielefeld University; 2006.In this thesis we present a new numerical method for computing some or all Lyapunov exponents of a discrete dynamical system. It is called hybrid because it combines the classical QR method with more recent methods that use spatial integration with respect to an invariant ergodic measure. We also investigate error expansions for the approximation of the exponents given by hybrid method.In dieser Arbeit wird ein neues numerisches Verfahren präsentiert, das uns ermöglicht, eine beliebige Anzahl von Liapunow-Exponenten eines diskreten dynamischen Systems zu berechnen. Da hierbei die diskrete QR-Methode mit der räumlichen Integration bezüglich eines invarianten ergodischen Maßes kombiniert wird, nennen wir die Methode hybrid. Es wird auch die Fehlerentwicklung der Approximation der Exponenten mittels der hybriden Methode untersucht

Error analysis of a hybrid method for computing Lyapunov exponents

Beyn W-J, Lust A. Error analysis of a hybrid method for computing Lyapunov exponents. Numerische Mathematik. 2013;123(2):189-217.In a previous paper (Beyn and Lust in Numer Math 113:357-375, 2009) we suggested a numerical method for computing all Lyapunov exponents of a dynamical system by spatial integration with respect to an ergodic measure. The method extended an earlier approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223-237, 1999) for the largest Lyapunov exponent by integrating the diagonal entries from the -decomposition of the Jacobian for an iterated map. In this paper we provide an asymptotic error analysis of the method for the case in which all Lyapunov exponents are simple. We employ Oseledec multiplicative ergodic theorem and impose certain hyperbolicity conditions on the invariant subspaces that belong to neighboring exponents. The resulting error expansion shows that one step of extrapolation is enough to obtain exponential decay of errors